diff --git a/docs/pdf/Makefile b/docs/pdf/Makefile index 2c928afa..e2d56ea9 100644 --- a/docs/pdf/Makefile +++ b/docs/pdf/Makefile @@ -87,7 +87,6 @@ TOPFILE = userguide.tex SECFILE = title.tex abstract.tex overview.tex conventions.tex distribution.tex \ building.tex background.tex gettingstarted.tex userinterface.tex \ errors.tex bibliography.tex license.tex -# advanced.tex listofroutines.tex FIGDIR = figures XPDFFLAGS = diff --git a/docs/pdf/abstract.tex b/docs/pdf/abstract.tex index ab4e28ba..fed6057d 100644 --- a/docs/pdf/abstract.tex +++ b/docs/pdf/abstract.tex @@ -12,10 +12,15 @@ context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) computational framework and can be used in conjuction with the Krylov solvers available in this framework. MLD2P4 enables the user to easily specify different aspects of a generic algebraic multilevel Schwarz preconditioner, thus allowing to search -for the ``best'' preconditioner for the problem at hand. The package has been designed -employing object-oriented techniques, using Fortran 95 and MPI, with interfaces to -additional third party libraries such as UMFPACK, SuperLU and SuperLU\_Dist, that -can be exploited in building multi-level preconditioners. +for the ``best'' preconditioner for the problem at hand. + +The package has been designed employing object-oriented techniques, +using Fortran 95, with interfaces to additional third party libraries +such as UMFPACK, SuperLU and SuperLU\_Dist, that +can be exploited in building multi-level preconditioners. The parallel +implementation is based on a Single Program Multiple Data (SPMD) +paradigm for distributed-memory architectures; the inter-process data +communication is based on MPI and is managed mainly through PSBLAS. This guide provides a brief description of the functionalities and the user interface of MLD2P4. diff --git a/docs/pdf/advanced.tex b/docs/pdf/advanced.tex deleted file mode 100644 index d5bb941d..00000000 --- a/docs/pdf/advanced.tex +++ /dev/null @@ -1,12 +0,0 @@ -\section{Advanced Use}\label{sec:advanced} - - - MLD2P4 software architecture \\ - - preconditioner data structure (descrizione "dettagliata") + possibilita' di settare singolarmente - i vari livelli (possibilita' accennata solamente nella precedente descrizione di precset) \\ - - descrizione routine medium level (con introduzione sulle potenzialita' di ampliamento (?), offerte - da queto strato software) \\ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "userguide" -%%% End: diff --git a/docs/pdf/background.tex b/docs/pdf/background.tex index a6abf84a..b1594f94 100644 --- a/docs/pdf/background.tex +++ b/docs/pdf/background.tex @@ -5,7 +5,7 @@ \emph{Domain Decomposition} (DD) preconditioners, coupled with Krylov iterative solvers, are widely used in the parallel solution of large and sparse linear systems. These preconditioners are based on the divide and conquer technique: the matrix -to be preconditioned is divided into submatrices, a ``local linear system'' +to be preconditioned is divided into submatrices, a ``local'' linear system involving each submatrix is (approximately) solved, and the local solutions are used to build a preconditioner for the whole original matrix. This process often corresponds to dividing a physical domain associated to the original matrix @@ -61,7 +61,7 @@ starting from the original matrix. The algebraic approach is based on the \emph{ aggregation} algorithm \cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}. A decoupled version of this algorithm is implemented, where the smoothed aggregation is applied locally to each submatrix \cite{TUMINARO_TONG}. In the next two subsections we provide -a brief description of the multi-level Schwarz preconditioners and on the smoothed +a brief description of the multi-level Schwarz preconditioners and of the smoothed aggregation technique as implemented in MLD2P4. For further details the user is referred to \cite{para_04,aaecc_07,apnum_07,dd2_96}. @@ -203,7 +203,7 @@ One more variant of two-level hybrid preconditioner is obtained by applying the smoother before and after the coarse-level correction. In this case, the preconditioner is symmetric if $A$, $M_{1L}$ and $M_{C}$ are symmetric. -As previously noted, on parallel computers the number of sumatrices usually matches +As previously noted, on parallel computers the number of submatrices usually matches the number of available processors. When the size of the system to be preconditioned is very large, the use of many processors, i.e.\ of many small submatrices, often leads to a large coarse-level system, whose solution may be computationally expensive. @@ -220,7 +220,7 @@ For a detailed descrition of them, the reader is referred to \cite[Chapter 3]{dd2_96}. The algorithm for the application of a multi-level hybrid post-smoothed preconditioner $M$ to a vector $v$, i.e.\ for the -computation of $w=M^{-1}v$ is reported, for +computation of $w=M^{-1}v$, is reported, for example, in Figure~\ref{fig:mlhpost_alg}. Here the number of levels is denoted by $nlev$ and the levels are numbered in increasing order starting from the finest one, i.e.\ the finest level is level 1; the coarse matrix diff --git a/docs/pdf/bibliography.tex b/docs/pdf/bibliography.tex index f05597d5..d68f9c64 100644 --- a/docs/pdf/bibliography.tex +++ b/docs/pdf/bibliography.tex @@ -28,12 +28,13 @@ in Scientific Computing, Lecture Notes in Computer Science, Springer, 2005, 593--602. % \bibitem{aaecc_07} A.~Buttari, P.~D'Ambra, D.~di~Serafino, S.~Filippone, -{\em 2LEV-D2P4: a package of high-performance preconditioners}, +{\em 2LEV-D2P4: a package of high-performance preconditioners +for scientific and engineering applications}, Applicable Algebra in Engineering, Communications and Computing, -18, 3, May, 2007, 223--239. +18, 3, 2007, 223--239. %Published online: 13 February 2007, {\tt http://dx.doi.org/10.1007/s00200-007-0035-z} % -\bibitem{apnum_07} P.~D'Ambra, S.~Filippone, D.~Di~Serafino, +\bibitem{apnum_07} P.~D'Ambra, S.~Filippone, D.~di~Serafino, {\em On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners}, Applied Numerical Mathematics, Elsevier Science, 57, 11-12, 2007, 1181-1196. @@ -105,8 +106,8 @@ BIT Numerical Mathematics, 43, 2003, 945--959. % \bibitem{PSBLASGUIDE} S.~Filippone, A.~Buttari, -{\em PSBLAS-2.1 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library}, -xxxxx. +{\em PSBLAS-2.3 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library}, +available from \texttt{http://www.ce.uniroma2.it/psblas/}. % \bibitem{psblas_00} S.~Filippone, M.~Colajanni, @@ -114,7 +115,8 @@ S.~Filippone, M.~Colajanni, Computation on Sparse Matrices}, ACM Transactions on Mathematical Software, 26, 4, 2000, 527--550. \bibitem{SUPERLUDIST} -X.~S.~Li, J.~W.~Demmel, {\em SuperLU\_DIST: A Scalable Distributed-memory Sparse Direct Solver for Unsymmetric Linear Systems}, +X.~S.~Li, J.~W.~Demmel, {\em SuperLU\_DIST: A Scalable Distributed-memory +Sparse Direct Solver for Unsymmetric Linear Systems}, ACM Transactions on Mathematical Software, 29, 2, 2003, 110--140. % %\bibitem{KIVA3PSBLAS} @@ -177,7 +179,7 @@ in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000. \bibitem{VANEK_MANDEL_BREZINA} P.~Van{\v e}k, J.~Mandel and M.~Brezina, {\em Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems}, -Computing, 1996, 56, 179-196. +Computing, 56, 1996, 179-196. % \end{thebibliography} diff --git a/docs/pdf/listofroutines.tex b/docs/pdf/listofroutines.tex deleted file mode 100644 index 3b508699..00000000 --- a/docs/pdf/listofroutines.tex +++ /dev/null @@ -1,10 +0,0 @@ -\section{List of Routines}\label{sec:routines} - - Elenco (ordine alfabetico) di tutte le routine, con rinvio (ipertestuale e num. pag.) alla descrizione - di ciascuna in qualche paragrafo precedente - (una specie di indice analitico, che rimanda alle routine descritte precedentemente nei rispettivi paragrafi) - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "userguide" -%%% End: diff --git a/docs/pdf/methods.tex b/docs/pdf/methods.tex deleted file mode 100644 index acb678fd..00000000 --- a/docs/pdf/methods.tex +++ /dev/null @@ -1,607 +0,0 @@ -\section{Iterative Methods} -\label{sec:methods} -\markboth{\underline{MLD2P4 User's and Reference Guide}} - {\underline{\ref{sec:methods} Iterative Methods}} - -In this chapter we provide routines for preconditioners and iterative -methods. The interfaces for Krylov subspace methods are available in -the module \verb|mld_krylov_mod|. The installation process of MLD2P4 -ensures that these may be used as a drop-in replacement for the PSBLAS -methods; they are accessible under the PSBLAS names (see the PSBLAS -documentation), - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% -% Krylov Methods driver routine -% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subroutine{mld\_krylov \label{krylov}}{Krylov Methods Driver Routine} - -This subroutine is a driver that provides a general interface for all -the Krylov-Subspace family methods. - -The stopping criterion is the normwise backward error, in the infinity -norm, i.e. the iteration is stopped when -\[ err = \frac{\|r_i\|}{(\|A\|\|x_i\|+\|b\|)} < eps \] -or the 2-norm residual reduction -\[ err = \frac{\|r_i\|}{\|b\|_2} < eps \] -according to the value passed through the istop argument (see -later). In the above formulae, $x_i$ is the tentative solution and -$r_i=b-Ax_i$ the corresponding residual at the $i$-th iteration. - - -\syntax{call psb\_krylov}{method,a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop} - -\begin{description} -\item[Type:] Synchronous. -\item[\bf On Entry] -\item[method] a string that defines the iterative method to be - used. Supported values are: - \begin{description} - \item[CG]: the Conjugate Gradient method; - \item[CGS]:the Conjugate Gradient Stabilized method; - - \item[BICG]: the Bi-Conjugate Gradient method; - \item[BICGSTAB]: the Bi-Conjugate Gradient Stabilized method; - \item[BICGSTABL]: the Bi-Conjugate Gradient Stabilized method with restarting; - \item[RGMRES]: the Generalized Minimal Residual method with restarting. - \end{description} -\item[a] the local portion of global sparse matrix -$A$. \\ -Scope: {\bf local} \\ -Type: {\bf required}\\ -Intent: {\bf in}.\\ -Specified as: a structured data of type \spdata. -\item[prec] The data structure containing the preconditioner.\\ -Scope: {\bf local} \\ -Type: {\bf required}\\ -Intent: {\bf in}.\\ -Specified as: a structured data of type \precdata. -\item[b] The RHS vector. \\ -Scope: {\bf local} \\ -Type: {\bf required}\\ -Intent: {\bf in}.\\ -Specified as: a rank one array. -\item[x] The initial guess. \\ -Scope: {\bf local} \\ -Type: {\bf required}\\ -Intent: {\bf inout}.\\ -Specified as: a rank one array. -\item[eps] The stopping tolerance. \\ -Scope: {\bf global} \\ -Type: {\bf required}\\ -Intent: {\bf in}.\\ -Specified as: a real number. -\item[desc\_a] contains data structures for communications.\\ -Scope: {\bf local} \\ -Type: {\bf required}\\ -Intent: {\bf in}.\\ -Specified as: a structured data of type \descdata. -\item[itmax] The maximum number of iterations to perform.\\ -Scope: {\bf global} \\ -Type: {\bf optional}\\ -Intent: {\bf in}.\\ -Default: $itmax = 1000$.\\ -Specified as: an integer variable $itmax \ge 1$. -\item[itrace] If $>0$ print out an informational message about - convergence every $itrace$ iterations.\\ -Scope: {\bf global} \\ -Type: {\bf optional}\\ -Intent: {\bf in}.\\ -\item[irst] An integer specifying the restart parameter.\\ -Scope: {\bf global} \\ -Type: {\bf optional}.\\ -Intent: {\bf in}.\\ -Values: $irst>0$. This is employed for the BiCGSTABL or RGMRES -methods, otherwise it is ignored. - -\item[istop] An integer specifying the stopping criterion.\\ -Scope: {\bf global} \\ -Type: {\bf optional}.\\ -Intent: {\bf in}.\\ -Values: 1: use the normwise backward error, 2: use the scaled 2-norm -of the residual. Default: 1. -\item[\bf On Return] -\item[x] The computed solution. \\ -Scope: {\bf local} \\ -Type: {\bf required}\\ -Intent: {\bf inout}.\\ -Specified as: a rank one array. -\item[iter] The number of iterations performed.\\ -Scope: {\bf global} \\ -Type: {\bf optional}\\ -Intent: {\bf out}.\\ -Returned as: an integer variable. -\item[err] The convergence estimate on exit.\\ -Scope: {\bf global} \\ -Type: {\bf optional}\\ -Intent: {\bf out}.\\ -Returned as: a real number. -\item[info] Error code.\\ -Scope: {\bf local} \\ -Type: {\bf required} \\ -Intent: {\bf out}.\\ -An integer value; 0 means no error has been detected. -\end{description} - -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% % -%% % CG -%% % -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%% \subroutine{psb\_cg \label{cg}}{CG Iterative Method} - -%% This subroutine implements the CG method with restarting. The -%% stopping criterion is the normwise backward error, in the infinity -%% norm, i.e. the iteration is stopped when -%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \] -%% or -%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \] -%% according to the value passed through the istop argument (see later). - - -%% \syntax{call psb\_cg}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop} - -%% \begin{description} -%% \item[\bf On Entry] -%% \item[a] the local portion of global sparse matrix -%% $A$. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \spdata. -%% \item[prec] The data structure containing the preconditioner.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \precdata. -%% \item[b] The RHS vector. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[x] The initial guess. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[eps] The stopping tolerance. \\ -%% Scope: {\bf global} \\ -%% Type: {\bf required}\\ -%% Specified as: a real number. -%% \item[desc\_a] contains data structures for communications.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \descdata. -%% \item[itmax] The maximum number of iterations to perform.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Default: $itmax = 1000$.\\ -%% Specified as: an integer variable $itmax \ge 1$. -%% \item[itrace] If $>0$ print out an informational message about -%% convergence every $itrace$ iterations.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[istop] An integer specifying the stopping criterion.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ - -%% \item[\bf On Return] -%% \item[x] The computed solution. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[iter] The number of iterations performed.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: an integer variable. -%% \item[err] The error estimate on exit.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: a real number. -%% \item[info] Error code.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required} \\ -%% An integer value; 0 means no error has been detected. -%% \end{description} -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% % -%% % CGS -%% % -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%% \subroutine{psb\_cgs \label{cgs}}{CGS Iterative Method} - -%% This subroutine implements the CGS method with restarting. The -%% stopping criterion is the normwise backward error, in the infinity -%% norm, i.e. the iteration is stopped when -%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \] -%% or -%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \] -%% according to the value passed through the istop argument (see later). - - -%% \syntax{call psb\_cgs}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop} - -%% \begin{description} -%% \item[\bf On Entry] -%% \item[a] the local portion of global sparse matrix -%% $A$. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \spdata. -%% \item[prec] The data structure containing the preconditioner.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \precdata. -%% \item[b] The RHS vector. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[x] The initial guess. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[eps] The stopping tolerance. \\ -%% Scope: {\bf global} \\ -%% Type: {\bf required}\\ -%% Specified as: a real number. -%% \item[desc\_a] contains data structures for communications.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \descdata. -%% \item[itmax] The maximum number of iterations to perform.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Default: $itmax = 1000$.\\ -%% Specified as: an integer variable $itmax \ge 1$. -%% \item[itrace] If $>0$ print out an informational message about -%% convergence every $itrace$ iterations.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[istop] An integer specifying the stopping criterion.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ - -%% \item[\bf On Return] -%% \item[x] The computed solution. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[iter] The number of iterations performed.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: an integer variable. -%% \item[err] The error estimate on exit.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: a real number. -%% \item[info] Error code.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required} \\ -%% An integer value; 0 means no error has been detected. -%% \end{description} - -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% % -%% % BiCG -%% % -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% \subroutine{psb\_bicg \label{bicg}}{BiCG Iterative Method} - -%% This subroutine implements the BiCG method with restarting. The -%% stopping criterion is the normwise backward error, in the infinity -%% norm, i.e. the iteration is stopped when -%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \] -%% or -%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \] -%% according to the value passed through the istop argument (see later). - - -%% \syntax{call psb\_bicg}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop} - -%% \begin{description} -%% \item[\bf On Entry] -%% \item[a] the local portion of global sparse matrix -%% $A$. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \spdata. -%% \item[prec] The data structure containing the preconditioner.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \precdata. -%% \item[b] The RHS vector. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[x] The initial guess. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[eps] The stopping tolerance. \\ -%% Scope: {\bf global} \\ -%% Type: {\bf required}\\ -%% Specified as: a real number. -%% \item[desc\_a] contains data structures for communications.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \descdata. -%% \item[itmax] The maximum number of iterations to perform.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Default: $itmax = 1000$.\\ -%% Specified as: an integer variable $itmax \ge 1$. -%% \item[itrace] If $>0$ print out an informational message about -%% convergence every $itrace$ iterations.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[istop] An integer specifying the stopping criterion.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ - -%% \item[\bf On Return] -%% \item[x] The computed solution. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[iter] The number of iterations performed.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: an integer variable. -%% \item[err] The error estimate on exit.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: a real number. -%% \item[info] Error code.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required} \\ -%% An integer value; 0 means no error has been detected. -%% \end{description} - - -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% % -%% % BiCGSTAB -%% % -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% \subroutine{psb\_bicgstab \label{bicgstab}}{BiCGSTAB Iterative Method} - -%% This subroutine implements the BiCGSTAB method with restarting. The -%% stopping criterion is the normwise backward error, in the infinity -%% norm, i.e. the iteration is stopped when -%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \] -%% or -%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \] -%% according to the value passed through the istop argument (see later). - - -%% \syntax{call psb\_bicgstab}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop} - -%% \begin{description} -%% \item[\bf On Entry] -%% \item[a] the local portion of global sparse matrix -%% $A$. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \spdata. -%% \item[prec] The data structure containing the preconditioner.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \precdata. -%% \item[b] The RHS vector. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[x] The initial guess. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[eps] The stopping tolerance. \\ -%% Scope: {\bf global} \\ -%% Type: {\bf required}\\ -%% Specified as: a real number. -%% \item[desc\_a] contains data structures for communications.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \descdata. -%% \item[itmax] The maximum number of iterations to perform.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Default: $itmax = 1000$.\\ -%% Specified as: an integer variable $itmax \ge 1$. -%% \item[itrace] If $>0$ print out an informational message about -%% convergence every $itrace$ iterations.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[istop] An integer specifying the stopping criterion.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ - -%% \item[\bf On Return] -%% \item[x] The computed solution. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[iter] The number of iterations performed.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: an integer variable. -%% \item[err] The error estimate on exit.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: a real number. -%% \item[info] Error code.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required} \\ -%% An integer value; 0 means no error has been detected. -%% \end{description} - - -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% % -%% % BiCGSTAB(L) -%% % -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% \subroutine{psb\_bicgstabl \label{bicgstabl}}{BiCGSTAB-$l$ Iterative Method} - -%% This subroutine implements the BiCGSTAB-$l$ method with restarting. The -%% stopping criterion is the normwise backward error, in the infinity -%% norm, i.e. the iteration is stopped when -%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \] -%% or -%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \] -%% according to the value passed through the istop argument (see later). - - -%% \syntax{call psb\_bicgstab}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop} - -%% \begin{description} -%% \item[\bf On Entry] -%% \item[a] the local portion of global sparse matrix -%% $A$. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \spdata. -%% \item[prec] The data structure containing the preconditioner.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \precdata. -%% \item[b] The RHS vector. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[x] The initial guess. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[eps] The stopping tolerance. \\ -%% Scope: {\bf global} \\ -%% Type: {\bf required}\\ -%% Specified as: a real number. -%% \item[desc\_a] contains data structures for communications.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \descdata. -%% \item[itmax] The maximum number of iterations to perform.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Default: $itmax = 1000$.\\ -%% Specified as: an integer variable $itmax \ge 1$. -%% \item[itrace] If $>0$ print out an informational message about -%% convergence every $itrace$ iterations.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[irst] An integer specifying the restarting iteration.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[istop] An integer specifying the stopping criterion.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ - -%% \item[\bf On Return] -%% \item[x] The computed solution. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[iter] The number of iterations performed.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: an integer variable. -%% \item[err] The error estimate on exit.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: a real number. -%% \item[info] Error code.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required} \\ -%% An integer value; 0 means no error has been detected. -%% \end{description} - - -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% % -%% % GMRES -%% % -%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% \subroutine{psb\_gmres \label{gmres}}{GMRES Iterative Method} - -%% This subroutine implements the GMRES method with restarting. The -%% stopping criterion is the normwise backward error, in the infinity -%% norm, i.e. the iteration is stopped when -%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \] -%% or -%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \] -%% according to the value passed through the istop argument (see later). - - -%% \syntax{call psb\_gmres}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop} - -%% \begin{description} -%% \item[\bf On Entry] -%% \item[a] the local portion of global sparse matrix -%% $A$. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \spdata. -%% \item[prec] The data structure containing the preconditioner.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \precdata. -%% \item[b] The RHS vector. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[x] The initial guess. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[eps] The stopping tolerance. \\ -%% Scope: {\bf global} \\ -%% Type: {\bf required}\\ -%% Specified as: a real number. -%% \item[desc\_a] contains data structures for communications.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a structured data of type \descdata. -%% \item[itmax] The maximum number of iterations to perform.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Default: $itmax = 1000$.\\ -%% Specified as: an integer variable $itmax \ge 1$. -%% \item[itrace] If $>0$ print out an informational message about -%% convergence every $itrace$ iterations.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[irst] An integer specifying the restart iteration.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% \item[istop] An integer specifying the stopping criterion.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ - -%% \item[\bf On Return] -%% \item[x] The computed solution. \\ -%% Scope: {\bf local} \\ -%% Type: {\bf required}\\ -%% Specified as: a rank one array. -%% \item[iter] The number of iterations performed.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: an integer variable. -%% \item[err] The error estimate on exit.\\ -%% Scope: {\bf global} \\ -%% Type: {\bf optional}\\ -%% Returned as: a real number. -%% \item[info] Error code.\\ -%% Scope: {\bf local} \\ -%% Type: {\bf required} \\ -%% An integer value; 0 means no error has been detected. -%% \end{description} - - - - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "userguide" -%%% End: diff --git a/docs/pdf/overview.tex b/docs/pdf/overview.tex index 1a675ad4..e23c2965 100644 --- a/docs/pdf/overview.tex +++ b/docs/pdf/overview.tex @@ -19,7 +19,8 @@ These preconditioners have the following general features: \begin{itemize} \item both \emph{additive and hybrid multilevel} variants are implemented, i.e.\ variants that are additive among the levels and inside each level, and variants -that are multiplicative among the levels and additive inside each level; the basic Additive Schwarz (AS) preconditioners are obtained by considering only one level; +that are multiplicative among the levels and additive inside each level; +the basic Additive Schwarz (AS) preconditioners are obtained by considering only one level; \item a \emph{purely algebraic} approach is used to generate a sequence of coarse-level corrections to a basic AS preconditioner, without explicitly using any information on the geometry of the original problem (e.g.\ the @@ -70,7 +71,20 @@ by expert users to build new versions of multi-level Schwarz preconditioners. We provide here a description of the upper-layer routines, but not of the medium-layer ones. -This guide is organized as follows. \textbf{ORGANIZZAZIONE DELLA GUIDA} +This guide is organized as follows. The notational conventions used in +the guide and in the naming of the MLD2P4 routines are reported in +Section~\ref{sec:conventions}. Information on the distribution of the source code and the +related license is given in Section~\ref{sec:distribution}, while details on the configuration +and installation of package are given in Section~\ref{sec:building}. A description of +multi-level Schwarz preconditioners based on smoothed aggregation is provided +in Section~\ref{sec:background}, to help the users in choosing among the different preconditioners +implemented in MLD2P4. The basics for building and applying the preconditioners +with the Krylov solvers implemented in PSBLAS are reported in Section~\ref{sec:started}, where the +Fortran 95 codes of a few sample programs are also shown. A reference guide for +the upper-layer routines of MLD2P4, that are the user interface, is provided +in Section~\ref{sec:userinterface}. The error handling mechanism used by the package is briefly described +in Section~\ref{sec:errors}. The copyright terms concerning the distribution and modification +of MLD2P4 are reported in Appendix~\ref{sec:license}. %%% Local Variables: %%% mode: latex diff --git a/docs/pdf/userguide.tex b/docs/pdf/userguide.tex index 50a834da..ae5d1e03 100644 --- a/docs/pdf/userguide.tex +++ b/docs/pdf/userguide.tex @@ -35,7 +35,7 @@ % /URI (http://ce.uniroma2.it/psblas) } -% \setlength\oddsidemargin{.7in} +\setlength\textwidth{1.15\textwidth} % \setlength\evensidemargin{.7in} % \newlength{\centeroffset} % \setlength{\centeroffset}{0.5\oddsidemargin} diff --git a/docs/pdf/userinterface.tex b/docs/pdf/userinterface.tex index 4813103c..b09188f4 100644 --- a/docs/pdf/userinterface.tex +++ b/docs/pdf/userinterface.tex @@ -7,7 +7,7 @@ The basic user interface of MLD2P4 consists of six routines. The four routines \ for the setup and the application of any one-level and multi-level preconditioner implemented in the package. The routine \verb|mld_precfree| deallocates the preconditioner data structure, while -\verb|mld_prec\-descr| prints a description of the preconditioner setup by the user. +\verb|mld_precdescr| prints a description of the preconditioner setup by the user. For each routine, the same user interface is overloaded with respect to the real/complex case and the single/double precision; @@ -32,8 +32,8 @@ i.e. provides the constants \verb|psb_spk_| = \verb|kind(1.e0)| and \verb|psb_dpk_| = \verb|kind(1.d0)|; \item real parameters defining the preconditioner must be declared - according to the precision of the previous data structures - (see Section~\ref{sec:precset}). + according to the precision of the sparse matrix and preconditioner + data structures (see Section~\ref{sec:precset}). \end{itemize} A description of each routine is given in the remainder of this section. @@ -122,10 +122,10 @@ can be logically divided into four groups, i.e.\ parameters defining \item the aggregation algorithm; \item the coarse-space correction at the coarsest level. \end{enumerate} -A list of the parameters that can be set, along with allowed and +A list of the parameters that can be set, along with their allowed and default values, is given in Tables~\ref{tab:p_type}-\ref{tab:p_coarse}. For a detailed description of the meaning of the parameters, please -see Section~\ref{sec:background}. +refer to Section~\ref{sec:background}. % %Note that the routine allows to set different features of the %preconditioner at each level through the use of \verb|ilev|. @@ -144,19 +144,18 @@ see Section~\ref{sec:background}. \verb|mld_ml_type_| & \verb|character(len=*)| & \texttt{'ADD'} \ \ \ \texttt{'MULT'} & \texttt{'MULT'} - & basic multi-level framework: additive or multiplicative - among the levels (always additive inside a level) \\ + & Basic multi-level framework: additive or multiplicative + among the levels (always additive inside a level). \\ \hline \verb|mld_smoother_type_|& \verb|character(len=*)| & \texttt{'DIAG'} \ \ \ \texttt{'BJAC'} \ \ \ \texttt{'AS'} & \texttt{'AS'} - & basic one-level preconditioner (i.e.\ smoother) of the - multi-level preconditioner: diagonal, block Jacobi, - AS \\ + & Basic one-level preconditioner (i.e.\ smoother): diagonal, + block Jacobi, AS \\ \hline \verb|mld_smoother_pos_| & \verb|character(len=*)| & \texttt{'PRE'} \ \ \ \texttt{'POST'} \ \ \ \texttt{'TWOSIDE'} & \texttt{'POST'} - & ``position'' of the smoother: pre-smoother, post-smoother, - pre- and post-smoother \\ + & ``Position'' of the smoother: pre-smoother, post-smoother, + pre- and post-smoother. \\ \hline \end{tabular} \end{center} @@ -166,7 +165,7 @@ see Section~\ref{sec:background}. \begin{sidewaystable} \begin{center} -\begin{tabular}{|l|l|p{2.6cm}|l|p{7cm}|} +\begin{tabular}{|l|l|p{3.2cm}|l|p{7cm}|} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline @@ -174,35 +173,35 @@ see Section~\ref{sec:background}. \verb|mld_sub_ovr_| & \verb|integer| & any integer number $\ge 0$ & 1 - & Number of overlap layers. \\ + & Number of overlap layers. \\ \hline \verb|mld_sub_restr_| & \verb|character(len=*)| - & \texttt{'HALO'} \ \ \ \ \ \texttt{'NONE'} + & \texttt{'HALO'} \hspace{2.5cm} \texttt{'NONE'} & \texttt{'HALO'} & Type of restriction operator: \texttt{'HALO'} for taking into account the overlap, \texttt{'NONE'} - for neglecting it. \\ + for neglecting it. \\ \hline \verb|mld_sub_prol_| & \verb|character(len=*)| - & \texttt{'SUM'} \ \ \ \ \ \texttt{'NONE'} + & \texttt{'SUM'} \hspace{2.5cm} \texttt{'NONE'} & \texttt{'NONE'} & Type of prolongator operator: \texttt{'SUM'} for adding the contributions from the overlap, \texttt{'NONE'} - for neglecting them. \\ + for neglecting them. \\ \hline \verb|mld_sub_solve_| & \verb|character(len=*)| - & \texttt{'ILU'} \ \ \ \ \ \texttt{'MILU'} \ \ \ \ \ \texttt{'ILUT'} \ \ \ \ \ - \texttt{'UMF'} \ \ \ \ \ \texttt{'SLU'} + & \texttt{'ILU'} \hspace{2.5cm} \texttt{'MILU'} \hspace{2.5cm} \texttt{'ILUT'} + \hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'} & \texttt{'UMF'} - & Local solver: ILU($p$), MILU($p$), ILU($p,t$), LU from UMFPACK, LU from SuperLU, - plus triangular solve. \\ + & Local solver: ILU($p$), MILU($p$), ILU($p,t$), LU from UMFPACK, LU from SuperLU + (plus triangular solve). \\ \hline \verb|mld_sub_fillin_| & \verb|integer| - & any integer number $\ge 0$ + & Any~int.~num.~$\ge 0$ & 0 - & Fill-in level $p$ of the incomplete LU factorizations. \\ + & Fill-in level $p$ of the incomplete LU factorizations. \\ \hline \verb|mld_sub_thresh_| & \verb|real(|\emph{kind\_parameter}\verb|)| - & any real number $\ge 0$ + & Any~real~num.~$\ge 0$ & \texttt{0.e0} (or \texttt{0.d0}) - & Drop tolerance $t$ in the ILU($p,t$) factorization. \\ + & Drop tolerance $t$ in the ILU($p,t$) factorization. \\ \hline \verb|mld_sub_ren_| & \verb|character(len=*)| - & \texttt{'RENUM\_NONE'}, \texttt{'RENUM\_GLOBAL'} %, \texttt{'RENUM_GPS'} + & \texttt{'RENUM\_NONE'} \texttt{'RENUM\_GLOBAL'} %, \texttt{'RENUM_GPS'} & \texttt{'RENUM\_NONE'} & Row and column reordering of the local submatrices: no reordering, reordering according to the global numbering of the rows and columns of @@ -216,7 +215,7 @@ see Section~\ref{sec:background}. \begin{sidewaystable} \begin{center} -\begin{tabular}{|l|l|p{2.6cm}|l|p{7cm}|} +\begin{tabular}{|l|l|p{2.3cm}|p{2.6cm}|p{7cm}|} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline @@ -224,21 +223,25 @@ see Section~\ref{sec:background}. \verb|mld_aggr_alg_| & \verb|character(len=*)| & \texttt{'DEC'} & \texttt{'DEC'} - & Aggregation algorithm. Currently, only the decoupled aggregation is available. \\ + & Aggregation algorithm. Currently, only the decoupled aggregation is available. \\ \hline \verb|mld_aggr_kind_| & \verb|character(len=*)| - & \texttt{'SMOOTH'} \ \ \ \ \ \texttt{'RAW'} + & \texttt{'SMOOTH'} \hspace{2.5cm} \texttt{'RAW'} & \texttt{'SMOOTH'} - & Type of aggregation: smoothed or raw, i.e.\ using the tentative prolongator. \\ + & Type of aggregation: smoothed or raw, i.e.\ using the tentative prolongator. \\ \hline \verb|mld_aggr_thresh_| & \verb|real(|\emph{kind\_parameter}\verb|)| - & any real number $\in [0, 1]$ + & Any~real~num. $\in [0, 1]$ & \texttt{0.e0} (or \texttt{0.d0}) - & The threshold $\theta$ in the aggregation algorithm. \\ + & The threshold $\theta$ in the aggregation algorithm. \\ \hline \verb|mld_aggr_eig_| & \verb|character(len=*)| & \texttt{'A\_NORMI'} & \texttt{'A\_NORMI'} & Estimate of the maximum eigenvalue of $D^{-1}A$ for the smoothed aggregation. Currently, only the infinity norm of - the matrix is available. \\ + the matrix is available. \\ \hline +\verb|mld_aggr_damp_| & \verb|real(|\emph{kind\_parameter}\verb|)| + & Any~real~num. $>0$ + & \texttt{4.e0/3.e0} \hspace{1cm} (or \texttt{4.d0/3.d0}) + & The damping parameter $\omega$ in the aggregation algorithm. \\ \hline \end{tabular} \end{center} @@ -248,41 +251,41 @@ see Section~\ref{sec:background}. \begin{sidewaystable} \begin{center} -\begin{tabular}{|l|l|p{2.6cm}|l|p{7cm}|} +\begin{tabular}{|l|l|p{3.2cm}|l|p{7cm}|} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline %\multicolumn{5}{|c|}{\emph{coarse-space correction at the coarsest level}}\\ \hline \verb|mld_coarse_mat_| & \verb|character(len=*)| - & \texttt{'DISTR'} \ \ \ \ \ \texttt{'REPL'} + & \texttt{'DISTR'} \hspace{2.5cm} \texttt{'REPL'} & \texttt{'DISTR'} - & Coarsest matrix: distributed among the processors or replicated on each of them. \\ + & Coarsest matrix: distributed among the processors or replicated on each of them. \\ \hline \verb|mld_coarse_solve_| & \verb|character(len=*)| - & \texttt{'BJAC'} \ \ \ \ \ \texttt{'UMF'} \ \ \ \ \ \ \ \ \texttt{'SLU'} - \ \ \ \ \ \texttt{'SLUDIST'} + & \texttt{'BJAC'} \hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'} + \hspace{2.5cm} \texttt{'SLUDIST'} & \texttt{'BJAC'} & Solver used at the coarsest level: block Jacobi, sequential LU from UMFPACK, - sequential LU from SuperLU, or distributed LU from SuperLU\_Dist. + sequential LU from SuperLU, distributed LU from SuperLU\_Dist. If the coarsest matrix is distributed, only \texttt{'BJAC'} and \texttt{'SLUDIST'} - can be chosen; if it is replicated, only \emph{'BJAC'} or \texttt{'SLUDIST'} can - be selected. \\ + can be chosen; if it is replicated, only \texttt{'BJAC'} or \texttt{'SLUDIST'} can + be selected. \\ \hline \verb|mld_coarse_subsolve_| & \verb|character(len=*)| - & \texttt{'ILU'} \ \ \ \ \ \ \ \texttt{'MILU'} \ \ \ \ \ \ \ \ \texttt{'ILUT'} - \ \ \ \ \ \ \ \texttt{'UMF'} \ \ \ \ \ \ \ \texttt{'SLU'} + & \texttt{'ILU'} \hspace{2.5cm} \texttt{'MILU'} \hspace{2.5cm} \texttt{'ILUT'} + \hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'} & \texttt{'UMF'} & Solver for the diagonal blocks of the coarse matrix, in case the block Jacobi solver is chosen as coarsest-level solver: ILU($p$), MILU($p$), ILU($p,t$), LU from UMFPACK, - LU from SuperLU, plus triangular solve. \\ + LU from SuperLU, plus triangular solve. \\ \hline \verb|mld_coarse_sweeps_|& \verb|integer| - & any integer number $> 0$ + & Any~int.~num.~$> 0$ & \texttt{4} & Number of Block-Jacobi sweeps when 'BJAC' is used as coarsest-level solver. \\ \verb|mld_coarse_fillin_| & \verb|integer| - & any integer number $\ge 0$ + & Any~int.~num.~$\ge 0$ & \texttt{0} - & Fill-in level $p$ of the incomplete LU factorizations. \\ + & Fill-in level $p$ of the incomplete LU factorizations. \\ \hline \verb|mld_coarse_thresh_| & \verb|real(|\emph{kind\_parameter}\verb|)| - & any real number $\ge 0$ + & Any~real.~num.~$\ge 0$ & \texttt{0.d0} (or \texttt{0.e0}) & Drop tolerance $t$ in the ILU($p,t$) factorization. \\ \hline @@ -395,6 +398,7 @@ This routine deallocates the preconditioner data structure. \subsection{Subroutine mld\_precdescr\label{sec:precdescr}} \begin{center} +\verb|mld_precdescr(p,info)|\\ \verb|mld_precdescr(p,info,iout)|\\ \end{center} diff --git a/docs/userguide.pdf b/docs/userguide.pdf index fd106d0a..219a3b5c 100644 --- a/docs/userguide.pdf +++ b/docs/userguide.pdf @@ -141,14 +141,14 @@ endobj << /S /GoTo /D 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cm +1 0 0 1 -505.3752 -570.4927 cm BT -/F15 10.9091 Tf 202.4658 543.3944 Td[(7)]TJ +/F15 10.9091 Tf 505.3752 570.4927 Td[(7)]TJ ET -1 0 0 1 207.9204 543.3944 cm +1 0 0 1 510.8298 570.4927 cm 0 g 0 G -1 0 0 1 -207.9204 -543.3944 cm +1 0 0 1 -510.8298 -570.4927 cm BT -/F15 10.9091 Tf 207.9204 543.3944 Td[(,)]TJ +/F15 10.9091 Tf 510.8298 570.4927 Td[(,)]TJ ET -1 0 0 1 214.587 543.3944 cm +1 0 0 1 517.4965 570.4927 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -214.587 -543.3944 cm +1 0 0 1 -517.4965 -570.4927 cm BT -/F15 10.9091 Tf 214.587 543.3944 Td[(15)]TJ +/F15 10.9091 Tf 517.4965 570.4927 Td[(15)]TJ ET -1 0 0 1 225.4961 543.3944 cm +1 0 0 1 528.4056 570.4927 cm 0 g 0 G -1 0 0 1 -225.4961 -543.3944 cm +1 0 0 1 -528.4056 -570.4927 cm BT -/F15 10.9091 Tf 225.4961 543.3944 Td[(].)]TJ/F20 10.9091 Tf -66.8211 -14.0113 Td[(A)51(dditive)-324(Schw)-1(arz)]TJ/F15 10.9091 Tf 85.7786 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-297(are)-297(DD)-297(p)1(rec)-1(on)1(diti)1(oners)-297(usin)1(g)-297(o)28(v)28(e)-1(r)1(lap-)]TJ -102.7151 -13.5492 Td[(pi)1(ng)-429(sub)1(matrice)-1(s,)-452(i.e.)-429(with)-428(som)-1(e)-429(com)-1(mon)-428(ro)28(ws)-1(,)-452(to)-428(c)-1(ou)1(ple)-429(th)1(e)-429(lo)-28(cal)-429(in)1(for)1(-)]TJ 0 -13.5492 Td[(mation)-443(re)-1(l)1(ate)-1(d)-443(to)-444(th)1(e)-445(sub)1(m)-1(atr)1(ice)-1(s)-444(\050s)-1(ee,)-472(e.g.,)-471([)]TJ +/F15 10.9091 Tf 528.4056 570.4927 Td[(].)]TJ/F20 10.9091 Tf -369.7306 -14.0112 Td[(A)51(dditive)-468(Schw)-1(arz)]TJ/F15 10.9091 Tf 89.0586 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-454(are)-453(DD)-453(prec)-1(on)1(di)1(tioners)-453(using)-453(o)28(v)28(e)-1(r)1(lapp)1(ing)-453(sub)1(-)]TJ -105.9951 -13.5492 Td[(matrice)-1(s,)-403(i.)1(e)-1(.)-389(with)-388(s)-1(ome)-389(c)-1(omm)-1(on)-388(ro)28(ws)-1(,)-403(t)1(o)-389(c)-1(ou)1(ple)-389(the)-389(lo)-28(cal)-389(in)1(formation)-388(relate)-1(d)-388(to)-389(the)]TJ 0 -13.5492 Td[(submatr)1(ic)-1(es)-362(\050s)-1(ee)-1(,)-369(e.g.,)-368([)]TJ ET -1 0 0 1 367.3921 502.2847 cm +1 0 0 1 253.9509 529.3831 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -367.3921 -502.2847 cm +1 0 0 1 -253.9509 -529.3831 cm BT -/F15 10.9091 Tf 367.3921 502.2847 Td[(15)]TJ +/F15 10.9091 Tf 253.9509 529.3831 Td[(15)]TJ ET -1 0 0 1 378.3012 502.2847 cm +1 0 0 1 264.86 529.3831 cm 0 g 0 G -1 0 0 1 -378.3012 -502.2847 cm +1 0 0 1 -264.86 -529.3831 cm BT -/F15 10.9091 Tf 378.3012 502.2847 Td[(]\051.)-776(Th)1(e)-445(main)-443(m)-1(oti)1(v)55(at)1(ion)]TJ -236.5627 -13.5492 Td[(for)-339(c)27(h)1(o)-28(osing)-340(Ad)1(di)1(tiv)28(e)-340(Sc)27(h)29(w)27(ar)1(z)-341(p)1(rec)-1(on)1(diti)1(oners)-340(is)-340(their)-339(in)28(tri)1(nsic)-340(paral)1(le)-1(l)1(is)-1(m.)]TJ 0 -13.5492 Td[(A)-485(dra)28(wbac)28(k)-486(of)-485(th)1(e)-1(se)-486(p)1(re)-1(cond)1(ition)1(e)-1(r)1(s)-486(is)-486(th)1(at)-486(th)1(e)-486(n)28(um)28(b)-27(e)-1(r)-485(of)-485(iteration)1(s)-486(of)]TJ 0 -13.5492 Td[(th)1(e)-316(p)1(rec)-1(on)1(dit)1(ioned)-315(solv)28(ers)-315(generally)-314(gro)28(ws)-315(w)-1(i)1(th)-315(t)1(he)-315(n)28(u)1(m)27(b)-27(e)-1(r)-314(of)-314(s)-1(u)1(bmatrice)-1(s.)]TJ 0 -13.5492 Td[(This)-403(ma)28(y)-403(b)-27(e)-404(a)-403(seriou)1(s)-404(li)1(m)-1(i)1(tation)-402(on)-403(p)1(arallel)-403(com)-1(p)1(uters,)-420(s)-1(i)1(nce)-404(th)1(e)-403(n)28(um)28(b)-28(er)]TJ 0 -13.5492 Td[(of)-408(submatr)1(ic)-1(es)-409(usual)1(ly)-408(m)-1(atc)28(hes)-409(the)-409(n)29(um)27(b)-27(er)-408(of)-409(a)28(v)56(ailab)1(le)-409(p)1(ro)-28(ce)-1(ss)-1(or)1(s)-1(.)-669(Opti)1(-)]TJ 0 -13.5492 Td[(mal)-332(c)-1(on)28(v)28(ergence)-333(rates)-1(,)-332(i.)1(e)-1(.)-332(iterati)1(on)-332(n)28(um)28(b)-28(ers)-333(i)1(nd)1(e)-1(p)-27(e)-1(n)1(den)28(t)-332(of)-332(the)-332(n)28(um)28(b)-28(er)-332(of)]TJ 0 -13.5492 Td[(submatr)1(ic)-1(es)-1(,)-481(c)-1(an)-451(b)-28(e)-452(obtai)1(ned)-452(b)28(y)-452(correc)-1(t)1(ing)-452(th)1(e)-453(p)1(re)-1(cond)1(ition)1(e)-1(r)-451(thr)1(ough)-452(a)]TJ 0 -13.5492 Td[(suitab)1(le)-280(app)1(ro)28(ximation)-279(of)-279(the)-280(ori)1(ginal)-279(lin)1(e)-1(ar)-279(s)-1(y)1(s)-1(tem)-280(in)-279(a)-280(c)-1(oar)1(s)-1(e)-280(space)-1(,)-290(whi)1(c)27(h)]TJ 0 -13.5492 Td[(glob)1(ally)-333(coupl)1(e)-1(s)-334(t)1(he)-334(in)1(for)1(m)-1(ati)1(on)-333(related)-333(to)-334(t)1(he)-334(singl)1(e)-334(s)-1(u)1(bmatri)1(c)-1(es)-1(.)]TJ/F20 10.9091 Tf 16.9365 -14.0113 Td[(Two-l)1(evel)-256(Schwarz)]TJ/F15 10.9091 Tf 90.3282 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-223(are)-222(obtain)1(e)-1(d)-221(b)28(y)-222(c)-1(om)28(bi)1(nin)1(g)-223(b)1(as)-1(i)1(c)-223(\050on)1(e)-1(-)]TJ -107.2647 -13.5492 Td[(lev)28(e)-1(l)1(\051)-235(S)1(c)27(h)28(w)28(arz)-235(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-235(w)-1(i)1(th)-234(a)-235(coarse)-1(-lev)28(e)-1(l)-234(c)-1(or)1(rec)-1(ti)1(on.)-411(In)-235(t)1(his)-235(con)28(text,)]TJ 0 -13.5492 Td[(th)1(e)-410(one-le)-1(v)28(el)-409(p)1(rec)-1(on)1(diti)1(oner)-409(is)-409(often)-409(called)-409(`sm)-1(o)-27(other'.)-671(Di\013e)-1(r)1(e)-1(n)29(t)-409(t)28(w)27(o-lev)28(e)-1(l)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-376(are)-376(ob)1(tain)1(e)-1(d)-375(b)28(y)-375(v)56(aryi)1(ng)-375(the)-376(c)28(hoice)-376(of)-375(th)1(e)-376(s)-1(mo)-28(oth)1(e)-1(r)-375(an)1(d)-375(of)]TJ 0 -13.5492 Td[(th)1(e)-359(coarse)-1(-lev)28(e)-1(l)-358(corr)1(e)-1(ction,)-363(and)-358(t)1(he)-358(w)27(a)28(y)-358(th)1(e)-1(y)-358(ar)1(e)-359(com)27(b)1(ined)-358([)]TJ +/F15 10.9091 Tf 264.86 529.3831 Td[(]\051.)-530(Th)1(e)-363(main)-361(m)-1(oti)1(v)55(at)1(ion)-362(f)1(or)-362(c)28(ho)-28(osing)-362(Ad)1(di)1(tiv)28(e)-362(Sc)27(h)29(w)27(arz)-362(p)1(re)-1(-)]TJ -123.1215 -13.5492 Td[(condi)1(tioners)-361(is)-362(t)1(heir)-361(in)29(trin)1(s)-1(ic)-361(par)1(allelism)-1(.)-527(A)-361(dr)1(a)27(wb)1(ac)27(k)-361(of)-361(th)1(e)-1(se)-362(p)1(rec)-1(on)1(dit)1(ioners)-361(is)-362(th)1(at)]TJ 0 -13.5492 Td[(th)1(e)-389(n)28(u)1(m)27(b)-27(e)-1(r)-388(of)-387(iteration)1(s)-389(of)-388(th)1(e)-389(p)1(rec)-1(on)1(diti)1(oned)-388(s)-1(ol)1(v)28(e)-1(r)1(s)-389(generally)-388(gr)1(o)27(ws)-388(w)-1(i)1(th)-388(th)1(e)-389(n)28(u)1(m)-1(-)]TJ 0 -13.5492 Td[(b)-27(e)-1(r)-370(of)-369(s)-1(u)1(bmatri)1(c)-1(es)-1(.)-554(T)-1(h)1(is)-370(m)-1(a)28(y)-370(b)-27(e)-371(a)-370(se)-1(r)1(ious)-370(limitation)-369(on)-370(p)1(arallel)-370(com)-1(p)1(uters,)-379(s)-1(in)1(c)-1(e)-370(the)]TJ 0 -13.5492 Td[(n)28(u)1(m)27(b)-27(e)-1(r)-403(of)-402(s)-1(u)1(bmatrices)-404(usual)1(ly)-403(m)-1(at)1(c)27(hes)-404(th)1(e)-404(n)28(u)1(m)27(b)-27(e)-1(r)-402(of)-403(a)28(v)55(ai)1(lable)-403(pr)1(o)-28(c)-1(ess)-1(ors.)-654(Op)1(timal)]TJ 0 -13.5492 Td[(con)28(v)28(e)-1(r)1(ge)-1(n)1(c)-1(e)-262(rates,)-276(i.e.)-262(it)1(e)-1(rati)1(on)-262(n)29(um)28(b)-28(ers)-262(in)1(dep)-28(end)1(e)-1(n)28(t)-261(of)-262(t)1(he)-262(n)28(um)28(b)-27(e)-1(r)-261(of)-262(sub)1(m)-1(at)1(rice)-1(s,)-276(can)]TJ 0 -13.5492 Td[(b)-27(e)-370(obtai)1(ned)-369(b)28(y)-369(c)-1(or)1(re)-1(ctin)1(g)-370(th)1(e)-370(pr)1(e)-1(cond)1(ition)1(e)-1(r)-369(th)1(rou)1(gh)-369(a)-370(suit)1(able)-370(ap)1(pr)1(o)27(x)1(im)-1(at)1(ion)-369(of)-369(the)]TJ 0 -13.5492 Td[(ori)1(ginal)-323(l)1(inear)-323(syste)-1(m)-323(in)-322(a)-323(c)-1(oar)1(s)-1(e)-323(s)-1(p)1(ac)-1(e,)-325(whi)1(c)27(h)-323(gl)1(oball)1(y)-323(c)-1(ou)1(ples)-323(the)-323(inf)1(ormation)-322(related)]TJ 0 -13.5492 Td[(to)-333(the)-333(s)-1(in)1(gle)-334(sub)1(m)-1(atr)1(ice)-1(s.)]TJ/F20 10.9091 Tf 16.9365 -14.0113 Td[(Two-l)1(evel)-256(Schwarz)]TJ/F15 10.9091 Tf 90.3282 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-223(are)-222(obtain)1(e)-1(d)-221(b)28(y)-222(c)-1(om)28(bi)1(nin)1(g)-223(b)1(as)-1(i)1(c)-223(\050on)1(e)-1(-lev)28(e)-1(l\051)-222(S)1(c)27(h)28(w)28(arz)]TJ -107.2647 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-368(with)-366(a)-368(coarse)-1(-lev)28(e)-1(l)-366(c)-1(orr)1(e)-1(ction)1(.)-546(In)-367(t)1(his)-367(c)-1(on)28(t)1(e)-1(xt,)-375(th)1(e)-368(on)1(e)-1(-lev)28(e)-1(l)-367(p)1(rec)-1(on)1(di-)]TJ 0 -13.5492 Td[(tion)1(e)-1(r)-422(i)1(s)-423(oft)1(e)-1(n)-422(called)-422(`smo)-28(other')1(.)-711(Di\013e)-1(r)1(e)-1(n)28(t)-422(t)28(w)28(o-)-1(l)1(e)-1(v)28(el)-422(pr)1(e)-1(cond)1(ition)1(e)-1(rs)-422(are)-423(ob)1(tain)1(e)-1(d)-421(b)28(y)]TJ 0 -13.5492 Td[(v)56(aryi)1(ng)-310(t)1(he)-310(c)27(h)1(oice)-310(of)-310(t)1(he)-310(sm)-1(o)-28(ot)1(her)-310(an)1(d)-309(of)-309(the)-310(coarse)-1(-lev)28(e)-1(l)-309(c)-1(or)1(rec)-1(tion)1(,)-314(and)-309(th)1(e)-310(w)27(a)28(y)-309(they)]TJ 0 -13.5492 Td[(are)-399(com)27(b)1(in)1(e)-1(d)-398([)]TJ ET -1 0 0 1 431.3782 312.1338 cm +1 0 0 1 213.1882 352.7814 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -431.3782 -312.1338 cm +1 0 0 1 -213.1882 -352.7814 cm BT -/F15 10.9091 Tf 431.3782 312.1338 Td[(15)]TJ +/F15 10.9091 Tf 213.1882 352.7814 Td[(15)]TJ ET -1 0 0 1 442.2873 312.1338 cm +1 0 0 1 224.0973 352.7814 cm 0 g 0 G -1 0 0 1 -442.2873 -312.1338 cm +1 0 0 1 -224.0973 -352.7814 cm BT -/F15 10.9091 Tf 442.2873 312.1338 Td[(].)-518(The)-358(s)-1(ame)]TJ -300.5488 -13.5492 Td[(reas)-1(on)1(in)1(g)-295(can)-294(b)-28(e)-294(appl)1(ie)-1(d)-293(s)-1(tar)1(ting)-294(fr)1(om)-295(the)-294(c)-1(oar)1(s)-1(e-)-1(l)1(e)-1(v)28(el)-295(syste)-1(m,)-302(i.)1(e)-1(.)-294(a)-294(c)-1(oarse-)]TJ 0 -13.5492 Td[(space)-422(correc)-1(t)1(ion)-421(can)-421(b)-27(e)-421(bui)1(lt)-421(fr)1(om)-422(t)1(his)-421(s)-1(y)1(s)-1(tem)-1(,)-442(th)28(u)1(s)-422(ob)1(taini)1(ng)]TJ/F20 10.9091 Tf 309.5887 0 Td[(mu)1(lti-l)1(evel)]TJ/F15 10.9091 Tf -309.5887 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-1(.)]TJ 16.9365 -14.0113 Td[(It)-309(is)-309(w)27(or)1(th)-309(not)1(ing)-309(th)1(at)-309(optimal)-309(p)1(rec)-1(on)1(diti)1(oners)-310(d)1(o)-309(not)-309(n)1(e)-1(ce)-1(ss)-1(ar)1(ily)-309(corr)1(e)-1(-)]TJ -16.9365 -13.5492 Td[(sp)-28(on)1(d)-371(to)-370(m)-1(i)1(nim)28(um)-371(exe)-1(cuti)1(on)-371(ti)1(m)-1(es)-1(.)-556(Ind)1(e)-1(ed,)-380(to)-370(obtain)-370(e\013)-1(ec)-1(t)1(iv)28(e)-371(m)27(u)1(lti-lev)27(el)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-278(a)-277(trad)1(e)-1(o\013)-277(b)-27(e)-1(t)28(w)28(e)-1(en)-277(op)1(timalit)28(y)-277(of)-277(con)28(v)28(ergence)-278(and)-276(the)-277(c)-1(ost)-277(of)]TJ 0 -13.5492 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Td[(where)-347(th)1(e)-348(smo)-28(othed)-346(aggregation)-346(is)-347(app)1(lied)-346(lo)-28(cally)-346(to)-347(eac)27(h)-346(s)-1(u)1(bmatri)1(x)-347([)]TJ ET -1 0 0 1 217.7888 555.0952 cm +1 0 0 1 430.4914 636.2967 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -217.7888 -555.0952 cm +1 0 0 1 -430.4914 -636.2967 cm BT -/F15 10.9091 Tf 217.7888 555.0952 Td[(18)]TJ +/F15 10.9091 Tf 430.4914 636.2967 Td[(18)]TJ ET -1 0 0 1 228.6979 555.0952 cm +1 0 0 1 441.4005 636.2967 cm 0 g 0 G -1 0 0 1 -228.6979 -555.0952 cm +1 0 0 1 -441.4005 -636.2967 cm BT -/F15 10.9091 Tf 228.6979 555.0952 Td[(].)-417(In)-253(the)-253(next)-253(t)27(w)28(o)-253(s)-1(u)1(bse)-1(ction)1(s)-254(w)27(e)-253(pro)28(vi)1(de)-254(a)-253(b)1(rief)]TJ -134.7801 -13.5492 Td[(desc)-1(ri)1(ption)-316(of)-316(th)1(e)-317(m)27(u)1(lti-lev)28(e)-1(l)-316(Sc)28(h)28(w)27(ar)1(z)-317(pr)1(e)-1(cond)1(ition)1(e)-1(rs)-317(an)1(d)-316(on)-316(the)-317(smo)-28(othed)]TJ 0 -13.5492 Td[(aggregation)-355(tec)27(h)1(niq)1(ue)-356(as)-356(imp)1(le)-1(men)28(te)-1(d)-355(i)1(n)-355(MLD2P4.)-510(F)83(or)-355(f)1(ur)1(ther)-355(details)-356(th)1(e)]TJ 0 -13.5492 Td[(user)-333(is)-334(referr)1(e)-1(d)-333(to)-333([)]TJ +/F15 10.9091 Tf 441.4005 636.2967 Td[(].)-484(In)-346(the)-347(n)1(e)-1(xt)]TJ -347.4827 -13.5492 Td[(t)28(w)27(o)-248(s)-1(u)1(bse)-1(ction)1(s)-250(w)28(e)-249(pr)1(o)27(v)1(ide)-249(a)-249(b)1(rief)-248(des)-1(crip)1(tion)-248(of)-248(the)-249(m)28(ulti)1(-)-1(lev)28(el)-249(Sc)28(h)28(w)28(arz)-249(pr)1(e)-1(cond)1(ition)1(e)-1(rs)]TJ 0 -13.5492 Td[(and)-389(of)-389(th)1(e)-390(s)-1(mo)-28(oth)1(e)-1(d)-389(aggr)1(e)-1(gati)1(on)-389(te)-1(c)28(hni)1(que)-390(as)-390(i)1(m)-1(p)1(lem)-1(en)28(ted)-389(in)-389(MLD2P)1(4.)-613(F)83(or)-389(f)1(ur)1(ther)]TJ 0 -13.5492 Td[(detail)1(s)-334(the)-333(use)-1(r)-333(is)-333(re)-1(f)1(e)-1(r)1(red)-333(to)-333([)]TJ ET -1 0 0 1 184.7664 514.4476 cm +1 0 0 1 238.1604 595.6491 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -184.7664 -514.4476 cm +1 0 0 1 -238.1604 -595.6491 cm BT -/F15 10.9091 Tf 184.7664 514.4476 Td[(2)]TJ +/F15 10.9091 Tf 238.1604 595.6491 Td[(2)]TJ ET -1 0 0 1 190.2209 514.4476 cm +1 0 0 1 243.6149 595.6491 cm 0 g 0 G -1 0 0 1 -190.2209 -514.4476 cm +1 0 0 1 -243.6149 -595.6491 cm BT -/F15 10.9091 Tf 190.2209 514.4476 Td[(,)]TJ +/F15 10.9091 Tf 243.6149 595.6491 Td[(,)]TJ ET -1 0 0 1 196.8876 514.4476 cm +1 0 0 1 250.2816 595.6491 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -196.8876 -514.4476 cm +1 0 0 1 -250.2816 -595.6491 cm BT -/F15 10.9091 Tf 196.8876 514.4476 Td[(3)]TJ +/F15 10.9091 Tf 250.2816 595.6491 Td[(3)]TJ ET -1 0 0 1 202.3421 514.4476 cm +1 0 0 1 255.7361 595.6491 cm 0 g 0 G -1 0 0 1 -202.3421 -514.4476 cm +1 0 0 1 -255.7361 -595.6491 cm BT -/F15 10.9091 Tf 202.3421 514.4476 Td[(,)]TJ +/F15 10.9091 Tf 255.7361 595.6491 Td[(,)]TJ ET -1 0 0 1 209.0088 514.4476 cm +1 0 0 1 262.4028 595.6491 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -209.0088 -514.4476 cm +1 0 0 1 -262.4028 -595.6491 cm BT -/F15 10.9091 Tf 209.0088 514.4476 Td[(4)]TJ +/F15 10.9091 Tf 262.4028 595.6491 Td[(4)]TJ ET -1 0 0 1 214.4634 514.4476 cm +1 0 0 1 267.8574 595.6491 cm 0 g 0 G -1 0 0 1 -214.4634 -514.4476 cm +1 0 0 1 -267.8574 -595.6491 cm BT -/F15 10.9091 Tf 214.4634 514.4476 Td[(,)]TJ +/F15 10.9091 Tf 267.8574 595.6491 Td[(,)]TJ ET -1 0 0 1 221.13 514.4476 cm +1 0 0 1 274.524 595.6491 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -221.13 -514.4476 cm +1 0 0 1 -274.524 -595.6491 cm BT -/F15 10.9091 Tf 221.13 514.4476 Td[(15)]TJ +/F15 10.9091 Tf 274.524 595.6491 Td[(15)]TJ ET -1 0 0 1 232.0391 514.4476 cm +1 0 0 1 285.4331 595.6491 cm 0 g 0 G -1 0 0 1 -232.0391 -514.4476 cm +1 0 0 1 -285.4331 -595.6491 cm BT -/F15 10.9091 Tf 232.0391 514.4476 Td[(].)]TJ/F19 11.9552 Tf -138.1213 -28.7309 Td[(5.)-1(1)-1125(M)-1(ult)-1(i-l)-1(ev)32(e)-1(l)-375(Sc)31(h)31(w)31(arz)-375(Pre)-1(condi)-1(ti)-1(oner)-1(s)]TJ/F15 10.9091 Tf 0 -20.5946 Td[(The)-443(Mu)1(ltilev)28(el)-443(pr)1(e)-1(cond)1(ition)1(e)-1(rs)-443(impl)1(e)-1(me)-1(n)29(te)-1(d)-442(in)-442(MLD2P4)-442(are)-443(obtain)1(e)-1(d)-442(b)28(y)]TJ 0 -13.5492 Td[(com)27(bi)1(nin)1(g)-438(AS)-436(pr)1(e)-1(cond)1(ition)1(e)-1(rs)-437(w)-1(i)1(th)-437(coars)-1(e-s)-1(p)1(ac)-1(e)-438(corr)1(e)-1(ction)1(s)-1(;)-489(th)1(e)-1(r)1(e)-1(f)1(ore)-438(w)28(e)]TJ 0 -13.5492 Td[(\014r)1(s)-1(t)-333(p)1(ro)28(vid)1(e)-334(a)-334(sk)28(etc)27(h)-333(of)-333(th)1(e)-334(AS)-333(p)1(re)-1(cond)1(ition)1(e)-1(r)1(s)-1(.)]TJ 16.9365 -13.5492 Td[(Giv)28(en)-396(th)1(e)-397(li)1(near)-396(s)-1(y)1(s)-1(tem)-397(\050)]TJ +/F15 10.9091 Tf 285.4331 595.6491 Td[(].)]TJ/F19 11.9552 Tf -191.5153 -29.7481 Td[(5.)-1(1)-1125(M)-1(ult)-1(i-l)-1(ev)32(e)-1(l)-375(Sc)31(h)31(w)31(arz)-375(Pre)-1(condi)-1(ti)-1(oner)-1(s)]TJ/F15 10.9091 Tf 0 -20.7766 Td[(The)-270(Mu)1(ltil)1(e)-1(v)28(el)-270(p)1(rec)-1(on)1(diti)1(oners)-270(impleme)-1(n)28(ted)-269(in)-269(MLD2P)1(4)-270(are)-270(ob)1(tained)-269(b)28(y)-270(com)27(b)1(in)1(ing)-269(AS)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-316(with)-314(c)-1(oar)1(s)-1(e-)-1(space)-316(corr)1(e)-1(ctions;)-321(th)1(e)-1(refor)1(e)-316(w)28(e)-316(\014)1(rst)-315(pro)28(vi)1(de)-315(a)-315(s)-1(k)28(etc)27(h)-315(of)-314(the)]TJ 0 -13.5492 Td[(AS)-333(p)1(re)-1(cond)1(ition)1(e)-1(r)1(s)-1(.)]TJ 16.9365 -13.6429 Td[(Giv)28(en)-302(th)1(e)-303(lin)1(e)-1(ar)-302(syste)-1(m)-302(\050)]TJ ET -1 0 0 1 234.7343 424.4745 cm +1 0 0 1 230.6386 504.3831 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -234.7343 -424.4745 cm +1 0 0 1 -230.6386 -504.3831 cm BT -/F15 10.9091 Tf 234.7343 424.4745 Td[(1)]TJ +/F15 10.9091 Tf 230.6386 504.3831 Td[(1)]TJ ET -1 0 0 1 240.1888 424.4745 cm +1 0 0 1 236.0931 504.3831 cm 0 g 0 G -1 0 0 1 -240.1888 -424.4745 cm +1 0 0 1 -236.0931 -504.3831 cm BT -/F15 10.9091 Tf 240.1888 424.4745 Td[(\051,)-412(wh)1(e)-1(r)1(e)]TJ/F24 10.9091 Tf 43.9978 0 Td[(A)]TJ/F15 10.9091 Tf 12.3552 0 Td[(=)-382(\050)]TJ/F24 10.9091 Tf 16.9006 0 Td[(a)]TJ/F25 7.9701 Tf 5.7664 -1.6364 Td[(ij)]TJ/F15 10.9091 Tf 7.2653 1.6364 Td[(\051)]TJ/F27 10.9091 Tf 8.4158 0 Td[(2)-382(<)]TJ/F25 7.9701 Tf 19.3249 3.9588 Td[(n)]TJ/F28 7.9701 Tf 5.1382 0 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 9.9586 -3.9588 Td[(is)-396(a)-397(n)1(onsingu)1(lar)]TJ -281.9802 -13.5492 Td[(sparse)-443(m)-1(atr)1(ix)-443(wit)1(h)-443(a)-442(s)-1(ymme)-1(tr)1(ic)-443(non)1(z)-1(ero)-443(p)1(atte)-1(r)1(n,)-470(l)1(e)-1(t)]TJ/F24 10.9091 Tf 263.5147 0 Td[(G)]TJ/F15 10.9091 Tf 13.5969 0 Td[(=)-460(\050)]TJ/F24 10.9091 Tf 17.747 0 Td[(W)27(;)-166(E)]TJ/F15 10.9091 Tf 23.5303 0 Td[(\051)-443(b)-27(e)-443(the)]TJ -318.3889 -13.5492 Td[(adj)1(ac)-1(ency)-384(grap)1(h)-384(of)]TJ/F24 10.9091 Tf 95.3368 0 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(,)-397(where)]TJ/F24 10.9091 Tf 39.4646 0 Td[(W)]TJ/F15 10.9091 Tf 15.7763 0 Td[(=)]TJ/F27 10.9091 Tf 12.4429 0 Td[(f)]TJ/F15 10.9091 Tf 5.4545 0 Td[(1)]TJ/F24 10.9091 Tf 5.4546 0 Td[(;)]TJ/F15 10.9091 Tf 4.8484 0 Td[(2)]TJ/F24 10.9091 Tf 5.4546 0 Td[(;)-166(:)-167(:)-166(:)-167(;)-166(n)]TJ/F27 10.9091 Tf 30.7903 0 Td[(g)]TJ/F15 10.9091 Tf 9.6476 0 Td[(and)]TJ/F24 10.9091 Tf 21.7688 0 Td[(E)]TJ/F15 10.9091 Tf 12.6398 0 Td[(=)]TJ/F27 10.9091 Tf 12.4429 0 Td[(f)]TJ/F15 10.9091 Tf 5.4546 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(i;)-166(j)]TJ/F15 10.9091 Tf 13.7237 0 Td[(\051)-363(:)]TJ/F24 10.9091 Tf 15.1888 0 Td[(a)]TJ/F25 7.9701 Tf 5.7664 -1.6364 Td[(ij)]TJ/F27 10.9091 Tf 11.2233 1.6364 Td[(6)]TJ/F15 10.9091 Tf 0 0 Td[(=)-363(0)]TJ/F27 10.9091 Tf 17.8974 0 Td[(g)]TJ/F15 10.9091 Tf -353.2005 -13.5492 Td[(are)-457(the)-457(v)28(e)-1(r)1(te)-1(x)-457(set)-457(and)-457(th)1(e)-458(edge)-457(s)-1(et)-457(of)]TJ/F24 10.9091 Tf 193.5472 0 Td[(G)]TJ/F15 10.9091 Tf 8.5773 0 Td[(,)-488(resp)-28(ec)-1(ti)1(v)28(e)-1(ly)84(.)-816(Tw)28(o)-457(v)28(e)-1(r)1(tic)-1(es)-458(ar)1(e)]TJ -202.1244 -13.5492 Td[(called)-354(adjacen)28(t)-355(i)1(f)-354(there)-355(is)-355(an)-354(edge)-355(conn)1(e)-1(ctin)1(g)-355(th)1(e)-1(m.)-508(F)84(or)-354(an)28(y)-354(in)28(tege)-1(r)]TJ/F24 10.9091 Tf 329.5943 0 Td[(\016)-351(>)]TJ/F15 10.9091 Tf 20.5758 0 Td[(0,)]TJ -350.1702 -13.5492 Td[(a)]TJ/F24 10.9091 Tf 9.0405 0 Td[(\016)]TJ/F15 10.9091 Tf 5.2614 0 Td[(-o)27(v)28(erl)1(ap)-329(p)1(artit)1(ion)-329(of)]TJ/F24 10.9091 Tf 99.3034 0 Td[(W)]TJ/F15 10.9091 Tf 15.4041 0 Td[(can)-329(b)-27(e)-329(de\014n)1(e)-1(d)-328(rec)-1(u)1(rsiv)28(ely)-329(as)-329(f)1(ollo)28(ws)-1(.)-443(G)1(iv)28(e)-1(n)-328(a)-329(0-)]TJ -129.0094 -13.5492 Td[(o)28(v)28(e)-1(r)1(lap)-234(\050or)-235(n)1(on-o)27(v)28(erl)1(appi)1(ng\051)-235(p)1(arti)1(tion)-235(of)]TJ/F24 10.9091 Tf 192.0208 0 Td[(W)]TJ/F15 10.9091 Tf 11.8182 0 Td[(,)-254(i.e.)-235(a)-235(se)-1(t)-234(of)]TJ/F24 10.9091 Tf 57.6294 0 Td[(m)]TJ/F15 10.9091 Tf 12.14 0 Td[(di)1(s)-1(j)1(oin)28(t)-235(n)1(onem)-1(p)1(t)28(y)]TJ -273.6084 -13.5492 Td[(se)-1(ts)]TJ/F24 10.9091 Tf 20.4102 0 Td[(W)]TJ/F22 7.9701 Tf 11.8182 3.9588 Td[(0)]TJ/F25 7.9701 Tf -1.5151 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 9.2777 3.0555 Td[(\032)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F15 10.9091 Tf 14.5314 0 Td[(suc)27(h)-248(th)1(at)]TJ/F27 10.9091 Tf 46.3962 0 Td[([)]TJ/F25 7.9701 Tf 7.2727 3.9588 Td[(m)]TJ 0 -7.0143 Td[(i)]TJ/F22 7.9701 Tf 2.8832 0 Td[(=1)]TJ/F24 10.9091 Tf 11.3188 3.0555 Td[(W)]TJ/F22 7.9701 Tf 11.8182 3.9588 Td[(0)]TJ/F25 7.9701 Tf -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 9.2778 3.0555 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F15 10.9091 Tf 11.8182 0 Td[(,)-265(a)]TJ/F24 10.9091 Tf 14.0959 0 Td[(\016)]TJ/F15 10.9091 Tf 5.2613 0 Td[(-o)27(v)28(erl)1(ap)-249(p)1(artiti)1(on)-249(of)]TJ/F24 10.9091 Tf 96.6852 0 Td[(W)]TJ/F15 10.9091 Tf 14.5314 0 Td[(is)-249(ob)1(tained)]TJ -307.3963 -13.5492 Td[(b)28(y)-324(c)-1(on)1(s)-1(i)1(derin)1(g)-325(th)1(e)-326(sets)]TJ/F24 10.9091 Tf 111.953 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 9.0896 3.0555 Td[(\033)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 4.5878 Td[(\016)]TJ/F28 7.9701 Tf 4.046 0 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -12.1477 -7.8446 Td[(i)]TJ/F15 10.9091 Tf 20.4213 3.2568 Td[(obt)1(ained)-324(b)28(y)-325(i)1(nclud)1(ing)-324(the)-325(v)28(erti)1(c)-1(es)-325(that)]TJ -173.585 -13.5492 Td[(are)-333(adjacen)28(t)-333(to)-334(an)29(y)-334(v)28(ertex)-333(in)]TJ/F24 10.9091 Tf 141.5759 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 4.5878 Td[(\016)]TJ/F28 7.9701 Tf 4.046 0 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -12.1477 -7.8446 Td[(i)]TJ/F15 10.9091 Tf 16.88 3.2568 Td[(.)]TJ -151.8224 -15.1758 Td[(Let)]TJ/F24 10.9091 Tf 19.8889 0 Td[(n)]TJ/F25 7.9701 Tf 6.548 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.5239 3.0555 Td[(b)-27(e)-366(t)1(he)-365(s)-1(ize)-365(of)]TJ/F24 10.9091 Tf 68.1009 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 10.0391 3.0555 Td[(and)]TJ/F24 10.9091 Tf 21.5555 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0842 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 8.2309 3.0555 Td[(2)-330(<)]TJ/F25 7.9701 Tf 18.7541 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F28 7.9701 Tf 4.0896 2.5822 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 9.6161 -3.9588 Td[(th)1(e)-366(r)1(e)-1(striction)-364(op)-27(e)-1(rat)1(or)-365(th)1(at)]TJ -222.5944 -14.9745 Td[(maps)-255(a)-254(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 67.8145 0 Td[(v)]TJ/F27 10.9091 Tf 8.7095 0 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F15 10.9091 Tf 8.4128 -3.9588 Td[(on)28(to)-254(the)-255(v)28(ector)]TJ/F24 10.9091 Tf 73.5114 0 Td[(v)]TJ/F25 7.9701 Tf 5.6793 3.9588 Td[(\016)]TJ -0.3914 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 7.9658 3.0555 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 7.3641 -1.3766 Td[(con)28(tain)1(ing)-254(the)-255(comp)-28(on)1(e)-1(n)28(t)1(s)-255(of)]TJ/F24 10.9091 Tf -220.5678 -13.5492 Td[(v)]TJ/F15 10.9091 Tf 8.5224 0 Td[(corres)-1(p)-27(ond)1(in)1(g)-261(to)-261(t)1(he)-261(v)28(e)-1(r)1(tice)-1(s)-261(in)]TJ/F24 10.9091 Tf 150.4886 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 6.0593 3.0555 Td[(.)-420(The)-261(tr)1(ansp)-28(ose)-261(of)]TJ/F24 10.9091 Tf 89.0226 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 7.4715 3.0555 Td[(is)-261(a)-260(prol)1(ongation)]TJ -280.1505 -14.9745 Td[(op)-27(e)-1(r)1(ator)-422(f)1(rom)]TJ/F27 10.9091 Tf 71.7152 0 Td[(<)]TJ/F25 7.9701 Tf 7.8787 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 9.1877 -1.3766 Td[(to)]TJ/F27 10.9091 Tf 14.2969 0 Td[(<)]TJ/F25 7.9701 Tf 7.8788 3.9588 Td[(n)]TJ/F15 10.9091 Tf 5.6363 -3.9588 Td[(.)-709(The)-422(matrix)]TJ/F24 10.9091 Tf 70.606 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 9.1803 3.0555 Td[(=)]TJ/F24 10.9091 Tf 13.1211 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F24 10.9091 Tf 4.6284 3.0555 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 4.6284 3.0555 Td[(\051)]TJ/F25 7.9701 Tf 4.2424 3.9588 Td[(T)]TJ/F27 10.9091 Tf 11.2407 -3.9588 Td[(2)-425(<)]TJ/F25 7.9701 Tf 19.7878 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F28 7.9701 Tf 4.0895 2.5822 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 9.1876 -1.3766 Td[(can)-422(b)-27(e)]TJ -326.4793 -13.5492 Td[(considered)-333(as)-334(a)-333(res)-1(tr)1(ic)-1(ti)1(on)-333(of)]TJ/F24 10.9091 Tf 140.5759 0 Td[(A)]TJ/F15 10.9091 Tf 11.8182 0 Td[(corres)-1(p)-27(ond)1(in)1(g)-334(to)-333(th)1(e)-334(s)-1(et)]TJ/F24 10.9091 Tf 119.2728 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5151 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 6.0592 3.0555 Td[(.)]TJ -271.0927 -13.5492 Td[(The)]TJ/F20 10.9091 Tf 22.4243 0 Td[(class)1(ic)51(al)-357(o)-1(ne-l)1(evel)-357(A)-1(S)]TJ/F15 10.9091 Tf 106.7234 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)-333(i)1(s)-334(de\014n)1(e)-1(d)-333(b)29(y)]TJ/F24 10.9091 Tf -31.6901 -29.1687 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(AS)]TJ/F15 10.9091 Tf 15.5385 3.4341 Td[(=)]TJ/F25 7.9701 Tf 15.6486 13.6364 Td[(m)]TJ/F30 10.9091 Tf -4.1335 -3.2727 Td[(X)]TJ/F25 7.9701 Tf 1.0269 -23.4509 Td[(i)]TJ/F22 7.9701 Tf 2.8831 0 Td[(=1)]TJ/F15 10.9091 Tf 11.8476 13.0872 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(R)]TJ/F25 7.9701 Tf 8.3675 4.5043 Td[(\016)]TJ -0.0843 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.6284 2.6969 Td[(\051)]TJ/F25 7.9701 Tf 4.2424 4.5043 Td[(T)]TJ/F15 10.9091 Tf 6.6045 -4.5043 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 4.5043 Td[(\016)]TJ 0 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.5441 2.6969 Td[(\051)]TJ/F28 7.9701 Tf 4.2424 4.5043 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -4.5043 Td[(R)]TJ/F25 7.9701 Tf 8.3675 4.5043 Td[(\016)]TJ -0.0843 -7.2012 Td[(i)]TJ/F24 10.9091 Tf 4.6284 2.6969 Td[(;)]TJ/F15 10.9091 Tf -241.2307 -32.5604 Td[(where)]TJ/F24 10.9091 Tf 31.8064 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.4414 3.0555 Td[(is)-357(as)-1(sume)-1(d)-357(t)1(o)-358(b)-27(e)-358(n)1(onsingu)1(lar.)-516(Its)-357(app)1(lication)-357(to)-357(a)-357(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 276.8281 0 Td[(v)]TJ/F27 10.9091 Tf 9.1444 0 Td[(2)-317(<)]TJ/F25 7.9701 Tf 18.6166 3.9588 Td[(n)]TJ/F15 10.9091 Tf -353.0187 -17.508 Td[(with)1(in)-333(a)-333(Kryl)1(o)27(v)-333(solv)28(er)-333(re)-1(q)1(uir)1(e)-1(s)-334(t)1(he)-334(f)1(ollo)28(wing)-333(th)1(re)-1(e)-333(s)-1(teps:)]TJ +/F15 10.9091 Tf 236.0931 504.3831 Td[(\051,)-308(where)]TJ/F24 10.9091 Tf 41.8461 0 Td[(A)]TJ/F15 10.9091 Tf 11.212 0 Td[(=)-278(\050)]TJ/F24 10.9091 Tf 15.7576 0 Td[(a)]TJ/F25 7.9701 Tf 5.7664 -1.6364 Td[(ij)]TJ/F15 10.9091 Tf 7.2653 1.6364 Td[(\051)]TJ/F27 10.9091 Tf 7.2727 0 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F28 7.9701 Tf 5.1382 0 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 8.9346 -3.9588 Td[(is)-303(a)-302(n)1(onsingu)1(lar)-302(s)-1(p)1(arse)-303(matrix)]TJ -270.1365 -13.5492 Td[(with)-321(a)-321(symme)-1(tr)1(ic)-322(n)1(onze)-1(r)1(o)-321(pattern)1(,)-324(let)]TJ/F24 10.9091 Tf 184.5267 0 Td[(G)]TJ/F15 10.9091 Tf 11.6075 0 Td[(=)-278(\050)]TJ/F24 10.9091 Tf 15.7575 0 Td[(W)27(;)-166(E)]TJ/F15 10.9091 Tf 23.5303 0 Td[(\051)-321(b)-27(e)-322(th)1(e)-322(ad)1(jace)-1(n)1(c)-1(y)-321(gr)1(aph)-320(of)]TJ/F24 10.9091 Tf 134.3786 0 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(,)-323(w)-1(h)1(e)-1(r)1(e)]TJ/F24 10.9091 Tf -377.9824 -13.5492 Td[(W)]TJ/F15 10.9091 Tf 15.4193 0 Td[(=)]TJ/F27 10.9091 Tf 12.0859 0 Td[(f)]TJ/F15 10.9091 Tf 5.4546 0 Td[(1)]TJ/F24 10.9091 Tf 5.4545 0 Td[(;)]TJ/F15 10.9091 Tf 4.8485 0 Td[(2)]TJ/F24 10.9091 Tf 5.4546 0 Td[(;)-166(:)-167(:)-166(:)-167(;)-166(n)]TJ/F27 10.9091 Tf 30.7903 0 Td[(g)]TJ/F15 10.9091 Tf 9.4334 0 Td[(and)]TJ/F24 10.9091 Tf 21.5546 0 Td[(E)]TJ/F15 10.9091 Tf 12.2829 0 Td[(=)]TJ/F27 10.9091 Tf 12.086 0 Td[(f)]TJ/F15 10.9091 Tf 5.4545 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(i;)-166(j)]TJ/F15 10.9091 Tf 13.7236 0 Td[(\051)-330(:)]TJ/F24 10.9091 Tf 14.4749 0 Td[(a)]TJ/F25 7.9701 Tf 5.7665 -1.6364 Td[(ij)]TJ/F27 10.9091 Tf 10.8664 1.6364 Td[(6)]TJ/F15 10.9091 Tf 0 0 Td[(=)-330(0)]TJ/F27 10.9091 Tf 17.5405 0 Td[(g)]TJ/F15 10.9091 Tf 9.4334 0 Td[(are)-365(th)1(e)-365(v)28(e)-1(rt)1(e)-1(x)-364(s)-1(et)-365(an)1(d)-365(th)1(e)-365(e)-1(d)1(ge)-366(set)-365(of)]TJ/F24 10.9091 Tf 184.4767 0 Td[(G)]TJ/F15 10.9091 Tf 8.5772 0 Td[(,)]TJ -409.4208 -13.5492 Td[(res)-1(p)-27(ec)-1(ti)1(v)28(e)-1(ly)84(.)-466(T)-1(w)28(o)-341(v)28(erti)1(c)-1(es)-341(are)-341(c)-1(al)1(le)-1(d)-340(ad)1(jace)-1(n)28(t)-340(if)-340(there)-341(is)-341(an)-340(edge)-341(c)-1(on)1(nec)-1(t)1(ing)-340(them)-1(.)-466(F)84(or)]TJ 0 -13.5492 Td[(an)28(y)-238(in)29(te)-1(ger)]TJ/F24 10.9091 Tf 54.621 0 Td[(\016)-316(>)]TJ/F15 10.9091 Tf 19.8067 0 Td[(0,)-257(a)]TJ/F24 10.9091 Tf 19.3437 0 Td[(\016)]TJ/F15 10.9091 Tf 5.2614 0 Td[(-o)27(v)28(erl)1(ap)-238(par)1(tition)-237(of)]TJ/F24 10.9091 Tf 96.3406 0 Td[(W)]TJ/F15 10.9091 Tf 14.4165 0 Td[(can)-238(b)-28(e)-238(de\014n)1(e)-1(d)-237(re)-1(cur)1(s)-1(i)1(v)28(e)-1(ly)-238(as)-238(foll)1(o)27(ws.)-413(Gi)1(v)28(e)-1(n)]TJ -209.7899 -13.5492 Td[(a)-344(0-o)28(v)28(e)-1(r)1(lap)-343(\050or)-343(non-o)28(v)28(e)-1(r)1(lapp)1(in)1(g\051)-344(par)1(tition)-343(of)]TJ/F24 10.9091 Tf 216.2495 0 Td[(W)]TJ/F15 10.9091 Tf 11.8182 0 Td[(,)-346(i.e.)-343(a)-344(se)-1(t)-343(of)]TJ/F24 10.9091 Tf 63.3773 0 Td[(m)]TJ/F15 10.9091 Tf 13.3269 0 Td[(di)1(s)-1(j)1(oin)28(t)-343(nonempt)28(y)-343(s)-1(ets)]TJ/F24 10.9091 Tf -304.7719 -13.5492 Td[(W)]TJ/F22 7.9701 Tf 11.8182 3.9588 Td[(0)]TJ/F25 7.9701 Tf -1.5152 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 9.2778 3.0555 Td[(\032)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F15 10.9091 Tf 14.8569 0 Td[(suc)27(h)-278(th)1(at)]TJ/F27 10.9091 Tf 47.047 0 Td[([)]TJ/F25 7.9701 Tf 7.2728 3.9588 Td[(m)]TJ 0 -7.0143 Td[(i)]TJ/F22 7.9701 Tf 2.8831 0 Td[(=1)]TJ/F24 10.9091 Tf 11.3188 3.0555 Td[(W)]TJ/F22 7.9701 Tf 11.8182 3.9588 Td[(0)]TJ/F25 7.9701 Tf -1.5151 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 9.2777 3.0555 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F15 10.9091 Tf 11.8182 0 Td[(,)-289(a)]TJ/F24 10.9091 Tf 14.6817 0 Td[(\016)]TJ/F15 10.9091 Tf 5.2614 0 Td[(-o)27(v)28(erl)1(ap)-279(p)1(arti)1(tion)-278(of)]TJ/F24 10.9091 Tf 97.6615 0 Td[(W)]TJ/F15 10.9091 Tf 14.8569 0 Td[(is)-279(ob)1(tained)-278(b)28(y)-278(c)-1(on)1(s)-1(i)1(derin)1(g)]TJ -289.8501 -13.5492 Td[(th)1(e)-307(s)-1(ets)]TJ/F24 10.9091 Tf 39.5379 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 9.0896 3.0555 Td[(\033)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 4.5878 Td[(\016)]TJ/F28 7.9701 Tf 4.046 0 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -12.1477 -7.8446 Td[(i)]TJ/F15 10.9091 Tf 20.2247 3.2568 Td[(obt)1(ained)-306(b)28(y)-307(i)1(nclud)1(ing)-306(the)-307(v)28(erti)1(c)-1(es)-307(that)-306(are)-307(adj)1(ac)-1(en)28(t)-306(to)-307(an)28(y)-306(v)28(ertex)]TJ -100.9733 -13.5492 Td[(in)]TJ/F24 10.9091 Tf 12.7273 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 4.5878 Td[(\016)]TJ/F28 7.9701 Tf 4.046 0 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -12.1477 -7.8446 Td[(i)]TJ/F15 10.9091 Tf 16.88 3.2568 Td[(.)]TJ -22.9738 -15.2694 Td[(Let)]TJ/F24 10.9091 Tf 18.79 0 Td[(n)]TJ/F25 7.9701 Tf 6.548 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 7.425 3.0555 Td[(b)-27(e)-265(th)1(e)-265(size)-265(of)]TJ/F24 10.9091 Tf 63.7053 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.9402 3.0555 Td[(and)]TJ/F24 10.9091 Tf 20.4566 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0842 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 7.6586 3.0555 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F28 7.9701 Tf 4.0895 2.5822 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 8.5172 -3.9588 Td[(th)1(e)-265(res)-1(t)1(riction)-264(op)-27(erator)-264(th)1(at)-264(maps)-265(a)-264(v)28(ec)-1(t)1(or)]TJ/F24 10.9091 Tf -211.5596 -14.9745 Td[(v)]TJ/F27 10.9091 Tf 8.9929 0 Td[(2)-303(<)]TJ/F25 7.9701 Tf 18.4651 3.9588 Td[(n)]TJ/F15 10.9091 Tf 9.4427 -3.9588 Td[(on)28(to)-349(t)1(he)-349(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 76.601 0 Td[(v)]TJ/F25 7.9701 Tf 5.6793 3.9588 Td[(\016)]TJ -0.3914 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 8.2491 3.0555 Td[(2)-303(<)]TJ/F25 7.9701 Tf 18.4651 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 8.394 -1.3766 Td[(con)28(tain)1(ing)-349(th)1(e)-350(comp)-28(on)1(e)-1(n)29(ts)-350(of)]TJ/F24 10.9091 Tf 144.9832 0 Td[(v)]TJ/F15 10.9091 Tf 9.4857 0 Td[(corres)-1(p)-27(ond)1(in)1(g)-349(to)-349(the)]TJ -313.5049 -14.9745 Td[(v)28(ertice)-1(s)-369(in)]TJ/F24 10.9091 Tf 52.9901 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 6.0593 3.0555 Td[(.)-551(The)-369(transp)-27(os)-1(e)-369(of)]TJ/F24 10.9091 Tf 94.0006 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.6538 3.0555 Td[(is)-369(a)-369(pr)1(olongati)1(on)-369(op)-27(e)-1(r)1(ator)-369(fr)1(om)]TJ/F27 10.9091 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Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F28 7.9701 Tf 4.0895 2.5822 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 5.6363 -3.9588 Td[(,)-327(w)-1(h)1(e)-1(r)1(e)]TJ 40.4397 2.7576 Td[(~)]TJ/F24 10.9091 Tf -2.3656 -2.7576 Td[(R)]TJ/F22 7.9701 Tf 8.3674 3.9588 Td[(0)]TJ/F25 7.9701 Tf -0.0842 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.3762 3.0555 Td[(is)-327(ob)1(tain)1(e)-1(d)-326(b)29(y)-326(z)-1(eroin)1(g)-327(th)1(e)-327(ro)28(ws)-327(of)]TJ/F24 10.9091 Tf 164.4935 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F15 10.9091 Tf -354.0266 -10.4937 Td[(corres)-1(p)-27(ond)1(in)1(g)-334(to)-333(th)1(e)-334(v)28(e)-1(r)1(tice)-1(s)-334(i)1(n)]TJ/F24 10.9091 Tf 154.4547 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 6.0593 3.0555 Td[(n)]TJ/F24 10.9091 Tf 5.4546 0 Td[(W)]TJ/F22 7.9701 Tf 11.8182 3.9588 Td[(0)]TJ/F25 7.9701 Tf -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 6.2475 3.0555 Td[(:)]TJ/F24 10.9091 Tf -81.7001 -34.0334 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5879 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.022 Td[(R)-6(AS)]TJ/F15 10.9091 Tf 21.8945 3.4341 Td[(=)]TJ/F25 7.9701 Tf 15.6487 13.6364 Td[(m)]TJ/F30 10.9091 Tf -4.1336 -3.2727 Td[(X)]TJ/F25 7.9701 Tf 1.0269 -23.4509 Td[(i)]TJ/F22 7.9701 Tf 2.8831 0 Td[(=1)]TJ/F15 10.9091 Tf 11.8476 13.0872 Td[(\050)]TJ 6.608 2.7576 Td[(~)]TJ/F24 10.9091 Tf -2.3655 -2.7576 Td[(R)]TJ/F22 7.9701 Tf 8.3674 4.5043 Td[(0)]TJ/F25 7.9701 Tf -0.0843 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.8166 2.6969 Td[(\051)]TJ/F25 7.9701 Tf 4.2424 4.5043 Td[(T)]TJ/F15 10.9091 Tf 6.6045 -4.5043 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 4.5043 Td[(\016)]TJ 0 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.5441 2.6969 Td[(\051)]TJ/F28 7.9701 Tf 4.2424 4.5043 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -4.5043 Td[(R)]TJ/F25 7.9701 Tf 8.3675 4.5043 Td[(\016)]TJ -0.0843 -7.2012 Td[(i)]TJ/F24 10.9091 Tf 4.6284 2.6969 Td[(:)]TJ/F15 10.9091 Tf -244.5028 -35.1789 Td[(Anal)1(ogously)83(,)-404(t)1(he)-390(AS)-390(v)56(arian)29(t)-390(c)-1(all)1(e)-1(d)]TJ/F20 10.9091 Tf 171.8214 0 Td[(AS)-410(w)-1(ith)-410(Harmonic)-410(ex)1(t)-1(ensi)1(o)-1(n)-410(\050ASH\051)]TJ/F15 10.9091 Tf 179.5003 0 Td[(is)]TJ -351.3217 -13.5492 Td[(de\014n)1(e)-1(d)-332(b)28(y)]TJ/F24 10.9091 Tf 110.5505 -22.1401 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(AS)-56(H)]TJ/F15 10.9091 Tf 23.0374 3.4341 Td[(=)]TJ/F25 7.9701 Tf 15.6487 13.6364 Td[(m)]TJ/F30 10.9091 Tf -4.1335 -3.2727 Td[(X)]TJ/F25 7.9701 Tf 1.0268 -23.4509 Td[(i)]TJ/F22 7.9701 Tf 2.8832 0 Td[(=1)]TJ/F15 10.9091 Tf 11.8476 13.0872 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 4.5043 Td[(\016)]TJ -0.0842 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.6283 2.6969 Td[(\051)]TJ/F25 7.9701 Tf 4.2425 4.5043 Td[(T)]TJ/F15 10.9091 Tf 6.6045 -4.5043 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 4.5043 Td[(\016)]TJ 0 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.5441 2.6969 Td[(\051)]TJ/F28 7.9701 Tf 4.2425 4.5043 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 7.0979 -1.7467 Td[(~)]TJ/F24 10.9091 Tf -2.3656 -2.7576 Td[(R)]TJ/F22 7.9701 Tf 8.3674 4.5043 Td[(0)]TJ/F25 7.9701 Tf -0.0842 -7.2012 Td[(i)]TJ/F24 10.9091 Tf 4.8165 2.6969 Td[(:)]TJ/F15 10.9091 Tf -245.0742 -30.2336 Td[(W)84(e)-376(n)1(ote)-375(that)-375(f)1(or)]TJ/F24 10.9091 Tf 85.1745 0 Td[(\016)]TJ/F15 10.9091 Tf 9.046 0 Td[(=)-347(0)-375(t)1(he)-375(thr)1(e)-1(e)-375(v)56(arian)28(t)1(s)-376(of)-374(the)-375(AS)-374(pr)1(e)-1(cond)1(ition)1(e)-1(r)-374(are)-375(all)]TJ -94.2205 -13.5492 Td[(equal)-333(to)-333(the)-333(blo)-27(c)27(k-Jac)-1(ob)1(i)-333(pr)1(e)-1(cond)1(ition)1(e)-1(r.)]TJ 16.9365 -13.8371 Td[(As)-338(already)-337(obse)-1(rv)28(ed,)-338(the)-338(con)28(v)28(e)-1(r)1(ge)-1(n)1(c)-1(e)-338(rate)-338(of)-337(the)-338(one-le)-1(v)28(el)-338(S)1(c)27(h)28(w)28(arz)-338(pr)1(e)-1(-)]TJ -16.9365 -13.5492 Td[(condi)1(tioned)-387(i)1(te)-1(r)1(ativ)28(e)-388(solv)28(ers)-388(d)1(e)-1(t)1(e)-1(ri)1(orates)-388(as)-387(the)-387(n)28(um)28(b)-28(er)]TJ/F24 10.9091 Tf 272.681 0 Td[(m)]TJ/F15 10.9091 Tf 13.8011 0 Td[(of)-387(p)1(artit)1(ions)-387(of)]TJ/F24 10.9091 Tf -286.4821 -13.5492 Td[(W)]TJ/F15 10.9091 Tf 15.7048 0 Td[(in)1(c)-1(r)1(e)-1(ase)-1(s)-356([)]TJ +/F15 10.9091 Tf 512.1912 462.2978 Td[(,)]TJ ET -1 0 0 1 206.3299 331.402 cm +1 0 0 1 519.1302 462.2978 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -206.3299 -331.402 cm +1 0 0 1 -519.1302 -462.2978 cm BT -/F15 10.9091 Tf 206.3299 331.402 Td[(7)]TJ +/F15 10.9091 Tf 519.1302 462.2978 Td[(15)]TJ ET -1 0 0 1 211.7845 331.402 cm +1 0 0 1 530.0393 462.2978 cm 0 g 0 G -1 0 0 1 -211.7845 -331.402 cm +1 0 0 1 -530.0393 -462.2978 cm BT -/F15 10.9091 Tf 211.7845 331.402 Td[(,)]TJ -ET -1 0 0 1 218.7014 331.402 cm -0 1 0 rg 0 1 0 RG -1 0 0 1 -218.7014 -331.402 cm -BT -/F15 10.9091 Tf 218.7014 331.402 Td[(15)]TJ -ET -1 0 0 1 229.6105 331.402 cm -0 g 0 G -1 0 0 1 -229.6105 -331.402 cm -BT -/F15 10.9091 Tf 229.6105 331.402 Td[(].)-513(T)83(o)-356(redu)1(c)-1(e)-356(the)-357(d)1(e)-1(p)-27(end)1(e)-1(n)1(c)-1(y)-356(of)-356(th)1(e)-357(n)28(u)1(m)27(b)-27(e)-1(r)-356(of)-355(iteration)1(s)]TJ -87.872 -13.5492 Td[(on)-331(th)1(e)-332(d)1(e)-1(gr)1(e)-1(e)-331(of)-331(par)1(allelism)-332(w)28(e)-332(ma)27(y)-330(in)28(tro)-27(du)1(c)-1(e)-332(a)-331(gl)1(obal)-331(coupl)1(ing)-331(among)-331(th)1(e)]TJ 0 -13.5492 Td[(o)28(v)28(e)-1(r)1(lapp)1(ing)-370(part)1(ition)1(s)-372(b)28(y)-370(de\014n)1(ing)-370(a)-371(c)-1(oar)1(s)-1(e-s)-1(p)1(ac)-1(e)-371(app)1(ro)28(ximation)]TJ/F24 10.9091 Tf 311.3467 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 11.142 1.6887 Td[(of)-370(the)]TJ -330.6705 -13.5492 Td[(matrix)]TJ/F24 10.9091 Tf 35.9 0 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(.)-558(In)-371(a)-372(p)1(ure)-371(alge)-1(b)1(raic)-371(s)-1(ettin)1(g,)]TJ/F24 10.9091 Tf 143.8491 0 Td[(A)]TJ/F25 7.9701 Tf 8.1819 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 11.1482 1.6887 Td[(is)-372(u)1(s)-1(u)1(ally)-371(b)1(uilt)-371(with)-371(a)-371(Galerk)1(in)]TJ -207.261 -13.5492 Td[(app)1(roac)28(h.)-584(Gi)1(v)28(e)-1(n)-379(a)-380(s)-1(et)]TJ/F24 10.9091 Tf 112.0021 0 Td[(W)]TJ/F25 7.9701 Tf 10.303 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 11.2411 1.6887 Td[(of)]TJ/F20 10.9091 Tf 12.9321 0 Td[(c)51(o)51(arse)-400(vertic)51(es)]TJ/F15 10.9091 Tf 69.0486 0 Td[(,)-391(w)-1(i)1(th)-380(size)]TJ/F24 10.9091 Tf 53.8325 0 Td[(n)]TJ/F25 7.9701 Tf 6.548 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(,)-391(and)-379(a)-380(s)-1(u)1(itab)1(le)]TJ -283.0042 -13.5492 Td[(res)-1(t)1(riction)-333(op)-27(e)-1(r)1(ator)]TJ/F24 10.9091 Tf 96.2425 0 Td[(R)]TJ/F25 7.9701 Tf 8.2832 -1.6887 Td[(C)]TJ/F27 10.9091 Tf 10.127 1.6887 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 -1.3394 Td[(C)]TJ/F28 7.9701 Tf 6.1942 1.3394 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 5.6364 -3.9588 Td[(,)]TJ/F24 10.9091 Tf 6.6666 0 Td[(A)]TJ/F25 7.9701 Tf 8.1819 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7331 1.6887 Td[(is)-334(d)1(e)-1(\014)1(ned)-333(as)]TJ/F24 10.9091 Tf -37.0268 -25.3719 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.1271 1.6887 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(R)]TJ/F25 7.9701 Tf 8.2831 -1.6887 Td[(C)]TJ/F24 10.9091 Tf 7.0968 1.6887 Td[(AR)]TJ/F25 7.9701 Tf 16.5493 4.5043 Td[(T)]TJ -0.0843 -7.2012 Td[(C)]TJ/F15 10.9091 Tf -206.6136 -22.675 Td[(and)-388(th)1(e)-389(c)-1(oar)1(s)-1(e-le)-1(v)28(el)-388(c)-1(orr)1(e)-1(ction)-388(matrix)-388(to)-388(b)-28(e)-389(com)27(b)1(in)1(e)-1(d)-388(with)-388(a)-389(generi)1(c)-389(one-)]TJ 0 -13.5492 Td[(lev)28(e)-1(l)-333(AS)-333(p)1(rec)-1(on)1(diti)1(oner)]TJ/F24 10.9091 Tf 114.9092 0 Td[(M)]TJ/F22 7.9701 Tf 10.5834 -1.6887 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 9.8935 1.6887 Td[(is)-334(ob)1(tain)1(e)-1(d)-333(as)]TJ/F24 10.9091 Tf -5.7566 -25.3719 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 15.5385 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 4.5043 Td[(T)]TJ -0.0843 -7.2012 Td[(C)]TJ/F24 10.9091 Tf 7.0968 2.6969 Td[(A)]TJ/F28 7.9701 Tf 8.1819 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -6.5865 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 11.3188 3.4341 Td[(R)]TJ/F25 7.9701 Tf 8.2831 -1.6887 Td[(C)]TJ/F24 10.9091 Tf 7.0968 1.6887 Td[(;)]TJ/F15 10.9091 Tf -221.761 -25.3719 Td[(where)]TJ/F24 10.9091 Tf 30.9518 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.1395 1.6887 Td[(is)-279(as)-1(sumed)-279(to)-279(b)-27(e)-279(non)1(s)-1(in)1(gular)1(.)-426(T)-1(h)1(e)-279(appl)1(ic)-1(at)1(ion)-279(of)]TJ/F24 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Td[(App)1(lyi)1(ng)]TJ/F24 10.9091 Tf 46.932 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F28 7.9701 Tf 5.759 0 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(A)]TJ/F15 10.9091 Tf 10.1367 3.4341 Td[(to)-302(a)-302(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 54.1595 0 Td[(v)]TJ/F15 10.9091 Tf 8.9749 0 Td[(with)1(in)-302(a)-302(Kr)1(ylo)28(v)-302(s)-1(ol)1(v)28(e)-1(r)-302(corr)1(e)-1(sp)-28(on)1(ds)-302(to)-302(app)1(ly-)]TJ -147.3662 -14.9847 Td[(in)1(g)]TJ/F24 10.9091 Tf 18.1819 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 16.1446 3.4341 Td[(and)]TJ/F24 10.9091 Tf 21.2121 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 11.9104 3.4341 Td[(to)]TJ/F24 10.9091 Tf 13.3333 0 Td[(v)]TJ/F15 10.9091 Tf 9.3157 0 Td[(in)1(dep)-28(end)1(e)-1(n)29(tly)-333(and)-333(th)1(e)-1(n)-332(s)-1(u)1(m)-1(min)1(g)-334(u)1(p)-333(the)-333(re)-1(sul)1(ts)-1(.)]TJ -98.5623 -13.6281 Td[(In)-444(th)1(e)-445(m)27(u)1(ltip)1(licativ)28(e)-445(cas)-1(e,)-472(the)-444(c)-1(om)28(bin)1(ation)-444(can)-444(b)-28(e)-444(p)-28(erf)1(orme)-1(d)-444(b)28(y)-444(\014)1(rst)]TJ -16.9365 -13.5492 Td[(app)1(lyi)1(ng)-405(the)-406(sm)-1(o)-27(other)]TJ/F24 10.9091 Tf 113.3596 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 12.6968 3.4341 Td[(and)-405(t)1(hen)-405(the)-406(coarse)-1(-lev)28(e)-1(l)-405(correc)-1(ti)1(on)-405(op)-28(erator)]TJ/F24 10.9091 Tf -140.8739 -13.5492 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 12.5082 3.4341 Td[(:)]TJ/F24 10.9091 Tf 100.5161 -14.3911 Td[(w)]TJ/F15 10.9091 Tf 11.1338 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(v)-36(;)]TJ -45.7405 -13.7889 Td[(z)]TJ/F15 10.9091 Tf 8.5833 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(w)]TJ/F15 10.9091 Tf 10.5278 0 Td[(+)]TJ/F24 10.9091 Tf 10.9091 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 12.5082 3.4341 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(v)]TJ/F27 10.9091 Tf 8.1035 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(Aw)]TJ/F15 10.9091 Tf 16.2854 0 Td[(\051;)]TJ -227.7748 -19.3886 Td[(th)1(is)-268(corres)-1(p)-27(ond)1(s)-268(to)-267(the)-267(follo)28(win)1(g)]TJ/F20 10.9091 Tf 155.735 0 Td[(tw)-1(o-l)1(evel)-297(hy)1(brid)-297(pr)51(e-)1(smo)51(othe)51(d)]TJ/F15 10.9091 Tf 142.6701 0 Td[(Sc)28(h)28(w)28(arz)-268(p)1(re)-1(-)]TJ -298.4051 -13.5492 Td[(condi)1(tioner:)]TJ/F24 10.9091 Tf 82.315 -13.7858 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5879 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2342 0 Td[(LH)]TJ/F28 7.9701 Tf 13.3293 0 Td[(\000)]TJ/F25 7.9701 Tf 6.5866 0 Td[(P)-147(R)-7(E)]TJ/F15 10.9091 Tf 23.1536 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 14.9324 3.4341 Td[(+)]TJ/F30 10.9091 Tf 10.9091 8.8364 Td[(\000)]TJ/F24 10.9091 Tf 5 -8.8364 Td[(I)]TJ/F27 10.9091 Tf 8.0757 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 12.5082 3.4341 Td[(A)]TJ/F30 10.9091 Tf 8.1818 8.8364 Td[(\001)]TJ/F24 10.9091 Tf 6.8182 -8.8364 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(:)]TJ/F15 10.9091 Tf -273.3098 -20.3009 Td[(On)-351(the)-352(other)-352(h)1(and)1(,)-357(b)28(y)-351(appl)1(yin)1(g)-352(the)-352(s)-1(mo)-28(oth)1(e)-1(r)-351(after)-352(th)1(e)-353(coars)-1(e-le)-1(v)28(el)-352(correc-)]TJ 0 -13.5492 Td[(tion)1(,)-333(i.e.)-333(b)28(y)-333(c)-1(ompu)1(tin)1(g)]TJ/F24 10.9091 Tf 123.6076 -24.0371 Td[(w)]TJ/F15 10.9091 Tf 11.1338 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7758 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 12.5081 3.4341 Td[(v)-36(;)]TJ -45.7404 -13.7889 Td[(z)]TJ/F15 10.9091 Tf 8.5833 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(w)]TJ/F15 10.9091 Tf 10.5278 0 Td[(+)]TJ/F24 10.9091 Tf 10.909 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 8.274 3.4341 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(v)]TJ/F27 10.9091 Tf 8.1035 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(Aw)]TJ/F15 10.9091 Tf 16.2854 0 Td[(\051)]TJ/F24 10.9091 Tf 4.2424 0 Td[(;)]TJ/F15 10.9091 Tf -232.0172 -23.8323 Td[(th)1(e)]TJ/F20 10.9091 Tf 18.7879 0 Td[(tw)-1(o-l)1(evel)-357(h)-1(y)1(bri)1(d)-358(p)51(ost-smo)51(othe)51(d)]TJ/F15 10.9091 Tf 148.7565 0 Td[(Sc)28(h)28(w)28(arz)-334(p)1(re)-1(cond)1(ition)1(e)-1(r)-333(i)1(s)-334(obtai)1(ned:)]TJ/F24 10.9091 Tf -87.8707 -24.7447 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7758 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2341 0 Td[(LH)]TJ/F28 7.9701 Tf 13.3294 0 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(P)-147(O)-30(S)-56(T)]TJ/F15 10.9091 Tf 28.4361 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 10.6982 3.4341 Td[(+)]TJ/F30 10.9091 Tf 10.9091 8.8364 Td[(\000)]TJ/F24 10.9091 Tf 5 -8.8364 Td[(I)]TJ/F27 10.9091 Tf 8.0757 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(A)]TJ/F30 10.9091 Tf 8.1818 8.8364 Td[(\001)]TJ/F24 10.9091 Tf 6.8182 -8.8364 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 12.5082 3.4341 Td[(:)]TJ/F15 10.9091 Tf -275.951 -24.7447 Td[(On)1(e)-287(more)-286(v)56(ari)1(an)28(t)-286(of)-285(t)28(w)27(o-lev)28(e)-1(l)-285(h)28(yb)1(rid)-285(p)1(re)-1(cond)1(ition)1(e)-1(r)-285(is)-286(ob)1(tained)-285(b)28(y)-286(ap)1(ply)1(ing)]TJ 0 -13.5492 Td[(th)1(e)-366(sm)-1(o)-28(ot)1(her)-365(b)-28(efor)1(e)-366(an)1(d)-365(aft)1(e)-1(r)-365(th)1(e)-366(coarse)-1(-lev)28(e)-1(l)-365(corr)1(e)-1(ction.)-539(In)-365(th)1(is)-366(cas)-1(e,)-373(th)1(e)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)-333(i)1(s)-334(s)-1(y)1(m)-1(me)-1(t)1(ric)-334(i)1(f)]TJ/F24 10.9091 Tf 146.2729 0 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(,)]TJ/F24 10.9091 Tf 6.6667 0 Td[(M)]TJ/F22 7.9701 Tf 10.5833 -1.6887 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 9.8935 1.6887 Td[(and)]TJ/F24 10.9091 Tf 21.2122 0 Td[(M)]TJ/F25 7.9701 Tf 10.5833 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7332 1.6887 Td[(are)-333(s)-1(ymme)-1(tr)1(ic)-1(.)]TJ -211.4246 -13.6281 Td[(As)-451(previou)1(s)-1(l)1(y)-451(noted,)-480(on)-450(par)1(allel)-451(c)-1(ompu)1(ters)-452(t)1(he)-451(n)28(um)28(b)-28(er)-451(of)-450(s)-1(u)1(m)-1(at)1(rice)-1(s)]TJ -16.9365 -13.5492 Td[(usual)1(ly)-391(matc)27(h)1(e)-1(s)-392(t)1(he)-392(n)29(um)28(b)-28(er)-391(of)-391(a)28(v)56(ailab)1(le)-392(p)1(ro)-28(ce)-1(ss)-1(or)1(s)-1(.)-617(When)-391(th)1(e)-392(siz)-1(e)-391(of)-391(th)1(e)]TJ 0 -13.5492 Td[(sys)-1(tem)-384(to)-384(b)-27(e)-384(pr)1(e)-1(cond)1(ition)1(e)-1(d)-383(is)-384(v)28(ery)-384(l)1(arge,)-396(the)-384(use)-384(of)-383(m)-1(an)28(y)-383(pr)1(o)-28(ce)-1(ss)-1(or)1(s)-1(,)-396(i.e.)]TJ 0 -13.5492 Td[(of)-293(man)28(y)-293(sm)-1(al)1(l)-293(s)-1(u)1(bmatri)1(c)-1(es)-1(,)-301(of)1(te)-1(n)-292(le)-1(ad)1(s)-294(to)-293(a)-293(lar)1(ge)-294(coarse)-1(-lev)28(e)-1(l)-293(syste)-1(m,)-301(whose)]TJ 0 -13.5492 Td[(soluti)1(on)-444(ma)27(y)-444(b)-27(e)-445(compu)1(tational)1(ly)-444(exp)-28(ensiv)28(e.)-777(On)-443(the)-444(other)-444(han)1(d,)-471(the)-444(use)]TJ 0 -13.5492 Td[(of)-383(few)-384(pr)1(o)-28(c)-1(es)-1(sors)-384(of)1(te)-1(n)-383(leads)-384(to)-383(lo)-28(cal)-384(sumatrice)-1(s)-384(th)1(at)-384(ar)1(e)-385(t)1(o)-28(o)-384(exp)-27(e)-1(n)1(s)-1(iv)28(e)-384(to)]TJ 0 -13.5492 Td[(b)-27(e)-299(p)1(ro)-28(ce)-1(ss)-1(ed)-298(on)-297(s)-1(in)1(gle)-298(pro)-27(c)-1(es)-1(sors,)-305(b)-28(ecause)-299(of)-298(mem)-1(or)1(y)-298(and)1(/or)-298(c)-1(omp)1(utin)1(g)-298(re-)]TJ 0 -13.5492 Td[(qu)1(irem)-1(en)28(ts.)-410(Therefor)1(e)-1(,)-250(it)-229(se)-1(em)-1(s)-229(natu)1(ral)-229(to)-230(u)1(s)-1(e)-230(a)-229(rec)-1(u)1(rsiv)28(e)-230(app)1(roac)27(h)1(,)-250(in)-229(whic)28(h)]TJ 0 -13.5492 Td[(th)1(e)-254(coarse)-1(-lev)28(e)-1(l)-252(correc)-1(ti)1(on)-253(i)1(s)-253(re)-1(-ap)1(plied)-252(s)-1(tar)1(tin)1(g)-253(fr)1(om)-253(the)-253(curr)1(e)-1(n)29(t)-253(coars)-1(e-le)-1(v)28(el)]TJ 0 -13.5492 Td[(sys)-1(tem.)-435(The)-304(c)-1(or)1(re)-1(sp)-27(ond)1(ing)-304(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-1(,)-310(called)]TJ/F20 10.9091 Tf 242.0819 0 Td[(mu)1(lti-l)1(evel)]TJ/F15 10.9091 Tf 53.5124 0 Td[(pr)1(e)-1(cond)1(ition)1(-)]TJ -295.5943 -13.5492 Td[(ers)-1(,)-322(can)-319(s)-1(ign)1(i\014)1(c)-1(an)28(tl)1(y)-320(r)1(e)-1(d)1(uce)-321(t)1(he)-320(com)-1(p)1(utati)1(onal)-319(c)-1(ost)-320(of)-319(pr)1(e)-1(cond)1(ition)1(ing)-319(with)]TJ 0 -13.5492 Td[(res)-1(p)-27(ec)-1(t)-460(to)-461(t)1(he)-461(t)28(w)27(o-lev)28(e)-1(l)-460(cas)-1(e)-461(\050see)-461([)]TJ +/F15 10.9091 Tf 93.9178 704.1363 Td[(th)1(is)-279(corresp)-28(on)1(ds)-278(to)-279(t)1(he)-279(f)1(ollo)28(win)1(g)]TJ/F20 10.9091 Tf 156.3223 0 Td[(tw)-1(o-l)1(evel)-307(hy)1(brid)-307(pr)52(e-smo)51(othe)51(d)]TJ/F15 10.9091 Tf 143.0037 0 Td[(Sc)28(h)28(w)28(arz)-279(p)1(rec)-1(on)1(diti)1(oner:)]TJ/F24 10.9091 Tf -190.113 -24.5723 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2342 0 Td[(LH)]TJ/F28 7.9701 Tf 13.3293 0 Td[(\000)]TJ/F25 7.9701 Tf 6.5866 0 Td[(P)-147(R)-7(E)]TJ/F15 10.9091 Tf 23.1536 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 14.9324 3.4341 Td[(+)]TJ/F30 10.9091 Tf 10.9091 8.8364 Td[(\000)]TJ/F24 10.9091 Tf 5 -8.8364 Td[(I)]TJ/F27 10.9091 Tf 8.0757 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 12.5082 3.4341 Td[(A)]TJ/F30 10.9091 Tf 8.1818 8.8364 Td[(\001)]TJ/F24 10.9091 Tf 6.8182 -8.8364 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(:)]TJ/F15 10.9091 Tf -300.2078 -24.5724 Td[(On)-368(th)1(e)-369(other)-368(han)1(d,)-377(b)28(y)-368(app)1(lyi)1(ng)-369(t)1(he)-369(sm)-1(o)-27(other)-369(af)1(ter)-369(t)1(he)-369(coars)-1(e-le)-1(v)28(el)-368(c)-1(or)1(rec)-1(tion)1(,)-377(i.e.)-368(b)28(y)]TJ 0 -13.5492 Td[(com)-1(p)1(utin)1(g)]TJ/F24 10.9091 Tf 150.5056 -12.9058 Td[(w)]TJ/F15 10.9091 Tf 11.1338 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7758 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 12.5081 3.4341 Td[(v)-36(;)]TJ -45.7404 -13.7889 Td[(z)]TJ/F15 10.9091 Tf 8.5833 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(w)]TJ/F15 10.9091 Tf 10.5278 0 Td[(+)]TJ/F24 10.9091 Tf 10.909 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 8.274 3.4341 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(v)]TJ/F27 10.9091 Tf 8.1035 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(Aw)]TJ/F15 10.9091 Tf 16.2854 0 Td[(\051)]TJ/F24 10.9091 Tf 4.2424 0 Td[(;)]TJ/F15 10.9091 Tf -258.9152 -19.1875 Td[(th)1(e)]TJ/F20 10.9091 Tf 18.7879 0 Td[(tw)-1(o-l)1(evel)-357(h)-1(y)1(bri)1(d)-358(p)51(ost-smo)51(othe)51(d)]TJ/F15 10.9091 Tf 148.7565 0 Td[(Sc)28(h)28(w)28(arz)-334(p)1(re)-1(cond)1(ition)1(e)-1(r)-333(i)1(s)-334(obtai)1(ned:)]TJ/F24 10.9091 Tf -60.9726 -24.5723 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2342 0 Td[(LH)]TJ/F28 7.9701 Tf 13.3293 0 Td[(\000)]TJ/F25 7.9701 Tf 6.5866 0 Td[(P)-147(O)-30(S)-56(T)]TJ/F15 10.9091 Tf 28.4361 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 10.6982 3.4341 Td[(+)]TJ/F30 10.9091 Tf 10.9091 8.8364 Td[(\000)]TJ/F24 10.9091 Tf 5 -8.8364 Td[(I)]TJ/F27 10.9091 Tf 8.0757 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(A)]TJ/F30 10.9091 Tf 8.1818 8.8364 Td[(\001)]TJ/F24 10.9091 Tf 6.8182 -8.8364 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 12.5082 3.4341 Td[(:)]TJ/F15 10.9091 Tf -302.8491 -24.5724 Td[(On)1(e)-223(more)-223(v)56(ari)1(an)28(t)-222(of)-222(t)28(w)27(o-lev)28(e)-1(l)-222(h)29(ybr)1(id)-222(p)1(re)-1(cond)1(ition)1(e)-1(r)-221(is)-223(ob)1(tained)-222(b)28(y)-222(ap)1(ply)1(ing)-222(th)1(e)-223(s)-1(mo)-28(oth)1(e)-1(r)]TJ 0 -13.5492 Td[(b)-27(e)-1(f)1(ore)-240(an)1(d)-239(aft)1(e)-1(r)-239(th)1(e)-240(coars)-1(e-le)-1(v)28(el)-239(correc)-1(ti)1(on.)-413(In)-239(th)1(is)-240(case)-1(,)-258(th)1(e)-240(p)1(re)-1(cond)1(ition)1(e)-1(r)-238(is)-240(symm)-1(etric)]TJ 0 -13.5492 Td[(if)]TJ/F24 10.9091 Tf 10 0 Td[(A)]TJ/F15 10.9091 Tf 8.1819 0 Td[(,)]TJ/F24 10.9091 Tf 6.6666 0 Td[(M)]TJ/F22 7.9701 Tf 10.5834 -1.6887 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 9.8935 1.6887 Td[(and)]TJ/F24 10.9091 Tf 21.2121 0 Td[(M)]TJ/F25 7.9701 Tf 10.5834 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7331 1.6887 Td[(are)-333(s)-1(ymme)-1(tr)1(ic)-1(.)]TJ -75.1517 -13.5706 Td[(As)-533(pr)1(e)-1(v)1(iously)-532(noted,)-582(on)-532(p)1(arallel)-532(c)-1(ompu)1(te)-1(r)1(s)-533(the)-533(n)29(um)27(b)-27(er)-532(of)-533(sub)1(matrice)-1(s)-533(u)1(s)-1(u)1(ally)]TJ -16.9365 -13.5492 Td[(matc)27(h)1(e)-1(s)-307(the)-307(n)28(u)1(m)27(b)-27(e)-1(r)-306(of)-307(a)28(v)56(ailab)1(le)-307(pr)1(o)-28(c)-1(ess)-1(ors.)-435(When)-307(th)1(e)-307(s)-1(ize)-307(of)-307(th)1(e)-307(s)-1(ystem)-308(to)-306(b)-28(e)-307(pr)1(e)-1(con-)]TJ 0 -13.5492 Td[(di)1(tioned)-300(i)1(s)-301(v)28(e)-1(r)1(y)-300(large,)-307(t)1(he)-301(u)1(s)-1(e)-300(of)-300(man)28(y)-300(p)1(ro)-28(ce)-1(ss)-1(or)1(s)-1(,)-306(i.e.)-300(of)-300(man)28(y)-300(sm)-1(all)-299(s)-1(u)1(bmatrice)-1(s,)-307(of)1(ten)]TJ 0 -13.5492 Td[(leads)-314(to)-314(a)-314(large)-314(coars)-1(e-le)-1(v)28(el)-314(sys)-1(t)1(e)-1(m,)-318(whose)-315(solu)1(tion)-314(ma)28(y)-314(b)-27(e)-315(comput)1(ational)1(ly)-314(exp)-28(ensiv)28(e.)]TJ 0 -13.5492 Td[(On)-384(th)1(e)-386(ot)1(her)-385(h)1(and)1(,)-398(th)1(e)-385(use)-385(of)-385(f)1(e)-1(w)-385(p)1(ro)-28(ces)-1(sors)-385(often)-384(le)-1(ad)1(s)-385(to)-385(lo)-27(c)-1(al)-384(s)-1(u)1(m)-1(at)1(rice)-1(s)-385(th)1(at)-385(are)]TJ 0 -13.5492 Td[(to)-28(o)-424(exp)-27(e)-1(n)1(s)-1(i)1(v)28(e)-425(to)-424(b)-27(e)-425(p)1(ro)-28(ces)-1(se)-1(d)-423(on)-424(single)-424(pr)1(o)-28(ce)-1(ss)-1(or)1(s)-1(,)-446(b)-28(ec)-1(au)1(s)-1(e)-424(of)-424(mem)-1(or)1(y)-424(and)1(/or)-424(com)-1(-)]TJ 0 -13.5492 Td[(pu)1(tin)1(g)-324(requi)1(re)-1(men)28(ts.)-442(Th)1(e)-1(r)1(e)-1(f)1(ore)-1(,)-325(it)-324(se)-1(ems)-325(n)1(atur)1(al)-324(to)-324(u)1(s)-1(e)-324(a)-324(rec)-1(u)1(rsiv)28(e)-324(appr)1(oac)27(h)1(,)-326(in)-323(whic)28(h)]TJ 0 -13.5492 Td[(th)1(e)-395(c)-1(oar)1(s)-1(e-)-1(l)1(e)-1(v)28(el)-394(c)-1(orr)1(e)-1(ction)-394(is)-394(re)-1(-ap)1(plied)-394(startin)1(g)-395(f)1(rom)-395(t)1(he)-395(curr)1(e)-1(n)29(t)-395(coarse)-1(-lev)28(e)-1(l)-394(sys)-1(t)1(e)-1(m.)]TJ 0 -13.5492 Td[(The)-321(c)-1(or)1(res)-1(p)-27(ond)1(ing)-321(p)1(rec)-1(on)1(dit)1(ioners,)-323(c)-1(all)1(e)-1(d)]TJ/F20 10.9091 Tf 202.455 0 Td[(mu)1(lti-l)1(evel)]TJ/F15 10.9091 Tf 53.6944 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-1(,)-323(can)-321(signi)1(\014can)28(tly)]TJ -256.1494 -13.5492 Td[(redu)1(c)-1(e)-282(the)-283(com)-1(p)1(utat)1(ional)-282(cos)-1(t)-282(of)-282(pr)1(e)-1(cond)1(ition)1(ing)-282(with)-282(r)1(e)-1(sp)-28(ec)-1(t)-282(to)-282(the)-282(t)28(w)27(o-lev)28(e)-1(l)-282(c)-1(ase)-283(\050se)-1(e)]TJ 0 -13.5492 Td[([)]TJ ET -1 0 0 1 265.1812 162.1685 cm +1 0 0 1 96.9481 370.2545 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -265.1812 -162.1685 cm +1 0 0 1 -96.9481 -370.2545 cm BT -/F15 10.9091 Tf 265.1812 162.1685 Td[(15)]TJ +/F15 10.9091 Tf 96.9481 370.2545 Td[(15)]TJ ET -1 0 0 1 276.0903 162.1685 cm +1 0 0 1 107.8572 370.2545 cm 0 g 0 G -1 0 0 1 -276.0903 -162.1685 cm +1 0 0 1 -107.8572 -370.2545 cm BT -/F15 10.9091 Tf 276.0903 162.1685 Td[(,)-492(Chap)1(te)-1(r)-460(3]\051.)-825(Add)1(itiv)28(e)-461(an)1(d)-460(h)28(ybr)1(id)]TJ -182.1725 -13.5492 Td[(m)27(u)1(ltil)1(e)-1(v)28(el)-325(pr)1(e)-1(cond)1(ition)1(e)-1(rs)-325(are)-325(obtain)1(e)-1(d)-324(as)-326(di)1(re)-1(ct)-325(exte)-1(n)1(s)-1(i)1(ons)-326(of)-324(the)-325(t)28(w)27(o-lev)28(e)-1(l)]TJ 0 -13.5492 Td[(coun)28(terp)1(arts.)-750(F)84(or)-435(a)-435(detailed)-435(desc)-1(ri)1(tion)-435(of)-434(them)-1(,)-460(th)1(e)-436(reader)-435(is)-435(referred)-435(to)]TJ 0 -13.5492 Td[([)]TJ +/F15 10.9091 Tf 107.8572 370.2545 Td[(,)-320(Chap)1(te)-1(r)-317(3])1(\051.)-439(Add)1(itiv)28(e)-317(and)-316(h)28(ybr)1(id)-317(m)28(ult)1(ilev)28(e)-1(l)-317(p)1(re)-1(cond)1(ition)1(e)-1(r)1(s)-318(ar)1(e)-318(obt)1(ained)-317(as)-317(dir)1(e)-1(ct)]TJ -13.9394 -13.5492 Td[(exte)-1(n)1(s)-1(i)1(ons)-359(of)-359(th)1(e)-360(t)28(w)28(o-)-1(lev)28(el)-359(c)-1(ou)1(n)28(terpar)1(ts)-1(.)-521(F)84(or)-359(a)-359(detailed)-359(d)1(e)-1(sc)-1(r)1(ition)-358(of)-359(th)1(e)-1(m,)-365(the)-359(reader)]TJ 0 -13.5492 Td[(is)-321(referred)-321(to)-321([)]TJ ET -1 0 0 1 96.9481 121.5209 cm +1 0 0 1 161.2428 343.1561 cm 0 1 0 rg 0 1 0 RG -1 0 0 1 -96.9481 -121.5209 cm +1 0 0 1 -161.2428 -343.1561 cm +BT +/F15 10.9091 Tf 161.2428 343.1561 Td[(15)]TJ +ET +1 0 0 1 172.1519 343.1561 cm +0 g 0 G +1 0 0 1 -172.1519 -343.1561 cm +BT +/F15 10.9091 Tf 172.1519 343.1561 Td[(,)-323(Chapter)-321(3])1(.)-441(Th)1(e)-322(algor)1(ith)1(m)-322(f)1(or)-321(the)-321(app)1(lication)-320(of)-321(a)-321(m)27(u)1(lti-lev)28(e)-1(l)-320(h)28(ybr)1(id)]TJ -78.2341 -13.5492 Td[(p)-27(os)-1(t-sm)-1(o)-28(ot)1(hed)-335(pr)1(e)-1(cond)1(ition)1(e)-1(r)]TJ/F24 10.9091 Tf 145.6759 0 Td[(M)]TJ/F15 10.9091 Tf 15.4287 0 Td[(to)-335(a)-335(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 55.2408 0 Td[(v)]TJ/F15 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Td[(to)-333(the)-333(next)-333(c)-1(oarser)-333(le)-1(v)28(el)]TJ/F24 9.9626 Tf -65.5183 -11.9552 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(R)]TJ/F10 6.9738 Tf 7.5645 -1.4944 Td[(l)]TJ/F24 9.9626 Tf 3.058 1.4944 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F13 6.9738 Tf 2.5599 0 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F41 9.9626 Tf -55.3714 -13.2954 Td[(endfor)]TJ/F15 9.9626 Tf 0 -17.6244 Td[(!)-444(app)1(ly)-333(the)-333(c)-1(oarses)-1(t-lev)28(e)-1(l)-333(correc)-1(ti)1(on)]TJ/F24 9.9626 Tf 0 -15.1773 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(nl)-6(ev)]TJ/F15 9.9626 Tf 18.7654 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(A)]TJ/F13 6.9738 Tf 7.472 4.2618 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F10 6.9738 Tf -6.2267 -7.2671 Td[(nl)-6(ev)]TJ/F24 9.9626 Tf 15.9981 3.0053 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(nl)-6(ev)]TJ/F41 9.9626 Tf -62.4652 -16.13 Td[(for)]TJ/F24 9.9626 Tf 17.7666 0 Td[(l)]TJ/F15 9.9626 Tf 5.936 0 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(nl)-20(ev)]TJ/F27 9.9626 Tf 21.1879 0 Td[(\000)]TJ/F15 9.9626 Tf 9.9626 0 Td[(1)]TJ/F24 9.9626 Tf 4.9813 0 Td[(;)]TJ/F15 9.9626 Tf 4.4278 0 Td[(1)]TJ/F24 9.9626 Tf 4.9814 0 Td[(;)]TJ/F27 9.9626 Tf 4.4278 0 Td[(\000)]TJ/F15 9.9626 Tf 7.7487 0 Td[(1)]TJ/F41 9.9626 Tf 8.8003 0 Td[(do)]TJ/F15 9.9626 Tf -90.7739 -14.7898 Td[(!)-444(tran)1(s)-1(f)1(e)-1(r)]TJ/F24 9.9626 Tf 44.1124 0 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F7 6.9738 Tf 2.5599 0 Td[(+1)]TJ/F15 9.9626 Tf 13.9062 1.4944 Td[(to)-333(the)-333(next)-333(\014ner)-333(lev)28(e)-1(l)]TJ/F24 9.9626 Tf -65.463 -11.9674 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(R)]TJ/F10 6.9738 Tf 7.6415 3.6153 Td[(T)]TJ -0.077 -6.4368 Td[(l)]TJ/F7 6.9738 Tf 2.5599 0 Td[(+1)]TJ/F24 9.9626 Tf 10.5854 2.8215 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F7 6.9738 Tf 2.5598 0 Td[(+1)]TJ/F15 9.9626 Tf 10.5854 1.4944 Td[(;)]TJ -59.9654 -14.855 Td[(!)-444(c)-1(ompu)1(te)-334(th)1(e)-334(res)-1(i)1(du)1(al)-334(at)-333(th)1(e)-334(c)-1(u)1(rr)1(e)-1(n)28(t)-333(lev)28(e)-1(l)]TJ/F24 9.9626 Tf 0 -12.3426 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F27 9.9626 Tf 5.2719 1.4944 Td[(\000)]TJ/F24 9.9626 Tf 9.9626 0 Td[(A)]TJ/F13 6.9738 Tf 7.472 4.2618 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F10 6.9738 Tf -6.2267 -7.2671 Td[(l)]TJ/F24 9.9626 Tf 10.696 3.0053 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 3.058 1.4944 Td[(;)]TJ -67.0102 -14.7898 Td[(!)-444(app)1(ly)-333(the)-333(basic)-334(Sc)28(h)28(w)28(arz)-334(p)1(re)-1(cond)1(ition)1(e)-1(r)-333(to)-333(th)1(e)-334(res)-1(i)1(dual)]TJ/F24 9.9626 Tf 0 -12.3426 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(M)]TJ/F13 6.9738 Tf 10.7513 4.2618 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F10 6.9738 Tf -7.3128 -7.2671 Td[(l)]TJ/F24 9.9626 Tf 11.7822 3.0053 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F15 9.9626 Tf -46.7782 -13.2954 Td[(!)-444(up)-27(date)]TJ/F24 9.9626 Tf 40.6809 0 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F24 9.9626 Tf -45.5654 -10.4608 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.2719 1.4944 Td[(+)]TJ/F24 9.9626 Tf 9.9626 0 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F41 9.9626 Tf -55.8022 -10.4608 Td[(endfor)]TJ/F24 9.9626 Tf 0 -14.7898 Td[(w)]TJ/F15 9.9626 Tf 10.1678 0 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(y)]TJ/F7 6.9738 Tf 4.8845 -1.4944 Td[(1)]TJ/F15 9.9626 Tf 4.4694 1.4944 Td[(;)]TJ ET -1 0 0 1 454.8257 443.4809 cm +1 0 0 1 481.7238 443.4809 cm q []0 d 0 J @@ -2437,117 +2668,180 @@ q 268.3164 0.1992 l S Q -1 0 0 1 -186.9078 -443.0824 cm +1 0 0 1 -51.4936 -18.5347 cm +0 g 0 G +1 0 0 1 -162.3123 -424.5477 cm BT -/F15 10.9091 Tf 141.7385 424.5478 Td[(Figu)1(re)-483(1:)-743(Ap)1(pl)1(ic)-1(ati)1(on)-482(of)-483(th)1(e)-483(m)27(u)1(lti-lev)28(e)-1(l)-482(h)28(yb)1(rid)-482(p)-27(os)-1(t)1(-)-1(sm)-1(o)-27(othed)-482(precondi)1(-)]TJ 0 -13.5492 Td[(tion)1(e)-1(r)1(.)]TJ +/F15 10.9091 Tf 162.3123 424.5477 Td[(Figu)1(re)-334(1:)-444(Ap)1(pli)1(c)-1(ation)-332(of)-333(the)-334(m)28(ul)1(ti-le)-1(v)28(el)-333(h)28(yb)1(rid)-333(p)-27(ost-)-1(smo)-28(othed)-333(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(.)]TJ ET -1 0 0 1 141.7385 399.0434 cm +1 0 0 1 533.6158 424.5477 cm 0 g 0 G -1 0 0 1 -141.7385 -399.0434 cm +1 0 0 1 -391.8773 -14.0763 cm +0 g 0 G +1 0 0 1 0 -32.1522 cm +0 g 0 G +1 0 0 1 -141.7385 -378.3192 cm BT -/F15 10.9091 Tf 141.7385 368.8931 Td[(p)-27(os)-1(t-sm)-1(o)-28(ot)1(hed)-371(p)1(rec)-1(on)1(diti)1(oner)]TJ/F24 10.9091 Tf 146.4537 0 Td[(M)]TJ/F15 10.9091 Tf 15.8177 0 Td[(to)-371(a)-370(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 56.4076 0 Td[(v)]TJ/F15 10.9091 Tf 5.6793 0 Td[(,)-380(i.)1(e)-1(.)-370(for)-371(t)1(he)-371(c)-1(ompu)1(tation)-370(of)]TJ/F24 10.9091 Tf -224.3583 -13.5492 Td[(w)]TJ/F15 10.9091 Tf 11.1728 0 Td[(=)]TJ/F24 10.9091 Tf 11.554 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -3.9588 Td[(v)]TJ/F15 10.9091 Tf 9.339 0 Td[(is)-336(r)1(e)-1(p)-27(orted,)-335(for)-335(e)-1(x)1(am)-1(p)1(le)-1(,)-335(in)-335(Figu)1(re)]TJ +/F15 10.9091 Tf 155.0719 378.3192 Td[(2.)]TJ ET -1 0 0 1 364.6541 355.3439 cm -0 0 1 rg 0 0 1 RG -1 0 0 1 -364.6541 -355.3439 cm +1 0 0 1 163.5567 378.3192 cm +0 g 0 G +1 0 0 1 -163.5567 -378.3192 cm BT -/F15 10.9091 Tf 364.6541 355.3439 Td[(1)]TJ +/F15 10.9091 Tf 169.0113 378.3192 Td[(construction)-333(of)-332(the)-334(p)1(rolon)1(gator)]TJ/F24 10.9091 Tf 152.5759 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(T)]TJ -0.0842 -7.1916 Td[(C)]TJ/F15 10.9091 Tf 7.0968 3.2328 Td[(;)]TJ ET -1 0 0 1 370.1086 355.3439 cm +1 0 0 1 141.7385 353.2676 cm 0 g 0 G -1 0 0 1 -370.1086 -355.3439 cm +1 0 0 1 -141.7385 -353.2676 cm BT -/F15 10.9091 Tf 370.1086 355.3439 Td[(.)-451(Here)-336(th)1(e)-336(n)28(u)1(m)27(b)-27(e)-1(r)-335(of)-335(lev)28(e)-1(l)1(s)]TJ -228.3701 -13.5492 Td[(is)-330(denoted)-330(b)28(y)]TJ/F24 10.9091 Tf 67.2272 0 Td[(nl)-20(ev)]TJ/F15 10.9091 Tf 24.3775 0 Td[(and)-329(the)-330(lev)28(e)-1(ls)-330(are)-330(n)28(um)28(b)-28(ered)-330(in)-329(increas)-1(i)1(ng)-330(ord)1(e)-1(r)-329(s)-1(tar)1(ting)]TJ -91.6047 -13.5492 Td[(fr)1(om)-335(th)1(e)-335(\014)1(nes)-1(t)-334(on)1(e)-1(,)-334(i.)1(e)-1(.)-334(th)1(e)-335(\014)1(nes)-1(t)-334(lev)28(e)-1(l)-334(i)1(s)-335(lev)28(e)-1(l)-333(1;)-335(th)1(e)-335(coarse)-335(matrix)-334(an)1(d)-334(th)1(e)]TJ 0 -13.5492 Td[(corres)-1(p)-27(ond)1(in)1(g)-271(b)1(as)-1(i)1(c)-271(p)1(re)-1(cond)1(ition)1(e)-1(r)-269(at)-270(e)-1(ac)27(h)-269(lev)28(e)-1(l)]TJ/F24 10.9091 Tf 228.2262 0 Td[(l)]TJ/F15 10.9091 Tf 6.4165 0 Td[(are)-270(denoted)-270(b)28(y)]TJ/F24 10.9091 Tf 72.507 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.7772 Td[(l)]TJ/F15 10.9091 Tf 6.067 1.7772 Td[(and)]TJ/F24 10.9091 Tf 20.5226 0 Td[(M)]TJ/F25 7.9701 Tf 10.5833 -1.7772 Td[(l)]TJ/F15 10.9091 Tf 3.1203 1.7772 Td[(,)]TJ -355.6247 -13.5492 Td[(res)-1(p)-27(ec)-1(ti)1(v)28(e)-1(ly)84(,)-333(with)]TJ/F24 10.9091 Tf 86.4546 0 Td[(A)]TJ/F22 7.9701 Tf 8.1818 -1.6364 Td[(1)]TJ/F15 10.9091 Tf 7.7626 1.6364 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(.)]TJ/F19 11.9552 Tf -122.096 -29.0398 Td[(5)-1(.2)-1125(Sm)-1(o)-32(ot)-1(hed)-375(Agg)-1(reg)-1(ati)-1(on)]TJ/F15 10.9091 Tf 0 -20.5946 Td[(In)-295(ord)1(e)-1(r)-295(to)-296(d)1(e)-1(\014)1(ne)-296(th)1(e)-296(res)-1(tr)1(ic)-1(ti)1(on)-295(op)-28(erator)]TJ/F24 10.9091 Tf 199.1784 0 Td[(R)]TJ/F25 7.9701 Tf 8.2832 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(,)-303(whi)1(c)27(h)-295(is)-296(use)-1(d)-295(to)-295(c)-1(omp)1(ute)-296(th)1(e)]TJ -214.5584 -13.5492 Td[(coars)-1(e-le)-1(v)28(el)-372(matri)1(x)]TJ/F24 10.9091 Tf 93.9916 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(,)-381(MLD2P)1(4)-372(use)-1(s)-372(th)1(e)]TJ/F20 10.9091 Tf 97.5121 0 Td[(smo)51(othe)51(d)-393(aggr)51(e)51(gation)]TJ/F15 10.9091 Tf 105.7818 0 Td[(algor)1(ithm)]TJ -312.564 -13.5492 Td[(desc)-1(ri)1(b)-28(ed)-314(in)-313([)]TJ +/F15 10.9091 Tf 155.0719 353.2676 Td[(3.)]TJ ET -1 0 0 1 205.3486 224.4143 cm +1 0 0 1 163.5567 353.2676 cm +0 g 0 G +1 0 0 1 -163.5567 -353.2676 cm +BT +/F15 10.9091 Tf 169.0113 353.2676 Td[(app)1(lication)-333(of)]TJ/F24 10.9091 Tf 68.7879 0 Td[(R)]TJ/F25 7.9701 Tf 8.2832 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7331 1.6887 Td[(and)]TJ/F24 10.9091 Tf 21.2122 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(T)]TJ -0.0843 -7.1916 Td[(C)]TJ/F15 10.9091 Tf 10.7332 3.2328 Td[(to)-333(bu)1(ild)]TJ/F24 10.9091 Tf 41.2122 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(.)]TJ -194.8598 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228.1688 224.4143 cm +1 0 0 1 301.1939 314.6668 cm 0 g 0 G -1 0 0 1 -228.1688 -224.4143 cm +1 0 0 1 -301.1939 -314.6668 cm BT -/F15 10.9091 Tf 228.1688 224.4143 Td[(].)-438(Th)1(e)-315(b)1(as)-1(ic)-314(idea)-314(of)-314(th)1(is)-314(algorit)1(hm)-314(is)-315(to)-314(b)1(uil)1(d)-314(a)-314(c)-1(oar)1(s)-1(e)-314(s)-1(et)]TJ -86.4303 -13.5492 Td[(of)-350(v)28(e)-1(r)1(tice)-1(s)]TJ/F24 10.9091 Tf 52.2849 0 Td[(W)]TJ/F25 7.9701 Tf 10.3031 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.9211 1.6887 Td[(b)28(y)-350(s)-1(u)1(itab)1(ly)-350(group)1(ing)-350(the)-351(v)28(erti)1(c)-1(es)-351(of)]TJ/F24 10.9091 Tf 174.0367 0 Td[(W)]TJ/F15 10.9091 Tf 15.6424 0 Td[(in)28(t)1(o)-351(di)1(s)-1(j)1(oin)28(t)-350(s)-1(u)1(bse)-1(ts)]TJ -263.1882 -13.5492 Td[(\050aggregates)-1(\051,)-318(an)1(d)-315(t)1(o)-315(de\014n)1(e)-315(the)-315(coarse)-1(-to-\014ne)-315(space)-315(tran)1(s)-1(f)1(e)-1(r)-314(op)-27(e)-1(r)1(ator)]TJ/F24 10.9091 Tf 328.3274 0 Td[(R)]TJ/F25 7.9701 Tf 8.3675 3.9588 Td[(T)]TJ -0.0843 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Td[(to)-333(a)-334(certain)-333(r)1(o)-28(ot)-333(v)28(e)-1(r)1(te)-1(x)]TJ/F24 10.9091 Tf 115.2425 0 Td[(r)]TJ/F27 10.9091 Tf 8.255 0 Td[(2)]TJ/F24 10.9091 Tf 10.303 0 Td[(W)]TJ/F15 10.9091 Tf 11.8182 0 Td[(,)-333(i.e.)]TJ/F24 10.9091 Tf -216.2056 -26.6219 Td[(N)]TJ/F25 7.9701 Tf 8.7652 -1.6364 Td[(r)]TJ/F15 10.9091 Tf 7.5849 1.6364 Td[(=)]TJ/F30 10.9091 Tf 11.5151 12.1092 Td[(n)]TJ/F24 10.9091 Tf 7.2728 -12.1092 Td[(s)]TJ/F27 10.9091 Tf 8.1439 0 Td[(2)]TJ/F24 10.9091 Tf 10.3029 0 Td[(W)]TJ/F15 10.9091 Tf 14.8485 0 Td[(:)]TJ/F27 10.9091 Tf 6.0606 0 Td[(j)]TJ/F24 10.9091 Tf 3.0303 0 Td[(a)]TJ/F25 7.9701 Tf 5.7664 -1.6364 Td[(r)-30(s)]TJ/F27 10.9091 Tf 8.4707 1.6364 Td[(j)]TJ/F24 10.9091 Tf 6.0605 0 Td[(>)-278(\022)]TJ/F30 10.9091 Tf 16.9394 9.8599 Td[(p)]TJ -ET -1 0 0 1 298.625 601.6956 cm -q -[]0 d -0 J -0.4364 w -0 0.2182 m -34.5348 0.2182 l -S -Q -1 0 0 1 -298.625 -601.6956 cm -BT -/F27 10.9091 Tf 298.625 591.8357 Td[(j)]TJ/F24 10.9091 Tf 3.0303 0 Td[(a)]TJ/F25 7.9701 Tf 5.7664 -1.6364 Td[(r)-30(r)]TJ/F24 10.9091 Tf 8.6114 1.6364 Td[(a)]TJ/F25 7.9701 Tf 5.7664 -1.6364 Td[(ss)]TJ/F27 10.9091 Tf 8.33 1.6364 Td[(j)]TJ/F30 10.9091 Tf 3.0303 12.1092 Td[(o)]TJ/F27 10.9091 Tf 9.6969 -12.1092 Td[([)-222(f)]TJ/F24 10.9091 Tf 15.1515 0 Td[(r)]TJ/F27 10.9091 Tf 5.2248 0 Td[(g)]TJ/F24 10.9091 Tf 7.2727 0 Td[(;)]TJ/F15 10.9091 Tf -276.5879 -27.228 Td[(for)-415(a)-415(giv)28(en)]TJ/F24 10.9091 Tf 56.9572 0 Td[(\022)]TJ/F27 10.9091 Tf 9.9459 0 Td[(2)]TJ/F15 10.9091 Tf 11.7943 0 Td[([0)]TJ/F24 10.9091 Tf 8.4849 0 Td[(;)]TJ/F15 10.9091 Tf 4.8484 0 Td[(1].)-690(Si)1(nce)-416(thi)1(s)-416(algori)1(thm)-416(h)1(as)-416(a)-415(s)-1(equen)28(ti)1(al)-416(n)1(atur)1(e)-1(,)-435(a)]TJ/F20 10.9091 Tf 252.1275 0 Td[(de-)]TJ -344.1582 -13.5492 Td[(c)51(oupl)1(e)51(d)]TJ/F15 10.9091 Tf 38.8545 0 Td[(v)28(ers)-1(i)1(on)-315(of)-314(it)-315(has)-315(b)-28(een)-315(c)27(h)1(os)-1(en,)-318(where)-315(e)-1(ac)27(h)-314(pr)1(o)-28(c)-1(es)-1(sor)]TJ/F24 10.9091 Tf 245.6359 0 Td[(i)]TJ/F15 10.9091 Tf 7.1949 0 Td[(in)1(dep)-28(end)1(e)-1(n)29(tly)]TJ -291.6853 -13.5492 Td[(app)1(lies)-450(the)-450(algori)1(thm)-450(to)-450(th)1(e)-451(se)-1(t)-449(of)-450(v)28(ertices)]TJ/F24 10.9091 Tf 216.1723 0 Td[(W)]TJ/F22 7.9701 Tf 11.8182 3.9588 Td[(0)]TJ/F25 7.9701 Tf -1.5151 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 11.1553 3.0555 Td[(ass)-1(ign)1(e)-1(d)-449(to)-450(it)-449(in)-450(th)1(e)-450(ini-)]TJ -237.6307 -13.5492 Td[(tial)-352(dat)1(a)-353(d)1(is)-1(tr)1(ibu)1(tion)1(.)-502(This)-353(v)28(ersion)-352(is)-353(em)27(b)1(arrass)-1(i)1(ngly)-352(par)1(allel,)-357(s)-1(i)1(nce)-353(it)-353(d)1(o)-28(es)]TJ 0 -13.5492 Td[(not)-392(r)1(e)-1(qu)1(ire)-393(an)29(y)-392(data)-392(c)-1(omm)27(u)1(ni)1(c)-1(ation)1(.)-621(On)-392(the)-392(other)-392(hand)1(,)-407(it)-392(ma)27(y)-392(p)1(ro)-28(d)1(uce)]TJ 0 -13.5492 Td[(non)1(-uni)1(form)-398(aggregate)-1(s)-398(near)-398(b)-27(oun)1(dary)-398(v)28(erti)1(c)-1(es)-1(,)-414(i.)1(e)-1(.)-398(n)1(e)-1(ar)-398(v)28(ertices)-399(adj)1(ac)-1(en)28(t)]TJ 0 -13.5492 Td[(to)-333(v)28(erti)1(c)-1(es)-334(i)1(n)-333(oth)1(e)-1(r)-332(pr)1(o)-28(ce)-1(ss)-1(or)1(s)-1(,)-333(an)1(d)-332(is)-333(s)-1(tr)1(ongly)-332(dep)-28(end)1(e)-1(n)29(t)-333(on)-332(the)-333(n)28(u)1(m)27(b)-27(e)-1(r)-332(of)]TJ 0 -13.5492 Td[(pr)1(o)-28(ce)-1(ss)-1(or)1(s)-276(and)-275(on)-275(th)1(e)-276(in)1(itial)-275(p)1(artition)1(in)1(g)-276(of)-275(th)1(e)-276(matrix)]TJ/F24 10.9091 Tf 259.7654 0 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(.)-425(Nev)28(e)-1(r)1(thele)-1(ss,)-287(thi)1(s)]TJ -267.9472 -13.5492 Td[(algor)1(ithm)-326(h)1(as)-327(b)-27(ee)-1(n)-325(c)27(h)1(os)-1(en)-326(f)1(or)-326(t)1(he)-326(impleme)-1(n)28(tat)1(ion)-326(i)1(n)-326(M)1(LD2P4,)-327(since)-326(it)-326(h)1(as)]TJ 0 -13.5492 Td[(b)-27(e)-1(en)-333(s)-1(h)1(o)28(w)-1(n)-332(to)-334(p)1(ro)-28(d)1(uce)-334(go)-28(o)-28(d)-332(re)-1(sul)1(ts)-334(in)-333(p)1(ractice)-334([)]TJ -ET -1 0 0 1 326.0089 442.6649 cm -0 1 0 rg 0 1 0 RG -1 0 0 1 -326.0089 -442.6649 cm -BT -/F15 10.9091 Tf 326.0089 442.6649 Td[(3)]TJ -ET -1 0 0 1 331.4635 442.6649 cm -0 g 0 G -1 0 0 1 -331.4635 -442.6649 cm -BT -/F15 10.9091 Tf 331.4635 442.6649 Td[(,)]TJ -ET -1 0 0 1 338.1301 442.6649 cm -0 1 0 rg 0 1 0 RG -1 0 0 1 -338.1301 -442.6649 cm -BT -/F15 10.9091 Tf 338.1301 442.6649 Td[(4)]TJ -ET -1 0 0 1 343.5847 442.6649 cm -0 g 0 G -1 0 0 1 -343.5847 -442.6649 cm -BT -/F15 10.9091 Tf 343.5847 442.6649 Td[(,)]TJ +/F42 10.9091 Tf 104.0692 740.0018 Td[(Mul)66(ti-level)-378(Domain)-378(Decomp)1(osition)-378(Ba)22(ck)23(gr)22(oun)-1(d)]TJ/F15 10.9091 Tf 391.3906 0 Td[(15)]TJ ET -1 0 0 1 350.2514 442.6649 cm -0 1 0 rg 0 1 0 RG -1 0 0 1 -350.2514 -442.6649 cm -BT -/F15 10.9091 Tf 350.2514 442.6649 Td[(18)]TJ -ET -1 0 0 1 361.1605 442.6649 cm +1 0 0 1 506.3689 740.0018 cm 0 g 0 G -1 0 0 1 -361.1605 -442.6649 cm +1 0 0 1 -506.3689 -740.0018 cm BT -/F15 10.9091 Tf 361.1605 442.6649 Td[(].)]TJ -250.3062 -13.5492 Td[(The)-343(p)1(rolon)1(gator)]TJ/F24 10.9091 Tf 81.47 0 Td[(P)]TJ/F25 7.9701 Tf 7.0038 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.2914 1.6887 Td[(=)]TJ/F24 10.9091 Tf 11.6795 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(T)]TJ -0.0843 -7.1916 Td[(C)]TJ/F15 10.9091 Tf 10.8318 3.2328 Td[(is)-343(b)1(ui)1(lt)-343(start)1(ing)-342(fr)1(om)-343(a)]TJ/F20 10.9091 Tf 113.0992 0 Td[(tent)-1(ative)-366(pr)52(o)-1(l)1(ongat)-1(or)]TJ/F24 10.9091 Tf -259.5953 -13.5492 Td[(P)]TJ/F27 10.9091 Tf 11.5492 0 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F28 7.9701 Tf 5.1382 0 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F26 5.9776 Tf 5.1382 -1.3394 Td[(C)]TJ/F15 10.9091 Tf 6.6924 -2.6194 Td[(,)-333(de\014n)1(e)-1(d)-333(as)]TJ/F24 10.9091 Tf 34.6342 -30.8987 Td[(P)]TJ/F15 10.9091 Tf 11.5492 0 Td[(=)-278(\050)]TJ/F24 10.9091 Tf 15.7575 0 Td[(p)]TJ/F25 7.9701 Tf 5.4886 -1.6364 Td[(ij)]TJ/F15 10.9091 Tf 7.2653 1.6364 Td[(\051)]TJ/F24 10.9091 Tf 4.2425 0 Td[(;)-1166(p)]TJ/F25 7.9701 Tf 21.2462 -1.6364 Td[(ij)]TJ/F15 10.9091 Tf 10.2955 1.6364 Td[(=)]TJ/F30 10.9091 Tf 11.5151 15.3819 Td[(\032)]TJ/F15 10.9091 Tf 13.1632 -9.1358 Td[(1)-1913(if)]TJ/F24 10.9091 Tf 35.7202 0 Td[(i)]TJ/F27 10.9091 Tf 6.7886 0 Td[(2)]TJ/F24 10.9091 Tf 10.303 0 Td[(V)]TJ/F25 7.9701 Tf 8.7878 5.3073 Td[(j)]TJ -2.4242 -8.7414 Td[(C)]TJ/F15 10.9091 Tf -59.1754 -10.1151 Td[(0)-1913(otherwise)]TJ/F24 10.9091 Tf 79.2607 7.3031 Td[(:)]TJ +/F15 10.9091 Tf 110.8543 704.1363 Td[(The)-277(pr)1(olongat)1(or)]TJ/F24 10.9091 Tf 80.0384 0 Td[(P)]TJ/F25 7.9701 Tf 7.0038 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.127 1.6887 Td[(=)]TJ/F24 10.9091 Tf 11.5152 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(T)]TJ -0.0843 -7.1916 Td[(C)]TJ/F15 10.9091 Tf 10.116 3.2328 Td[(is)-277(bu)1(ilt)-276(s)-1(tar)1(ting)-276(fr)1(om)-278(a)]TJ/F20 10.9091 Tf 109.5201 0 Td[(tent)-1(ative)-305(pr)51(olongator)]TJ/F24 10.9091 Tf 102.5945 0 Td[(P)]TJ/F27 10.9091 Tf 11.5492 0 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F28 7.9701 Tf 5.1382 0 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F26 5.9776 Tf 5.1382 -1.3394 Td[(C)]TJ/F15 10.9091 Tf 6.6924 -2.6194 Td[(,)]TJ -409.4208 -13.5492 Td[(de\014n)1(e)-1(d)-332(as)]TJ/F24 10.9091 Tf 114.8185 -17.8186 Td[(P)]TJ/F15 10.9091 Tf 11.5491 0 Td[(=)-278(\050)]TJ/F24 10.9091 Tf 15.7576 0 Td[(p)]TJ/F25 7.9701 Tf 5.4886 -1.6364 Td[(ij)]TJ/F15 10.9091 Tf 7.2653 1.6364 Td[(\051)]TJ/F24 10.9091 Tf 4.2425 0 Td[(;)-1166(p)]TJ/F25 7.9701 Tf 21.2462 -1.6364 Td[(ij)]TJ/F15 10.9091 Tf 10.2955 1.6364 Td[(=)]TJ/F30 10.9091 Tf 11.5151 15.3819 Td[(\032)]TJ/F15 10.9091 Tf 13.1632 -9.1358 Td[(1)-1913(if)]TJ/F24 10.9091 Tf 35.7202 0 Td[(i)]TJ/F27 10.9091 Tf 6.7886 0 Td[(2)]TJ/F24 10.9091 Tf 10.3029 0 Td[(V)]TJ/F25 7.9701 Tf 8.7879 5.3073 Td[(j)]TJ -2.4242 -8.7414 Td[(C)]TJ/F15 10.9091 Tf -59.1754 -10.1151 Td[(0)-1913(otherwise)]TJ/F24 10.9091 Tf 79.2607 7.3031 Td[(:)]TJ ET -1 0 0 1 438.6334 384.6678 cm +1 0 0 1 492.4295 672.7685 cm 0 g 0 G -1 0 0 1 -438.6334 -384.6678 cm +1 0 0 1 -492.4295 -672.7685 cm BT -/F15 10.9091 Tf 438.6334 384.6678 Td[(\0502\051)]TJ +/F15 10.9091 Tf 492.4295 672.7685 Td[(\0502\051)]TJ ET -1 0 0 1 452.5729 384.6678 cm +1 0 0 1 506.3689 672.7685 cm 0 g 0 G -1 0 0 1 -452.5729 -384.6678 cm +1 0 0 1 -506.3689 -672.7685 cm BT -/F24 10.9091 Tf 93.9178 352.5706 Td[(P)]TJ/F25 7.9701 Tf 7.0038 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7331 1.6887 Td[(is)-334(ob)1(tain)1(e)-1(d)-333(b)29(y)-334(ap)1(pl)1(ying)-333(to)]TJ/F24 10.9091 Tf 128.8487 0 Td[(P)]TJ/F15 10.9091 Tf 12.1553 0 Td[(a)-333(s)-1(mo)-28(oth)1(e)-1(r)]TJ/F24 10.9091 Tf 56.7576 0 Td[(S)]TJ/F27 10.9091 Tf 10.3484 0 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F28 7.9701 Tf 5.1382 0 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 5.6363 -3.9588 Td[(:)]TJ/F24 10.9091 Tf -106.1219 -24.5081 Td[(P)]TJ/F25 7.9701 Tf 7.0038 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.127 1.6887 Td[(=)]TJ/F24 10.9091 Tf 11.5152 0 Td[(S)-58(P)-28(;)]TJ +/F24 10.9091 Tf 93.9178 645.1545 Td[(P)]TJ/F25 7.9701 Tf 7.0038 -1.6886 Td[(C)]TJ/F15 10.9091 Tf 10.7331 1.6886 Td[(is)-334(ob)1(tain)1(e)-1(d)-333(b)29(y)-334(ap)1(pl)1(ying)-333(to)]TJ/F24 10.9091 Tf 128.8487 0 Td[(P)]TJ/F15 10.9091 Tf 12.1553 0 Td[(a)-333(s)-1(mo)-28(oth)1(e)-1(r)]TJ/F24 10.9091 Tf 56.7576 0 Td[(S)]TJ/F27 10.9091 Tf 10.3484 0 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F28 7.9701 Tf 5.1382 0 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 5.6363 -3.9588 Td[(:)]TJ/F24 10.9091 Tf -79.2239 -24.5081 Td[(P)]TJ/F25 7.9701 Tf 7.0038 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.127 1.6887 Td[(=)]TJ/F24 10.9091 Tf 11.5152 0 Td[(S)-58(P)-28(;)]TJ ET -1 0 0 1 438.6334 328.0625 cm +1 0 0 1 492.4295 620.6464 cm 0 g 0 G -1 0 0 1 -438.6334 -328.0625 cm +1 0 0 1 -492.4295 -620.6464 cm BT -/F15 10.9091 Tf 438.6334 328.0625 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Td[(=)-287(real)-287(dou)1(ble)-287(precis)-1(i)1(on;)]TJ/F48 10.9091 Tf 115.2483 0 Td[(c)]TJ/F15 10.9091 Tf 8.8611 0 Td[(=)-287(com)-1(p)1(le)-1(x)-287(sin)1(gle)-288(p)1(re)-1(cision;)]TJ/F48 10.9091 Tf -256.5343 -13.5492 Td[(z)]TJ/F15 10.9091 Tf 9.6138 0 Td[(=)-356(com)-1(p)1(le)-1(x)-356(d)1(oub)1(le)-357(p)1(re)-1(cision\051)1(.)-513(T)-1(h)1(is)-357(d)1(ata)-356(s)-1(tr)1(ucture)-356(is)-357(acc)-1(es)-1(sed)-356(b)28(y)-356(the)-356(use)-1(r)-356(on)1(ly)]TJ -9.6138 -13.5492 Td[(th)1(rough)-333(t)1(he)-334(M)1(LD2P4)-333(rou)1(tines)-1(,)-333(f)1(ollo)28(wing)-333(an)-333(ob)-55(j)1(e)-1(ct-)-1(or)1(ien)28(te)-1(d)-332(appr)1(oac)27(h)1(.)]TJ ET -1 0 0 1 141.7385 537.3426 cm +1 0 0 1 141.7385 555.4954 cm 0 g 0 G -1 0 0 1 -141.7385 -537.3426 cm +1 0 0 1 -141.7385 -555.4954 cm BT -/F15 10.9091 Tf 155.0719 537.3426 Td[(2.)]TJ +/F15 10.9091 Tf 155.0719 555.4954 Td[(2.)]TJ ET -1 0 0 1 163.5567 537.3426 cm +1 0 0 1 163.5567 555.4954 cm 0 g 0 G -1 0 0 1 -163.5567 -537.3426 cm +1 0 0 1 -163.5567 -555.4954 cm BT -/F20 10.9091 Tf 169.0113 537.3426 Td[(A)25(l)-50(lo)51(c)51(ate)-395(an)-1(d)-395(initi)1(a)-1(l)1(ize)-395(the)-395(pr)51(e)51(c)51(onditioner)-395(data)-396(str)1(uctur)52(e,)-405(ac)51(c)51(or)52(ding)-395(t)-1(o)]TJ 0 -13.5492 Td[(a)-449(pr)52(e)51(c)51(ond)-1(i)1(tioner)-449(ty)1(p)51(e)-449(chosen)-449(by)-448(the)-449(u)1(ser)]TJ/F15 10.9091 Tf 197.606 0 Td[(.)-740(This)-432(is)-432(p)-28(erf)1(orme)-1(d)-431(b)28(y)-432(th)1(e)]TJ -197.606 -13.5492 Td[(rou)1(tin)1(e)]TJ/F48 10.9091 Tf 39.7241 0 Td[(mld_precinit)]TJ/F15 10.9091 Tf 68.7265 0 Td[(,)-576(whi)1(c)27(h)-527(also)-528(se)-1(ts)-527(defaul)1(ts)-528(for)-527(eac)27(h)-527(pr)1(e)-1(cond)1(i-)]TJ -108.4506 -13.5492 Td[(tion)1(e)-1(r)-514(t)28(yp)-27(e)-515(s)-1(ele)-1(cted)-515(b)29(y)-515(th)1(e)-516(u)1(s)-1(er.)-988(The)-515(defau)1(lts)-515(ass)-1(o)-28(ciated)-514(to)-515(eac)27(h)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)-423(t)28(yp)-27(e)-424(ar)1(e)-424(giv)28(en)-423(in)-423(T)83(ab)1(le)]TJ +/F20 10.9091 Tf 169.0113 555.4954 Td[(A)25(l)-50(lo)51(c)51(ate)-364(an)-1(d)-364(ini)1(t)-1(i)1(aliz)1(e)-365(the)-364(pr)51(e)51(c)52(o)-1(nditi)1(o)-1(ner)-364(data)-364(structur)52(e,)-365(a)-1(c)52(c)51(or)51(ding)-364(to)-364(a)-365(pr)52(e)51(c)51(ond)-1(i)1(-)]TJ 0 -13.5492 Td[(tioner)-376(typ)51(e)-375(cho)-1(s)1(en)-377(by)-375(the)-376(user)]TJ/F15 10.9091 Tf 144.0045 0 Td[(.)-503(This)-353(is)-353(p)-28(erfor)1(m)-1(ed)-353(b)29(y)-353(the)-353(rou)1(tin)1(e)]TJ/F48 10.9091 Tf 169.417 0 Td[(mld_precinit)]TJ/F15 10.9091 Tf 68.7265 0 Td[(,)]TJ -382.148 -13.5492 Td[(whic)28(h)-406(al)1(s)-1(o)-405(s)-1(ets)-406(defaul)1(ts)-406(for)-405(e)-1(ac)28(h)-406(p)1(rec)-1(on)1(diti)1(oner)-406(t)28(yp)-27(e)-406(s)-1(elec)-1(ted)-405(b)28(y)-406(th)1(e)-406(use)-1(r)1(.)-662(The)]TJ 0 -13.5492 Td[(defau)1(lts)-404(ass)-1(o)-28(ciated)-404(to)-403(e)-1(ac)28(h)-404(p)1(re)-1(cond)1(ition)1(e)-1(r)-403(t)28(yp)-27(e)-405(ar)1(e)-404(giv)28(e)-1(n)-403(in)-403(T)83(abl)1(e)]TJ ET -1 0 0 1 361.0556 483.1458 cm +1 0 0 1 493.6411 514.8478 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -361.0556 -483.1458 cm +1 0 0 1 -493.6411 -514.8478 cm BT -/F15 10.9091 Tf 361.0556 483.1458 Td[(1)]TJ +/F15 10.9091 Tf 493.6411 514.8478 Td[(1)]TJ ET -1 0 0 1 366.5101 483.1458 cm +1 0 0 1 499.0956 514.8478 cm 0 g 0 G -1 0 0 1 -366.5101 -483.1458 cm +1 0 0 1 -499.0956 -514.8478 cm BT -/F15 10.9091 Tf 366.5101 483.1458 Td[(,)-446(wh)1(e)-1(re)-423(the)-424(stri)1(ngs)-424(u)1(s)-1(ed)-423(b)28(y)]TJ/F48 10.9091 Tf -197.4988 -13.5492 Td[(mld_precinit)]TJ/F15 10.9091 Tf 71.2379 0 Td[(to)-230(id)1(e)-1(n)28(ti)1(fy)-230(th)1(e)-231(pr)1(e)-1(cond)1(ition)1(e)-1(r)-230(t)28(y)1(p)-28(es)-231(are)-230(also)-231(gi)1(v)28(e)-1(n)1(.)-410(Note)]TJ -71.2379 -13.5492 Td[(th)1(at)-310(t)1(hes)-1(e)-310(stri)1(ngs)-310(ar)1(e)-310(v)56(alid)-309(al)1(s)-1(o)-309(if)-309(u)1(pp)-27(e)-1(r)1(c)-1(ase)-310(letters)-310(ar)1(e)-310(sub)1(s)-1(tit)1(uted)-309(b)28(y)]TJ 0 -13.5492 Td[(corres)-1(p)-27(ond)1(in)1(g)-334(lo)28(w)28(e)-1(r)1(c)-1(ase)-334(ones.)]TJ +/F15 10.9091 Tf 499.0956 514.8478 Td[(,)-421(where)-404(the)]TJ -330.0843 -13.5492 Td[(strin)1(gs)-306(use)-1(d)-304(b)28(y)]TJ/F48 10.9091 Tf 74.4455 0 Td[(mld_precinit)]TJ/F15 10.9091 Tf 72.0568 0 Td[(to)-305(iden)28(ti)1(fy)-305(th)1(e)-306(pr)1(e)-1(cond)1(ition)1(e)-1(r)-305(t)28(yp)-27(es)-306(are)-306(al)1(s)-1(o)-305(giv)28(en.)]TJ -146.5023 -13.5492 Td[(Note)-353(th)1(at)-352(thes)-1(e)-352(s)-1(tr)1(ings)-353(ar)1(e)-353(v)56(alid)-352(al)1(s)-1(o)-352(if)-352(u)1(pp)-27(e)-1(r)1(c)-1(ase)-353(letters)-353(are)-352(s)-1(u)1(bstitu)1(te)-1(d)-352(b)29(y)-353(cor-)]TJ 0 -13.5492 Td[(res)-1(p)-27(on)1(din)1(g)-334(l)1(o)27(w)28(e)-1(r)1(c)-1(ase)-334(on)1(e)-1(s.)]TJ ET -1 0 0 1 141.7385 418.9503 cm +1 0 0 1 141.7385 452.2668 cm 0 g 0 G -1 0 0 1 -141.7385 -418.9503 cm +1 0 0 1 -141.7385 -452.2668 cm BT -/F15 10.9091 Tf 155.0719 418.9503 Td[(3.)]TJ +/F15 10.9091 Tf 155.0719 452.2668 Td[(3.)]TJ ET -1 0 0 1 163.5567 418.9503 cm +1 0 0 1 163.5567 452.2668 cm 0 g 0 G -1 0 0 1 -163.5567 -418.9503 cm +1 0 0 1 -163.5567 -452.2668 cm BT -/F20 10.9091 Tf 169.0113 418.9503 Td[(Mo)51(di)1(fy)-423(the)-423(sele)52(ct)-1(e)51(d)-423(pr)52(e)51(c)51(ond)-1(i)1(tioner)-423(typ)51(e,)-439(by)-423(pr)52(op)51(erly)-422(setting)-423(p)-1(r)52(e)51(c)51(ondi-)]TJ 0 -13.5492 Td[(tioner)-472(p)51(a)-1(r)52(ameters.)]TJ/F15 10.9091 Tf 97.5819 0 Td[(This)-458(is)-458(p)-28(erf)1(orme)-1(d)-457(b)28(y)-458(the)-458(rou)1(tin)1(e)]TJ/F48 10.9091 Tf 167.7707 0 Td[(mld_precset)]TJ/F15 10.9091 Tf 62.9993 0 Td[(.)]TJ -328.3519 -13.5492 Td[(This)-375(rou)1(tine)-375(m)27(u)1(s)-1(t)-375(b)-27(e)-376(called)-375(on)1(ly)-375(if)-374(the)-375(use)-1(r)-375(w)28(an)28(ts)-375(to)-375(m)-1(o)-28(d)1(if)1(y)-375(the)-375(de-)]TJ 0 -13.5492 Td[(fau)1(lt)-253(v)55(al)1(ues)-254(of)-254(t)1(he)-254(par)1(am)-1(eters)-254(ass)-1(o)-28(ciated)-253(to)-254(th)1(e)-254(s)-1(elec)-1(ted)-253(precondi)1(tioner)]TJ 0 -13.5492 Td[(t)28(yp)-27(e)-1(,)-446(to)-425(ob)1(tain)-424(a)-424(v)56(ari)1(an)28(t)-424(of)-424(the)-424(precondi)1(tioner.)-717(E)1(xamples)-425(of)-423(use)-425(of)]TJ/F48 10.9091 Tf 0 -13.5492 Td[(mld_precset)]TJ/F15 10.9091 Tf 65.9152 0 Td[(are)-267(giv)28(e)-1(n)-266(in)-267(Sec)-1(t)1(ion)]TJ +/F20 10.9091 Tf 169.0113 452.2668 Td[(Mo)51(di)1(fy)-340(the)-341(sel)1(e)51(cte)51(d)-341(pr)51(e)52(c)51(on)-1(di)1(t)-1(i)1(on)-1(er)-340(typ)51(e,)-343(by)-340(pr)51(op)51(erl)1(y)-340(setting)-341(pr)52(e)51(c)51(on)-1(di)1(t)-1(i)1(on)-1(er)-340(p)51(ar)52(am-)]TJ 0 -13.5492 Td[(eters.)]TJ/F15 10.9091 Tf 30.6684 0 Td[(This)-266(i)1(s)-266(p)-28(erf)1(orme)-1(d)-265(b)28(y)-265(th)1(e)-266(routi)1(ne)]TJ/F48 10.9091 Tf 155.1694 0 Td[(mld_precset)]TJ/F15 10.9091 Tf 62.9993 0 Td[(.)-422(Th)1(is)-266(rou)1(tine)-266(m)28(ust)-266(b)-27(e)-266(called)]TJ -248.8371 -13.5492 Td[(onl)1(y)-365(i)1(f)-364(the)-365(u)1(s)-1(er)-364(w)27(an)28(ts)-365(t)1(o)-365(mo)-28(di)1(fy)-364(the)-365(d)1(e)-1(f)1(aul)1(t)-365(v)56(alu)1(e)-1(s)-365(of)-364(th)1(e)-365(paramete)-1(r)1(s)-365(as)-1(so)-28(ciated)]TJ 0 -13.5492 Td[(to)-468(t)1(he)-468(se)-1(lec)-1(t)1(e)-1(d)-467(pr)1(e)-1(cond)1(ition)1(e)-1(r)-467(t)28(yp)-27(e)-1(,)-501(to)-467(obtai)1(n)-468(a)-467(v)56(arian)28(t)-467(of)-467(the)-468(p)1(re)-1(cond)1(ition)1(e)-1(r)1(.)]TJ 0 -13.5492 Td[(Ex)1(am)-1(p)1(les)-396(of)-394(u)1(s)-1(e)-395(of)]TJ/F48 10.9091 Tf 95.9773 0 Td[(mld_precset)]TJ/F15 10.9091 Tf 67.3042 0 Td[(are)-395(giv)28(en)-394(in)-394(Sec)-1(t)1(ion)]TJ ET -1 0 0 1 329.6509 351.2043 cm +1 0 0 1 432.5731 398.07 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -329.6509 -351.2043 cm +1 0 0 1 -432.5731 -398.07 cm BT -/F15 10.9091 Tf 329.6509 351.2043 Td[(6.1)]TJ +/F15 10.9091 Tf 432.5731 398.07 Td[(6.1)]TJ ET -1 0 0 1 343.5903 351.2043 cm +1 0 0 1 446.5125 398.07 cm 0 g 0 G -1 0 0 1 -343.5903 -351.2043 cm +1 0 0 1 -446.5125 -398.07 cm BT -/F15 10.9091 Tf 343.5903 351.2043 Td[(;)-289(a)-267(c)-1(ompl)1(e)-1(te)-268(l)1(is)-1(t)-267(of)-267(al)1(l)-267(the)-268(p)1(rec)-1(on)1(-)]TJ -174.579 -13.5492 Td[(di)1(tioner)-390(par)1(am)-1(eters)-391(an)1(d)-390(their)-390(all)1(o)27(w)28(ed)-390(and)-390(d)1(e)-1(f)1(ault)-390(v)56(alu)1(e)-1(s)-391(i)1(s)-391(pr)1(o)27(v)1(ided)]TJ 0 -13.5492 Td[(in)-333(S)1(e)-1(ction)]TJ +/F15 10.9091 Tf 446.5125 398.07 Td[(;)-425(a)-395(complete)-395(list)-395(of)-394(all)]TJ -277.5012 -13.5492 Td[(th)1(e)-304(p)1(rec)-1(on)1(diti)1(oner)-303(p)1(arame)-1(ters)-303(an)1(d)-303(th)1(e)-1(i)1(r)-303(all)1(o)27(w)28(e)-1(d)-302(an)1(d)-303(d)1(e)-1(f)1(ault)-302(v)56(alues)-303(is)-303(pro)28(vi)1(ded)-303(i)1(n)]TJ 0 -13.5492 Td[(Section)]TJ ET -1 0 0 1 219.9204 324.106 cm +1 0 0 1 207.1931 370.9716 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -219.9204 -324.106 cm +1 0 0 1 -207.1931 -370.9716 cm BT -/F15 10.9091 Tf 219.9204 324.106 Td[(7)]TJ +/F15 10.9091 Tf 207.1931 370.9716 Td[(7)]TJ ET -1 0 0 1 225.375 324.106 cm +1 0 0 1 212.6477 370.9716 cm 0 g 0 G -1 0 0 1 -225.375 -324.106 cm +1 0 0 1 -212.6477 -370.9716 cm BT -/F15 10.9091 Tf 225.375 324.106 Td[(,)-333(T)83(abl)1(e)-1(s)]TJ +/F15 10.9091 Tf 212.6477 370.9716 Td[(,)-333(T)83(abl)1(e)-1(s)]TJ ET -1 0 0 1 266.3447 324.106 cm +1 0 0 1 253.6174 370.9716 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -266.3447 -324.106 cm +1 0 0 1 -253.6174 -370.9716 cm BT -/F15 10.9091 Tf 266.3447 324.106 Td[(2)]TJ +/F15 10.9091 Tf 253.6174 370.9716 Td[(2)]TJ ET -1 0 0 1 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Td[(F)84(or)-385(eac)27(h)-384(rou)1(tin)1(e)-1(,)-397(the)-385(same)-385(use)-1(r)-384(in)28(terf)1(ac)-1(e)-385(is)-385(o)28(v)28(erload)1(e)-1(d)-384(with)-384(res)-1(p)-27(ec)-1(t)-384(to)]TJ -16.9365 -13.5492 Td[(th)1(e)-353(real/com)-1(p)1(le)-1(x)-352(case)-353(and)-351(the)-353(sin)1(gle)-1(/d)1(oub)1(le)-353(p)1(rec)-1(ision)1(;)-362(ar)1(gume)-1(n)28(t)1(s)-353(with)-352(ap)1(-)]TJ 0 -13.5492 Td[(pr)1(opr)1(iate)-334(d)1(ata)-334(t)28(yp)-27(es)-334(m)27(u)1(s)-1(t)-333(b)-27(e)-334(passe)-1(d)-333(to)-333(th)1(e)-334(rou)1(tine,)-333(i.e.)]TJ +/F19 14.3462 Tf 93.9178 704.1363 Td[(7)-1125(User)-376(In)32(t)-1(er)-1(f)1(ace)]TJ/F15 10.9091 Tf 0 -30.694 Td[(The)-222(basic)-223(u)1(s)-1(er)-222(in)28(terf)1(ac)-1(e)-222(of)-222(MLD2P)1(4)-223(consists)-223(of)-222(six)-222(r)1(outin)1(e)-1(s.)-407(T)-1(h)1(e)-223(f)1(our)-222(r)1(outin)1(e)-1(s)]TJ/F48 10.9091 Tf 364.3237 0 Td[(mld_precinit)]TJ/F15 10.9091 Tf 68.7265 0 Td[(,)]TJ/F48 10.9091 Tf -433.0502 -13.5492 Td[(mld_precset)]TJ/F15 10.9091 Tf 62.9993 0 Td[(,)]TJ/F48 10.9091 Tf 7.5847 0 Td[(mld_precbld)]TJ/F15 10.9091 Tf 67.3701 0 Td[(and)]TJ/F48 10.9091 Tf 21.9466 0 Td[(mld_precaply)]TJ/F15 10.9091 Tf 73.0973 0 Td[(encapsulate)-401(all)-400(th)1(e)-402(f)1(un)1(c)-1(ti)1(onali)1(tie)-1(s)-401(f)1(or)]TJ -232.998 -13.5492 Td[(th)1(e)-447(s)-1(etu)1(p)-446(and)-445(the)-447(ap)1(pl)1(ic)-1(ati)1(on)-446(of)-446(an)28(y)-446(on)1(e)-1(-lev)28(e)-1(l)-445(and)-446(m)28(ulti)1(-)-1(l)1(e)-1(v)28(el)-446(precondi)1(tioner)-446(imp)1(le)-1(-)]TJ 0 -13.5492 Td[(me)-1(n)28(t)1(e)-1(d)-308(in)-309(t)1(he)-309(pac)27(k)56(age.)-436(T)-1(h)1(e)-309(routi)1(ne)]TJ/F48 10.9091 Tf 173.4573 0 Td[(mld_precfree)]TJ/F15 10.9091 Tf 72.0963 0 Td[(deallo)-27(c)-1(ates)-309(the)-309(pr)1(e)-1(cond)1(ition)1(e)-1(r)-308(data)]TJ -245.5536 -13.5492 Td[(structu)1(re)-1(,)-312(whil)1(e)]TJ/F48 10.9091 Tf 77.7976 0 Td[(mld_precdescr)]TJ/F15 10.9091 Tf 77.8092 0 Td[(pr)1(in)28(ts)-308(a)-307(des)-1(crip)1(tion)-307(of)-307(the)-308(p)1(rec)-1(on)1(diti)1(oner)-308(setup)-307(b)28(y)-307(the)]TJ -155.6068 -13.5492 Td[(user.)]TJ 16.9365 -16.8133 Td[(F)84(or)-222(e)-1(ac)28(h)-222(rou)1(tine,)-244(the)-222(s)-1(ame)-223(u)1(s)-1(er)-222(in)28(terf)1(ac)-1(e)-222(is)-223(o)28(v)28(e)-1(r)1(loaded)-222(with)-221(re)-1(sp)-27(e)-1(ct)-222(to)-222(the)-223(r)1(e)-1(al)1(/c)-1(omplex)]TJ -16.9365 -13.5492 Td[(cas)-1(e)-371(and)-370(th)1(e)-372(sin)1(gle)-1(/d)1(oub)1(le)-371(pr)1(e)-1(cis)-1(i)1(on;)-389(argu)1(m)-1(en)28(ts)-371(with)-370(app)1(rop)1(riate)-371(dat)1(a)-371(t)28(yp)-27(e)-1(s)-371(m)27(u)1(s)-1(t)-370(b)-28(e)]TJ 0 -13.5492 Td[(passe)-1(d)-333(to)-333(th)1(e)-334(rou)1(tine,)-333(i.e.)]TJ ET -1 0 0 1 141.7385 518.1011 cm +1 0 0 1 93.9178 526.2127 cm 0 g 0 G -1 0 0 1 -141.7385 -518.1011 cm +1 0 0 1 -93.9178 -526.2127 cm BT -/F27 10.9091 Tf 158.1022 518.1011 Td[(\017)]TJ +/F27 10.9091 Tf 110.2815 526.2127 Td[(\017)]TJ ET -1 0 0 1 163.5567 518.1011 cm +1 0 0 1 115.736 526.2127 cm 0 g 0 G -1 0 0 1 -163.5567 -518.1011 cm +1 0 0 1 -115.736 -526.2127 cm BT -/F15 10.9091 Tf 169.0113 518.1011 Td[(th)1(e)-324(sparse)-323(m)-1(atr)1(ix)-323(dat)1(a)-323(s)-1(tr)1(ucture,)-325(con)28(tain)1(ing)-323(th)1(e)-324(matr)1(ix)-323(to)-323(b)-27(e)-324(p)1(rec)-1(on)1(-)]TJ 0 -13.5492 Td[(di)1(tioned,)-307(m)27(u)1(s)-1(t)-301(b)-27(e)-302(of)-301(t)28(yp)-27(e)]TJ/F48 10.9091 Tf 122.626 0 Td[(mld_)]TJ/F20 10.9091 Tf 22.9088 0 Td[(x)]TJ/F48 10.9091 Tf 6.3742 0 Td[(spmat_type)]TJ/F15 10.9091 Tf 60.5591 0 Td[(with)]TJ/F20 10.9091 Tf 24.4991 0 Td[(x)]TJ/F15 10.9091 Tf 9.6611 0 Td[(=)]TJ/F48 10.9091 Tf 11.7718 0 Td[(s)]TJ/F15 10.9091 Tf 9.0142 0 Td[(for)-301(r)1(e)-1(al)-301(singl)1(e)]TJ -267.4143 -13.5492 Td[(pr)1(e)-1(cision,)]TJ/F20 10.9091 Tf 49.6866 0 Td[(x)]TJ/F15 10.9091 Tf 10.8993 0 Td[(=)]TJ/F48 10.9091 Tf 13.0099 0 Td[(d)]TJ/F15 10.9091 Tf 10.2523 0 Td[(for)-414(real)-415(d)1(oub)1(le)-415(prec)-1(i)1(s)-1(ion)1(,)]TJ/F20 10.9091 Tf 125.4437 0 Td[(x)]TJ/F15 10.9091 Tf 10.8993 0 Td[(=)]TJ/F48 10.9091 Tf 13.0099 0 Td[(c)]TJ/F15 10.9091 Tf 10.2522 0 Td[(for)-414(com)-1(p)1(le)-1(x)-414(s)-1(i)1(ngle)]TJ -243.4532 -13.5492 Td[(pr)1(e)-1(cision,)]TJ/F20 10.9091 Tf 48.5758 0 Td[(x)]TJ/F15 10.9091 Tf 10.0106 0 Td[(=)]TJ/F48 10.9091 Tf 12.1212 0 Td[(z)]TJ/F15 10.9091 Tf 9.3636 0 Td[(for)-333(complex)-333(dou)1(ble)-333(prec)-1(i)1(s)-1(ion)1(;)]TJ +/F15 10.9091 Tf 121.1906 526.2127 Td[(th)1(e)-464(sparse)-464(matri)1(x)-463(data)-463(stru)1(c)-1(tu)1(re,)-496(con)28(tain)1(in)1(g)-464(t)1(he)-464(matr)1(ix)-463(to)-463(b)-27(e)-464(pr)1(e)-1(cond)1(ition)1(e)-1(d)1(,)]TJ 0 -13.5492 Td[(m)27(u)1(s)-1(t)-423(b)-28(e)-424(of)-423(t)28(yp)-28(e)]TJ/F48 10.9091 Tf 82.7978 0 Td[(mld_)]TJ/F20 10.9091 Tf 22.9088 0 Td[(x)]TJ/F48 10.9091 Tf 6.3742 0 Td[(spmat_type)]TJ/F15 10.9091 Tf 61.8958 0 Td[(with)]TJ/F20 10.9091 Tf 25.8358 0 Td[(x)]TJ/F15 10.9091 Tf 10.9979 0 Td[(=)]TJ/F48 10.9091 Tf 13.1085 0 Td[(s)]TJ/F15 10.9091 Tf 10.3509 0 Td[(for)-423(real)-424(singl)1(e)-425(p)1(rec)-1(ision)1(,)]TJ/F20 10.9091 Tf 121.0749 0 Td[(x)]TJ/F15 10.9091 Tf 10.9979 0 Td[(=)]TJ/F48 10.9091 Tf 13.1085 0 Td[(d)]TJ/F15 10.9091 Tf -379.4511 -13.5492 Td[(for)-370(real)-371(dou)1(ble)-371(pr)1(e)-1(cision,)]TJ/F20 10.9091 Tf 123.4131 0 Td[(x)]TJ/F15 10.9091 Tf 10.4215 0 Td[(=)]TJ/F48 10.9091 Tf 12.5322 0 Td[(c)]TJ/F15 10.9091 Tf 9.7744 0 Td[(for)-370(c)-1(ompl)1(e)-1(x)-371(sin)1(gle)-372(p)1(rec)-1(i)1(s)-1(ion)1(,)]TJ/F20 10.9091 Tf 140.1102 0 Td[(x)]TJ/F15 10.9091 Tf 10.4214 0 Td[(=)]TJ/F48 10.9091 Tf 12.5322 0 Td[(z)]TJ/F15 10.9091 Tf 9.7745 0 Td[(for)-370(c)-1(ompl)1(e)-1(x)]TJ -328.9795 -13.5492 Td[(dou)1(bl)1(e)-334(pr)1(e)-1(cis)-1(i)1(on;)]TJ ET -1 0 0 1 141.7385 445.0143 cm +1 0 0 1 93.9178 449.9932 cm 0 g 0 G -1 0 0 1 -141.7385 -445.0143 cm +1 0 0 1 -93.9178 -449.9932 cm BT -/F27 10.9091 Tf 158.1022 445.0143 Td[(\017)]TJ +/F27 10.9091 Tf 110.2815 449.9932 Td[(\017)]TJ ET -1 0 0 1 163.5567 445.0143 cm +1 0 0 1 115.736 449.9932 cm 0 g 0 G -1 0 0 1 -163.5567 -445.0143 cm +1 0 0 1 -115.736 -449.9932 cm BT -/F15 10.9091 Tf 169.0113 445.0143 Td[(th)1(e)-404(p)1(rec)-1(on)1(diti)1(oner)-403(dat)1(a)-403(s)-1(tr)1(uctur)1(e)-404(m)28(ust)-403(b)-27(e)-404(of)-402(t)28(yp)-27(e)]TJ/F48 10.9091 Tf 247.524 0 Td[(mld_)]TJ/F20 10.9091 Tf 22.9088 0 Td[(x)]TJ/F48 10.9091 Tf 6.3742 0 Td[(prec_type)]TJ/F15 10.9091 Tf 51.5449 0 Td[(,)]TJ -328.3519 -13.5492 Td[(with)]TJ/F20 10.9091 Tf 24.8485 0 Td[(x)]TJ/F15 10.9091 Tf 10.0105 0 Td[(=)]TJ/F48 10.9091 Tf 12.1213 0 Td[(s)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F48 10.9091 Tf 6.6667 0 Td[(d)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F48 10.9091 Tf 6.6666 0 Td[(c)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F48 10.9091 Tf 6.6667 0 Td[(z)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)-333(ac)-1(cord)1(ing)-333(to)-333(the)-333(s)-1(p)1(arse)-334(m)-1(at)1(rix)-333(dat)1(a)-334(stru)1(c)-1(tu)1(re;)]TJ +/F15 10.9091 Tf 121.1906 449.9932 Td[(th)1(e)-338(p)1(re)-1(cond)1(ition)1(e)-1(r)-336(data)-337(stru)1(c)-1(tu)1(re)-337(m)27(u)1(s)-1(t)-337(b)-27(e)-338(of)-336(t)28(yp)-27(e)]TJ/F48 10.9091 Tf 241.7755 0 Td[(mld_)]TJ/F20 10.9091 Tf 22.9088 0 Td[(x)]TJ/F48 10.9091 Tf 6.3742 0 Td[(prec_type)]TJ/F15 10.9091 Tf 51.5449 0 Td[(,)-338(with)]TJ/F20 10.9091 Tf 31.6054 0 Td[(x)]TJ/F15 10.9091 Tf 10.0507 0 Td[(=)]TJ/F48 10.9091 Tf 12.1613 0 Td[(s)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F48 10.9091 Tf -382.148 -13.5492 Td[(d)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F48 10.9091 Tf 6.6667 0 Td[(c)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F48 10.9091 Tf 6.6667 0 Td[(z)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)-333(ac)-1(cord)1(ing)-333(to)-333(the)-333(s)-1(p)1(arse)-334(m)-1(at)1(rix)-333(dat)1(a)-334(stru)1(c)-1(tu)1(re;)]TJ ET -1 0 0 1 141.7385 399.0258 cm +1 0 0 1 93.9178 400.872 cm 0 g 0 G -1 0 0 1 -141.7385 -399.0258 cm +1 0 0 1 -93.9178 -400.872 cm BT -/F27 10.9091 Tf 158.1022 399.0258 Td[(\017)]TJ +/F27 10.9091 Tf 110.2815 400.872 Td[(\017)]TJ ET -1 0 0 1 163.5567 399.0258 cm +1 0 0 1 115.736 400.872 cm 0 g 0 G -1 0 0 1 -163.5567 -399.0258 cm +1 0 0 1 -115.736 -400.872 cm BT -/F15 10.9091 Tf 169.0113 399.0258 Td[(th)1(e)-418(arra)28(ys)-417(c)-1(on)28(tai)1(nin)1(g)-418(t)1(he)-418(v)28(ec)-1(tor)1(s)]TJ/F24 10.9091 Tf 165.0989 0 Td[(v)]TJ/F15 10.9091 Tf 10.2324 0 Td[(and)]TJ/F24 10.9091 Tf 22.1289 0 Td[(w)]TJ/F15 10.9091 Tf 12.6566 0 Td[(in)28(v)28(ol)1(v)28(e)-1(d)-417(i)1(n)-417(the)-418(p)1(rec)-1(on)1(di-)]TJ -210.1168 -13.5492 Td[(tion)1(e)-1(r)-356(app)1(lication)]TJ/F24 10.9091 Tf 88.4224 0 Td[(w)]TJ/F15 10.9091 Tf 11.5615 0 Td[(=)]TJ/F24 10.9091 Tf 11.9428 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -3.9588 Td[(v)]TJ/F15 10.9091 Tf 9.5723 0 Td[(m)27(u)1(s)-1(t)-356(b)-28(e)-357(of)-356(t)28(yp)-28(e)]TJ/F20 10.9091 Tf 79.875 0 Td[(typ)51(e)]TJ/F48 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ +/F15 10.9091 Tf 121.1906 400.872 Td[(th)1(e)-322(arr)1(a)27(y)1(s)-322(con)28(tain)1(ing)-321(th)1(e)-322(v)28(ec)-1(t)1(ors)]TJ/F24 10.9091 Tf 159.8466 0 Td[(v)]TJ/F15 10.9091 Tf 9.1819 0 Td[(and)]TJ/F24 10.9091 Tf 21.0784 0 Td[(w)]TJ/F15 10.9091 Tf 11.6062 0 Td[(in)28(v)28(ol)1(v)28(e)-1(d)-320(in)-321(th)1(e)-322(p)1(rec)-1(on)1(diti)1(oner)-321(app)1(lica-)]TJ -201.7131 -13.5492 Td[(tion)]TJ/F24 10.9091 Tf 21.3155 0 Td[(w)]TJ/F15 10.9091 Tf 11.1338 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -3.9588 Td[(v)]TJ/F15 10.9091 Tf 8.2069 0 Td[(m)27(u)1(s)-1(t)-231(b)-27(e)-233(of)-231(t)28(yp)-27(e)]TJ/F20 10.9091 Tf 74.4135 0 Td[(typ)51(e)]TJ/F48 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ ET -1 0 0 1 439.7227 385.4766 cm +1 0 0 1 317.1129 387.3228 cm q []0 d 0 J @@ -4150,11 +4273,11 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -439.7227 -385.4766 cm +1 0 0 1 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10.9091 Tf 5.7991 0 Td[('ml')]TJ/F15 10.9091 Tf 22.9089 0 Td[(,)-269(then)]TJ/F48 10.9091 Tf 29.9536 0 Td[(nlev)]TJ/F15 10.9091 Tf 22.9089 0 Td[(=2)-254(i)1(s)-255(ass)-1(u)1(me)-1(d)1(.)]TJ -227.6758 -13.5492 Td[(Oth)1(e)-1(r)1(w)-1(i)1(s)-1(e,)]TJ/F48 10.9091 Tf 54.6364 0 Td[(nlev)]TJ/F15 10.9091 Tf 26.5452 0 Td[(is)-334(i)1(gnored.)]TJ/F19 11.9552 Tf -133.13 -28.056 Td[(7.)-1(2)-1125(Subr)-1(out)-1(ine)-376(ml)-1(d)]TJ ET -1 0 0 1 216.9401 493.8428 cm +1 0 0 1 264.7607 493.8428 cm q []0 d 0 J @@ -4301,341 +4424,341 @@ q 4.0349 0.1992 l S Q -1 0 0 1 -216.9401 -493.8428 cm +1 0 0 1 -264.7607 -493.8428 cm BT -/F19 11.9552 Tf 220.975 493.8428 Td[(pre)-1(cset)]TJ +/F19 11.9552 Tf 268.7956 493.8428 Td[(pre)-1(cset)]TJ ET -1 0 0 1 187.6098 473.2482 cm +1 0 0 1 262.3285 473.2482 cm 0 g 0 G 0 g 0 G -1 0 0 1 -187.6098 -473.2482 cm +1 0 0 1 -262.3285 -473.2482 cm BT -/F48 10.9091 Tf 193.0644 473.2482 Td[(mld_precset\050p,what,val,info\051)]TJ/F15 10.9091 Tf -99.1466 -21.8621 Td[(This)-243(rou)1(tin)1(e)-244(se)-1(t)1(s)-244(th)1(e)-243(paramete)-1(r)1(s)-244(d)1(e)-1(\014)1(nin)1(g)-243(the)-243(p)1(re)-1(cond)1(ition)1(e)-1(r)1(.)-414(More)-243(p)1(re)-1(cise)-1(ly)84(,)]TJ 0 -13.5492 Td[(th)1(e)-334(par)1(am)-1(ete)-1(r)-333(i)1(den)28(ti\014)1(e)-1(d)-333(b)29(y)]TJ/F48 10.9091 Tf 133.6971 0 Td[(what)]TJ/F15 10.9091 Tf 26.5452 0 Td[(is)-334(assigned)-333(the)-333(v)55(al)1(ue)-334(con)28(tain)1(e)-1(d)-332(in)]TJ/F48 10.9091 Tf 163.2123 0 Td[(val)]TJ/F15 10.9091 Tf 17.1816 0 Td[(.)]TJ/F41 10.9091 Tf -340.6362 -28.2419 Td[(Argumen)32(ts)]TJ/F48 10.9091 Tf 5.9776 -19.6473 Td[(p)-3689(type\050mld_)]TJ/F20 10.9091 Tf 97.5157 0 Td[(x)]TJ/F48 10.9091 Tf 6.3742 0 Td[(prec_type\051,)-525(intent\050inout\051)]TJ/F15 10.9091 Tf 143.1803 0 Td[(.)]TJ -201.0994 -13.5492 Td[(The)-267(prec)-1(on)1(di)1(tioner)-267(dat)1(a)-268(stru)1(c)-1(tu)1(re.)-422(Note)-268(th)1(at)]TJ/F20 10.9091 Tf 216.6391 0 Td[(x)]TJ/F15 10.9091 Tf 9.2894 0 Td[(m)27(u)1(s)-1(t)-267(b)-27(e)-268(c)28(hose)-1(n)]TJ -225.9285 -13.5492 Td[(acc)-1(ord)1(in)1(g)-352(to)-352(the)-352(real/c)-1(ompl)1(e)-1(x,)-356(single/dou)1(bl)1(e)-353(p)1(re)-1(cision)-352(v)28(ersion)]TJ 0 -13.5492 Td[(of)-333(MLD)1(2P4)-333(un)1(der)-333(use)-1(.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(what)-2114(integer,)-525(intent\050in\051)]TJ/F15 10.9091 Tf 154.7878 0 Td[(.)]TJ -108.817 -13.5492 Td[(The)-338(n)28(u)1(m)27(b)-27(e)-1(r)-337(id)1(e)-1(n)29(tify)1(ing)-337(the)-338(p)1(arame)-1(ter)-338(t)1(o)-338(b)-27(e)-338(s)-1(et.)-457(A)-338(mnem)-1(on)1(ic)]TJ 0 -13.5492 Td[(constan)28(t)-378(has)-379(b)-27(ee)-1(n)-378(ass)-1(o)-27(c)-1(iated)-378(to)-378(e)-1(ac)28(h)-378(of)-378(th)1(e)-1(se)-379(n)28(u)1(m)27(b)-27(e)-1(r)1(s)-1(,)-389(as)-379(re-)]TJ 0 -13.5492 Td[(p)-27(orted)-333(in)-333(T)83(ab)1(le)-1(s)]TJ +/F48 10.9091 Tf 267.7831 473.2482 Td[(mld_precset\050p,what,val,info\051)]TJ/F15 10.9091 Tf -126.0446 -21.8621 Td[(This)-389(r)1(outin)1(e)-389(s)-1(ets)-389(the)-389(p)1(arame)-1(ters)-389(d)1(e)-1(\014)1(nin)1(g)-389(th)1(e)-389(precondi)1(tioner.)-610(Mor)1(e)-389(precis)-1(ely)84(,)-403(t)1(he)-389(pa-)]TJ 0 -13.5492 Td[(ramete)-1(r)-333(id)1(e)-1(n)29(ti\014ed)-333(b)28(y)]TJ/F48 10.9091 Tf 103.394 0 Td[(what)]TJ/F15 10.9091 Tf 26.5452 0 Td[(is)-334(assigned)-333(the)-333(v)55(al)1(ue)-334(con)28(tain)1(e)-1(d)-332(in)]TJ/F48 10.9091 Tf 163.2123 0 Td[(val)]TJ/F15 10.9091 Tf 17.1816 0 Td[(.)]TJ/F41 10.9091 Tf -310.3331 -28.2419 Td[(Argumen)32(ts)]TJ/F48 10.9091 Tf 5.9776 -19.6473 Td[(p)-3689(type\050mld_)]TJ/F20 10.9091 Tf 97.5157 0 Td[(x)]TJ/F48 10.9091 Tf 6.3742 0 Td[(prec_type\051,)-525(intent\050inout\051)]TJ/F15 10.9091 Tf 143.1803 0 Td[(.)]TJ -201.0994 -13.5492 Td[(The)-267(prec)-1(on)1(di)1(tioner)-267(dat)1(a)-268(stru)1(c)-1(tu)1(re.)-422(Note)-268(th)1(at)]TJ/F20 10.9091 Tf 216.6391 0 Td[(x)]TJ/F15 10.9091 Tf 9.2894 0 Td[(m)27(u)1(s)-1(t)-267(b)-27(e)-268(c)28(hose)-1(n)]TJ -225.9285 -13.5492 Td[(acc)-1(ord)1(in)1(g)-352(to)-352(the)-352(real/c)-1(ompl)1(e)-1(x,)-356(single/dou)1(bl)1(e)-353(p)1(re)-1(cision)-352(v)28(ersion)]TJ 0 -13.5492 Td[(of)-333(MLD)1(2P4)-333(un)1(der)-333(use)-1(.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(what)-2114(integer,)-525(intent\050in\051)]TJ/F15 10.9091 Tf 154.7878 0 Td[(.)]TJ -108.817 -13.5492 Td[(The)-338(n)28(u)1(m)27(b)-27(e)-1(r)-337(id)1(e)-1(n)29(tify)1(ing)-337(the)-338(p)1(arame)-1(ter)-338(t)1(o)-338(b)-27(e)-338(s)-1(et.)-457(A)-338(mnem)-1(on)1(ic)]TJ 0 -13.5492 Td[(constan)28(t)-378(has)-379(b)-27(ee)-1(n)-378(ass)-1(o)-27(c)-1(iated)-378(to)-378(e)-1(ac)28(h)-378(of)-378(th)1(e)-1(se)-379(n)28(u)1(m)27(b)-27(e)-1(r)1(s)-1(,)-389(as)-379(re-)]TJ 0 -13.5492 Td[(p)-27(orted)-333(in)-333(T)83(ab)1(le)-1(s)]TJ ET -1 0 0 1 227.7754 295.1034 cm +1 0 0 1 275.596 295.1034 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -227.7754 -295.1034 cm +1 0 0 1 -275.596 -295.1034 cm BT -/F15 10.9091 Tf 227.7754 295.1034 Td[(2)]TJ +/F15 10.9091 Tf 275.596 295.1034 Td[(2)]TJ ET -1 0 0 1 233.2299 295.1034 cm +1 0 0 1 281.0506 295.1034 cm 0 g 0 G -1 0 0 1 -233.2299 -295.1034 cm +1 0 0 1 -281.0506 -295.1034 cm BT -/F15 10.9091 Tf 233.2299 295.1034 Td[(-)]TJ +/F15 10.9091 Tf 281.0506 295.1034 Td[(-)]TJ ET -1 0 0 1 236.8663 295.1034 cm +1 0 0 1 284.687 295.1034 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -236.8663 -295.1034 cm +1 0 0 1 -284.687 -295.1034 cm BT -/F15 10.9091 Tf 236.8663 295.1034 Td[(5)]TJ +/F15 10.9091 Tf 284.687 295.1034 Td[(5)]TJ ET -1 0 0 1 242.3208 295.1034 cm +1 0 0 1 290.1415 295.1034 cm 0 g 0 G -1 0 0 1 -242.3208 -295.1034 cm +1 0 0 1 -290.1415 -295.1034 cm BT -/F15 10.9091 Tf 242.3208 295.1034 Td[(.)]TJ/F48 10.9091 Tf -142.4254 -13.5492 Td[(val)-2639(integer)]TJ/F20 10.9091 Tf 96.6324 0 Td[(or)]TJ/F48 10.9091 Tf 21.921 0 Td[(character\050len=*\051)]TJ/F20 10.9091 Tf 102.2065 0 Td[(or)]TJ/F48 10.9091 Tf 21.921 0 Td[(real\050psb_spk_\051)]TJ/F20 10.9091 Tf 90.752 0 Td[(or)]TJ/F48 10.9091 Tf -287.4621 -13.5492 Td[(real\050psb_dpk_\051)]TJ/F15 10.9091 Tf 80.181 0 Td[(,)]TJ/F48 10.9091 Tf 6.6666 0 Td[(intent\050in\051)]TJ/F15 10.9091 Tf 57.2721 0 Td[(.)]TJ -144.1197 -13.5492 Td[(The)-281(v)56(alu)1(e)-281(of)-280(th)1(e)-281(paramete)-1(r)-280(to)-280(b)-27(e)-281(s)-1(et.)-427(Th)1(e)-281(list)-280(of)-280(allo)28(w)27(ed)-280(v)56(alues)]TJ 0 -13.5492 Td[(and)-300(the)-301(c)-1(orr)1(e)-1(sp)-27(ondi)1(ng)-301(data)-301(t)28(yp)-27(es)-302(is)-301(giv)28(e)-1(n)-300(in)-301(T)83(ab)1(le)-1(s)]TJ +/F15 10.9091 Tf 290.1415 295.1034 Td[(.)]TJ/F48 10.9091 Tf -142.4254 -13.5492 Td[(val)-2639(integer)]TJ/F20 10.9091 Tf 96.6324 0 Td[(or)]TJ/F48 10.9091 Tf 21.921 0 Td[(character\050len=*\051)]TJ/F20 10.9091 Tf 102.2065 0 Td[(or)]TJ/F48 10.9091 Tf 21.921 0 Td[(real\050psb_spk_\051)]TJ/F20 10.9091 Tf 90.752 0 Td[(or)]TJ/F48 10.9091 Tf -287.4621 -13.5492 Td[(real\050psb_dpk_\051)]TJ/F15 10.9091 Tf 80.181 0 Td[(,)]TJ/F48 10.9091 Tf 6.6666 0 Td[(intent\050in\051)]TJ/F15 10.9091 Tf 57.2721 0 Td[(.)]TJ 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+/F15 10.9091 Tf 449.9263 240.9066 Td[(5)]TJ ET -1 0 0 1 407.5602 240.9066 cm +1 0 0 1 455.3809 240.9066 cm 0 g 0 G -1 0 0 1 -407.5602 -240.9066 cm +1 0 0 1 -455.3809 -240.9066 cm BT -/F15 10.9091 Tf 407.5602 240.9066 Td[(.)-434(W)1(hen)]TJ -261.694 -13.5492 Td[(th)1(e)-453(v)56(alue)-453(i)1(s)-453(of)-452(t)28(yp)-27(e)]TJ/F48 10.9091 Tf 101.3996 0 Td[(character\050len=*\051)]TJ/F15 10.9091 Tf 91.6353 0 Td[(,)-482(it)-452(is)-452(als)-1(o)-452(tr)1(e)-1(ated)-452(as)]TJ -193.0349 -13.5492 Td[(cas)-1(e)-334(i)1(nse)-1(n)1(s)-1(iti)1(v)28(e)-1(.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(info)-2114(integer,)-525(intent\050out\051)]TJ/F15 10.9091 Tf 160.515 0 Td[(.)]TJ -114.5442 -13.5492 Td[(Er)1(ror)-307(c)-1(o)-28(d)1(e)-1(.)-435(If)-308(n)1(o)-308(e)-1(r)1(ror,)-313(0)-307(is)-308(re)-1(t)1(urn)1(e)-1(d)1(.)-436(See)-308(Sec)-1(ti)1(on)]TJ +/F15 10.9091 Tf 455.3809 240.9066 Td[(.)-434(W)1(hen)]TJ -261.694 -13.5492 Td[(th)1(e)-453(v)56(alue)-453(i)1(s)-453(of)-452(t)28(yp)-27(e)]TJ/F48 10.9091 Tf 101.3996 0 Td[(character\050len=*\051)]TJ/F15 10.9091 Tf 91.6353 0 Td[(,)-482(it)-452(is)-452(als)-1(o)-452(tr)1(e)-1(ated)-452(as)]TJ -193.0349 -13.5492 Td[(cas)-1(e)-334(i)1(nse)-1(n)1(s)-1(iti)1(v)28(e)-1(.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(info)-2114(integer,)-525(intent\050out\051)]TJ/F15 10.9091 Tf 160.515 0 Td[(.)]TJ -114.5442 -13.5492 Td[(Er)1(ror)-307(c)-1(o)-28(d)1(e)-1(.)-435(If)-308(n)1(o)-308(e)-1(r)1(ror,)-313(0)-307(is)-308(re)-1(t)1(urn)1(e)-1(d)1(.)-436(See)-308(Sec)-1(ti)1(on)]TJ ET -1 0 0 1 384.2717 186.7099 cm +1 0 0 1 432.0924 186.7099 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -384.2717 -186.7099 cm +1 0 0 1 -432.0924 -186.7099 cm BT -/F15 10.9091 Tf 384.2717 186.7099 Td[(8)]TJ +/F15 10.9091 Tf 432.0924 186.7099 Td[(8)]TJ ET -1 0 0 1 389.7263 186.7099 cm +1 0 0 1 437.547 186.7099 cm 0 g 0 G -1 0 0 1 -389.7263 -186.7099 cm +1 0 0 1 -437.547 -186.7099 cm BT -/F15 10.9091 Tf 393.0848 186.7099 Td[(for)-307(details.)]TJ -299.167 -18.6103 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10.9091 Tf 629.5364 825.5652 Td[('HALO')]TJ +/F48 10.9091 Tf 625.6214 833.1368 Td[('HALO')]TJ ET -1 0 0 1 706.6173 794.402 cm +1 0 0 1 702.7023 801.9736 cm q []0 d 0 J @@ -5250,11 +5399,11 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -706.6173 -794.402 cm +1 0 0 1 -702.7023 -801.9736 cm BT -/F15 10.9091 Tf 712.7942 825.5652 Td[(T)28(yp)-28(e)-429(of)-429(res)-1(t)1(riction)-429(op)-27(e)-1(r)1(ator:)]TJ/F48 10.9091 Tf 146.3201 0 Td[('HALO')]TJ/F15 10.9091 Tf 39.0444 0 Td[(for)]TJ -185.3645 -13.5492 Td[(taki)1(ng)-478(in)29(to)-478(acc)-1(ou)1(n)28(t)-478(th)1(e)-479(o)28(v)28(erlap)1(,)]TJ/F48 10.9091 Tf 164.0619 0 Td[('NONE')]TJ/F15 10.9091 Tf -164.0619 -13.5492 Td[(for)-333(n)1(e)-1(gl)1(e)-1(ctin)1(g)-334(it)1(.)]TJ +/F15 10.9091 Tf 708.8791 833.1368 Td[(T)28(yp)-28(e)-429(of)-429(res)-1(t)1(riction)-429(op)-27(e)-1(r)1(ator:)]TJ/F48 10.9091 Tf 146.3202 0 Td[('HALO')]TJ/F15 10.9091 Tf 39.0444 0 Td[(for)]TJ -185.3646 -13.5492 Td[(taki)1(ng)-478(in)29(to)-478(acc)-1(ou)1(n)28(t)-478(th)1(e)-479(o)28(v)28(erlap)1(,)]TJ/F48 10.9091 Tf 164.062 0 Td[('NONE')]TJ/F15 10.9091 Tf -164.062 -13.5492 Td[(for)-333(n)1(e)-1(gl)1(e)-1(ctin)1(g)-334(it)1(.)]TJ ET -1 0 0 1 916.9977 794.402 cm +1 0 0 1 913.0827 801.9736 cm q []0 d 0 J @@ -5263,7 +5412,16 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -596.1309 -40.6476 cm +1 0 0 1 -612.9393 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +613.1385 0.1992 l +S +Q +1 0 0 1 -0.1993 -40.6476 cm q []0 d 0 J @@ -5272,11 +5430,11 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -320.8668 -753.7544 cm +1 0 0 1 -299.9441 -760.9275 cm BT -/F48 10.9091 Tf 327.0436 784.9176 Td[(mld_sub_prol_)]TJ +/F48 10.9091 Tf 306.121 792.0907 Td[(mld_sub_prol_)]TJ ET -1 0 0 1 418.7301 753.7544 cm +1 0 0 1 397.8075 760.9275 cm q []0 d 0 J @@ -5285,11 +5443,11 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -418.7301 -753.7544 cm +1 0 0 1 -397.8075 -760.9275 cm BT -/F48 10.9091 Tf 424.9069 784.9176 Td[(character\050len=*\051)]TJ +/F48 10.9091 Tf 403.9843 792.0907 Td[(character\050len=*\051)]TJ ET -1 0 0 1 537.7035 753.7544 cm +1 0 0 1 516.7808 760.9275 cm q []0 d 0 J @@ -5298,11 +5456,11 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -537.7035 -753.7544 cm +1 0 0 1 -516.7808 -760.9275 cm BT -/F48 10.9091 Tf 543.8803 784.9176 Td[('SUM')]TJ 0 -13.5492 Td[('NONE')]TJ +/F48 10.9091 Tf 522.9577 792.0907 Td[('SUM')]TJ 0 -13.5492 Td[('NONE')]TJ ET -1 0 0 1 623.3596 753.7544 cm +1 0 0 1 619.4446 760.9275 cm q []0 d 0 J @@ -5311,11 +5469,11 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -623.3596 -753.7544 cm +1 0 0 1 -619.4446 -760.9275 cm BT -/F48 10.9091 Tf 629.5364 784.9176 Td[('NONE')]TJ +/F48 10.9091 Tf 625.6214 792.0907 Td[('NONE')]TJ ET -1 0 0 1 706.6173 753.7544 cm +1 0 0 1 702.7023 760.9275 cm q []0 d 0 J @@ -5324,11 +5482,11 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -706.6173 -753.7544 cm +1 0 0 1 -702.7023 -760.9275 cm BT -/F15 10.9091 Tf 712.7942 784.9176 Td[(T)28(yp)-28(e)-416(of)-415(pr)1(olongator)-415(op)-27(e)-1(r)1(ator:)]TJ/F48 10.9091 Tf 152.1937 0 Td[('SUM')]TJ/F15 10.9091 Tf 33.1708 0 Td[(for)]TJ -185.3645 -13.5492 Td[(add)1(in)1(g)-252(the)-252(con)28(tr)1(ibu)1(tions)-252(f)1(rom)-252(th)1(e)-253(o)28(v)28(erlap)1(,)]TJ/F48 10.9091 Tf 0 -13.5492 Td[('NONE')]TJ/F15 10.9091 Tf 37.9996 0 Td[(for)-333(n)1(e)-1(gl)1(e)-1(ctin)1(g)-334(th)1(e)-1(m.)]TJ +/F15 10.9091 Tf 708.8791 792.0907 Td[(T)28(yp)-28(e)-416(of)-415(pr)1(olongator)-415(op)-27(e)-1(r)1(ator:)]TJ/F48 10.9091 Tf 152.1938 0 Td[('SUM')]TJ/F15 10.9091 Tf 33.1708 0 Td[(for)]TJ -185.3646 -13.5492 Td[(add)1(in)1(g)-252(the)-252(con)28(tr)1(ibu)1(tions)-252(f)1(rom)-252(th)1(e)-253(o)28(v)28(erlap)1(,)]TJ/F48 10.9091 Tf 0 -13.5492 Td[('NONE')]TJ/F15 10.9091 Tf 37.9997 0 Td[(for)-333(n)1(e)-1(gl)1(e)-1(ctin)1(g)-334(th)1(e)-1(m.)]TJ ET -1 0 0 1 916.9977 753.7544 cm +1 0 0 1 913.0827 760.9275 cm q []0 d 0 J @@ -5337,81 +5495,99 @@ q 0.1992 40.6476 l S Q -1 0 0 1 -596.1309 -54.1968 cm +1 0 0 1 -612.9393 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +613.1385 0.1992 l +S +Q +1 0 0 1 -0.1993 -67.746 cm q []0 d 0 J 0.3985 w 0.1992 0 m -0.1992 54.1968 l +0.1992 67.746 l S Q -1 0 0 1 -320.8668 -699.5576 cm +1 0 0 1 -299.9441 -692.783 cm BT -/F48 10.9091 Tf 327.0436 744.27 Td[(mld_sub_solve_)]TJ +/F48 10.9091 Tf 306.121 751.0446 Td[(mld_sub_solve_)]TJ ET -1 0 0 1 418.7301 699.5576 cm +1 0 0 1 397.8075 692.7831 cm q []0 d 0 J 0.3985 w 0.1992 0 m -0.1992 54.1968 l +0.1992 67.746 l S Q -1 0 0 1 -418.7301 -699.5576 cm +1 0 0 1 -397.8075 -692.7831 cm BT -/F48 10.9091 Tf 424.9069 744.27 Td[(character\050len=*\051)]TJ +/F48 10.9091 Tf 403.9843 751.0446 Td[(character\050len=*\051)]TJ ET -1 0 0 1 537.7035 699.5576 cm +1 0 0 1 516.7808 692.7831 cm q []0 d 0 J 0.3985 w 0.1992 0 m -0.1992 54.1968 l +0.1992 67.746 l S Q -1 0 0 1 -537.7035 -699.5576 cm +1 0 0 1 -516.7808 -692.7831 cm BT -/F48 10.9091 Tf 543.8803 744.27 Td[('ILU')]TJ 0 -13.5492 Td[('MILU')]TJ 0 -13.5492 Td[('ILUT')]TJ 0 -13.5492 Td[('UMF')-1506('SLU')]TJ +/F48 10.9091 Tf 522.9577 751.0446 Td[('ILU')]TJ 0 -13.5492 Td[('MILU')]TJ 0 -13.5492 Td[('ILUT')]TJ 0 -13.5492 Td[('UMF')]TJ 0 -13.5492 Td[('SLU')]TJ ET -1 0 0 1 623.3596 699.5576 cm +1 0 0 1 619.4446 692.7831 cm q []0 d 0 J 0.3985 w 0.1992 0 m -0.1992 54.1968 l +0.1992 67.746 l S Q -1 0 0 1 -623.3596 -699.5576 cm +1 0 0 1 -619.4446 -692.7831 cm BT -/F48 10.9091 Tf 629.5364 744.27 Td[('UMF')]TJ +/F48 10.9091 Tf 625.6214 751.0446 Td[('UMF')]TJ ET -1 0 0 1 706.6173 699.5576 cm +1 0 0 1 702.7023 692.7831 cm q []0 d 0 J 0.3985 w 0.1992 0 m -0.1992 54.1968 l +0.1992 67.746 l S Q -1 0 0 1 -706.6173 -699.5576 cm +1 0 0 1 -702.7023 -692.7831 cm BT -/F15 10.9091 Tf 712.7942 744.27 Td[(Lo)-28(cal)-326(solv)28(e)-1(r)1(:)-441(ILU\050)]TJ/F24 10.9091 Tf 87.8473 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)-327(MILU\050)]TJ/F24 10.9091 Tf 44.0257 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)-327(ILU\050)]TJ/F24 10.9091 Tf 34.0257 0 Td[(p;)-167(t)]TJ/F15 10.9091 Tf 14.2765 0 Td[(\051,)]TJ -191.1524 -13.5492 Td[(LU)-356(f)1(rom)-356(UM)1(FP)84(A)27(CK,)-355(LU)-355(from)-356(S)1(up)-27(e)-1(r)1(LU,)]TJ 0 -13.5492 Td[(pl)1(us)-334(tr)1(iangu)1(lar)-333(solv)28(e)-1(.)]TJ +/F15 10.9091 Tf 708.8791 751.0446 Td[(Lo)-28(cal)-326(solv)28(e)-1(r)1(:)-441(ILU\050)]TJ/F24 10.9091 Tf 87.8474 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)-327(MILU\050)]TJ/F24 10.9091 Tf 44.0257 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)-327(ILU\050)]TJ/F24 10.9091 Tf 34.0257 0 Td[(p;)-167(t)]TJ/F15 10.9091 Tf 14.2765 0 Td[(\051,)]TJ -191.1525 -13.5492 Td[(LU)-411(fr)1(om)-412(UM)1(FP)84(A)27(CK,)-411(LU)-411(f)1(rom)-411(Sup)-27(erLU)]TJ 0 -13.5492 Td[(\050p)1(lus)-334(t)1(rian)1(gular)-333(solv)28(e\051.)]TJ ET -1 0 0 1 916.9977 699.5576 cm +1 0 0 1 913.0827 692.7831 cm q []0 d 0 J 0.3985 w 0.1992 0 m -0.1992 54.1968 l +0.1992 67.746 l +S +Q +1 0 0 1 -612.9393 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +613.1385 0.1992 l S Q -1 0 0 1 -596.1309 -27.0983 cm +1 0 0 1 -0.1993 -27.0984 cm q []0 d 0 J @@ -5420,11 +5596,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -320.8668 -672.4593 cm +1 0 0 1 -299.9441 -665.2862 cm BT -/F48 10.9091 Tf 327.0436 690.0733 Td[(mld_sub_fillin_)]TJ +/F48 10.9091 Tf 306.121 682.9002 Td[(mld_sub_fillin_)]TJ ET -1 0 0 1 418.7301 672.4593 cm +1 0 0 1 397.8075 665.2862 cm q []0 d 0 J @@ -5433,11 +5609,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -418.7301 -672.4593 cm +1 0 0 1 -397.8075 -665.2862 cm BT -/F48 10.9091 Tf 424.9069 690.0733 Td[(integer)]TJ +/F48 10.9091 Tf 403.9843 682.9002 Td[(integer)]TJ ET -1 0 0 1 537.7035 672.4593 cm +1 0 0 1 516.7808 665.2862 cm q []0 d 0 J @@ -5446,11 +5622,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -537.7035 -672.4593 cm +1 0 0 1 -516.7808 -665.2862 cm BT -/F15 10.9091 Tf 543.8803 690.0733 Td[(an)28(y)-2225(in)29(te)-1(ger)]TJ 0 -13.5492 Td[(n)28(u)1(m)27(b)-27(e)-1(r)]TJ/F27 10.9091 Tf 39.7273 0 Td[(\025)]TJ/F15 10.9091 Tf 11.5151 0 Td[(0)]TJ +/F15 10.9091 Tf 522.9577 682.9002 Td[(An)28(y)-333(in)29(t.)-333(n)28(um.)]TJ/F27 10.9091 Tf 70.6061 0 Td[(\025)]TJ/F15 10.9091 Tf 11.5151 0 Td[(0)]TJ ET -1 0 0 1 623.3596 672.4593 cm +1 0 0 1 619.4446 665.2862 cm q []0 d 0 J @@ -5459,11 +5635,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -623.3596 -672.4593 cm +1 0 0 1 -619.4446 -665.2862 cm BT -/F15 10.9091 Tf 629.5364 690.0733 Td[(0)]TJ +/F15 10.9091 Tf 625.6214 682.9002 Td[(0)]TJ ET -1 0 0 1 706.6173 672.4593 cm +1 0 0 1 702.7023 665.2862 cm q []0 d 0 J @@ -5472,11 +5648,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -706.6173 -672.4593 cm +1 0 0 1 -702.7023 -665.2862 cm BT -/F15 10.9091 Tf 712.7942 690.0733 Td[(Fil)1(l-in)-276(lev)28(e)-1(l)]TJ/F24 10.9091 Tf 56.1768 0 Td[(p)]TJ/F15 10.9091 Tf 8.5013 0 Td[(of)-276(th)1(e)-277(in)1(c)-1(omp)1(le)-1(te)-276(LU)-276(factor-)]TJ -64.6781 -13.5492 Td[(ization)1(s)-1(.)]TJ +/F15 10.9091 Tf 708.8791 682.9002 Td[(Fil)1(l-in)-276(lev)28(e)-1(l)]TJ/F24 10.9091 Tf 56.1769 0 Td[(p)]TJ/F15 10.9091 Tf 8.5013 0 Td[(of)-276(th)1(e)-277(in)1(c)-1(omp)1(le)-1(te)-276(LU)-276(factor-)]TJ -64.6782 -13.5492 Td[(ization)1(s)-1(.)]TJ ET -1 0 0 1 916.9977 672.4593 cm +1 0 0 1 913.0827 665.2862 cm q []0 d 0 J @@ -5485,7 +5661,16 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -596.1309 -27.0984 cm +1 0 0 1 -612.9393 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +613.1385 0.1992 l +S +Q +1 0 0 1 -0.1993 -27.0984 cm q []0 d 0 J @@ -5494,11 +5679,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -320.8668 -645.3609 cm +1 0 0 1 -299.9441 -637.7893 cm BT -/F48 10.9091 Tf 327.0436 662.9749 Td[(mld_sub_thresh_)]TJ +/F48 10.9091 Tf 306.121 655.4033 Td[(mld_sub_thresh_)]TJ ET -1 0 0 1 418.7301 645.3609 cm +1 0 0 1 397.8075 637.7893 cm q []0 d 0 J @@ -5507,11 +5692,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -418.7301 -645.3609 cm +1 0 0 1 -397.8075 -637.7893 cm BT -/F48 10.9091 Tf 424.9069 662.9749 Td[(real\050)]TJ/F20 10.9091 Tf 28.6361 0 Td[(kind)]TJ +/F48 10.9091 Tf 403.9843 655.4033 Td[(real\050)]TJ/F20 10.9091 Tf 28.6361 0 Td[(kind)]TJ ET -1 0 0 1 474.2846 662.9749 cm +1 0 0 1 453.362 655.4033 cm q []0 d 0 J @@ -5520,11 +5705,11 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -474.2846 -662.9749 cm +1 0 0 1 -453.362 -655.4033 cm BT -/F20 10.9091 Tf 477.6301 662.9749 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5678 0 Td[(\051)]TJ +/F20 10.9091 Tf 456.7075 655.4033 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5678 0 Td[(\051)]TJ ET -1 0 0 1 537.7035 645.3609 cm +1 0 0 1 516.7808 637.7893 cm q []0 d 0 J @@ -5533,11 +5718,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -537.7035 -645.3609 cm +1 0 0 1 -516.7808 -637.7893 cm BT -/F15 10.9091 Tf 543.8803 662.9749 Td[(an)28(y)-668(r)1(e)-1(al)-668(n)28(u)1(m)-1(-)]TJ 0 -13.5492 Td[(b)-27(e)-1(r)]TJ/F27 10.9091 Tf 19.1212 0 Td[(\025)]TJ/F15 10.9091 Tf 11.5151 0 Td[(0)]TJ +/F15 10.9091 Tf 522.9577 655.4033 Td[(An)28(y)-333(real)-333(n)28(u)1(m)-1(.)]TJ/F27 10.9091 Tf 72.1515 0 Td[(\025)]TJ/F15 10.9091 Tf 11.5152 0 Td[(0)]TJ ET -1 0 0 1 623.3596 645.3609 cm +1 0 0 1 619.4446 637.7893 cm q []0 d 0 J @@ -5546,11 +5731,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -623.3596 -645.3609 cm +1 0 0 1 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741.4302 859.4382 Td[(Coarse)-1(st)-483(matrix)1(:)-743(distrib)1(uted)-482(am)-1(on)1(g)-483(the)]TJ 0 -13.5492 Td[(pr)1(o)-28(ce)-1(ss)-1(or)1(s)-334(or)-333(repli)1(c)-1(ated)-333(on)-333(eac)27(h)-333(of)-333(th)1(e)-1(m.)]TJ +/F15 10.9091 Tf 737.5152 860.2352 Td[(Coarse)-1(st)-483(matrix)1(:)-743(distrib)1(uted)-482(am)-1(on)1(g)-483(the)]TJ 0 -13.5492 Td[(pr)1(o)-28(ce)-1(ss)-1(or)1(s)-334(or)-333(repli)1(c)-1(ated)-333(on)-333(eac)27(h)-333(of)-333(th)1(e)-1(m.)]TJ ET -1 0 0 1 945.6337 841.8242 cm +1 0 0 1 941.7187 842.6212 cm q []0 d 0 J @@ -6425,7 +6742,16 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -624.767 -108.3936 cm +1 0 0 1 -641.5753 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +641.7746 0.1992 l +S +Q +1 0 0 1 -0.1993 -108.3936 cm q []0 d 0 J @@ -6434,11 +6760,11 @@ q 0.1992 108.3936 l S Q -1 0 0 1 -320.8667 -733.4306 cm +1 0 0 1 -299.9441 -733.8291 cm BT -/F48 10.9091 Tf 327.0436 832.3398 Td[(mld_coarse_solve_)]TJ +/F48 10.9091 Tf 306.121 832.7383 Td[(mld_coarse_solve_)]TJ ET -1 0 0 1 447.3661 733.4306 cm +1 0 0 1 426.4435 733.8291 cm q []0 d 0 J @@ -6447,11 +6773,11 @@ q 0.1992 108.3936 l S Q -1 0 0 1 -447.3661 -733.4306 cm +1 0 0 1 -426.4435 -733.8291 cm BT -/F48 10.9091 Tf 453.543 832.3398 Td[(character\050len=*\051)]TJ +/F48 10.9091 Tf 432.6204 832.7383 Td[(character\050len=*\051)]TJ ET -1 0 0 1 566.3395 733.4306 cm +1 0 0 1 545.4169 733.8291 cm q []0 d 0 J @@ -6460,11 +6786,11 @@ q 0.1992 108.3936 l S Q -1 0 0 1 -566.3395 -733.4306 cm +1 0 0 1 -545.4169 -733.8291 cm BT -/F48 10.9091 Tf 572.5163 832.3398 Td[('BJAC')]TJ 0 -13.5492 Td[('UMF')]TJ 0 -13.5492 Td[('SLU')]TJ 0 -13.5492 Td[('SLUDIST')]TJ +/F48 10.9091 Tf 551.5937 832.7383 Td[('BJAC')]TJ 0 -13.5492 Td[('UMF')]TJ 0 -13.5492 Td[('SLU')]TJ 0 -13.5492 Td[('SLUDIST')]TJ ET -1 0 0 1 651.9956 733.4306 cm +1 0 0 1 648.0806 733.8291 cm q []0 d 0 J @@ -6473,11 +6799,11 @@ q 0.1992 108.3936 l S Q -1 0 0 1 -651.9956 -733.4306 cm +1 0 0 1 -648.0806 -733.8291 cm BT -/F48 10.9091 Tf 658.1725 832.3398 Td[('BJAC')]TJ +/F48 10.9091 Tf 654.2575 832.7383 Td[('BJAC')]TJ ET -1 0 0 1 735.2534 733.4306 cm +1 0 0 1 731.3384 733.8291 cm q []0 d 0 J @@ -6486,11 +6812,11 @@ q 0.1992 108.3936 l S Q -1 0 0 1 -735.2534 -733.4306 cm +1 0 0 1 -731.3384 -733.8291 cm BT -/F15 10.9091 Tf 741.4302 832.3398 Td[(Sol)1(v)28(e)-1(r)-258(used)-259(at)-258(th)1(e)-259(c)-1(oarses)-1(t)-258(le)-1(v)28(el:)-407(b)1(lo)-28(c)28(k)-259(Ja-)]TJ 0 -13.5492 Td[(cobi,)-365(se)-1(q)1(uen)28(tial)-358(LU)-359(fr)1(om)-359(UMFP)84(A)28(CK,)-359(se)-1(-)]TJ 0 -13.5492 Td[(qu)1(e)-1(n)28(t)1(ial)-284(LU)-285(f)1(rom)-285(S)1(up)-27(e)-1(r)1(LU,)-285(or)-284(d)1(is)-1(tr)1(ibu)1(ted)]TJ 0 -13.5492 Td[(LU)-494(f)1(rom)-494(Su)1(p)-28(erLU)]TJ +/F15 10.9091 Tf 737.5152 832.7383 Td[(Sol)1(v)28(e)-1(r)-503(use)-1(d)-503(at)-504(the)-504(c)-1(oar)1(s)-1(es)-1(t)-503(le)-1(v)28(el:)-785(blo)-28(c)28(k)]TJ 0 -13.5492 Td[(Jacobi,)-517(se)-1(q)1(uen)28(tial)-480(LU)-480(from)-480(UMFP)84(A)28(CK,)]TJ 0 -13.5492 Td[(se)-1(qu)1(e)-1(n)29(tial)-369(LU)-368(from)-369(S)1(up)-27(e)-1(r)1(LU,)-369(d)1(is)-1(tr)1(ibu)1(ted)]TJ 0 -13.5492 Td[(LU)-494(f)1(rom)-494(Su)1(p)-28(erLU)]TJ ET -1 0 0 1 832.6154 791.6922 cm +1 0 0 1 828.7004 792.0907 cm q []0 d 0 J @@ -6499,11 +6825,11 @@ q 3.2728 0.1992 l S Q -1 0 0 1 -832.6154 -791.6922 cm +1 0 0 1 -828.7004 -792.0907 cm BT -/F15 10.9091 Tf 835.8882 791.6922 Td[(Dist.)-926(If)-493(the)-494(coars)-1(est)]TJ -94.458 -13.5492 Td[(matrix)-567(is)-568(distri)1(bu)1(te)-1(d)1(,)-627(on)1(ly)]TJ/F48 10.9091 Tf 140.2947 0 Td[('BJAC')]TJ/F15 10.9091 Tf 40.5547 0 Td[(and)]TJ/F48 10.9091 Tf -180.8494 -13.5492 Td[('SLUDIST')]TJ/F15 10.9091 Tf 57.0061 0 Td[(can)-501(b)-27(e)-501(c)27(hosen;)-584(if)-500(it)-501(is)-501(repl)1(i-)]TJ -57.0061 -13.5492 Td[(cate)-1(d)1(,)-368(onl)1(y)]TJ/F20 10.9091 Tf 56.739 0 Td[('BJA)26(C')]TJ/F15 10.9091 Tf 41.0251 0 Td[(or)]TJ/F48 10.9091 Tf 13.6651 0 Td[('SLUDIST')]TJ/F15 10.9091 Tf 55.4824 0 Td[(can)-361(b)-27(e)]TJ -166.9116 -13.5492 Td[(se)-1(lec)-1(t)1(e)-1(d)1(.)]TJ +/F15 10.9091 Tf 831.9731 792.0907 Td[(Dist.)-926(If)-493(the)-494(coars)-1(est)]TJ -94.4579 -13.5492 Td[(matrix)-567(is)-568(distri)1(bu)1(te)-1(d)1(,)-627(on)1(ly)]TJ/F48 10.9091 Tf 140.2946 0 Td[('BJAC')]TJ/F15 10.9091 Tf 40.5548 0 Td[(and)]TJ/F48 10.9091 Tf -180.8494 -13.5492 Td[('SLUDIST')]TJ/F15 10.9091 Tf 57.0061 0 Td[(can)-501(b)-27(e)-501(c)27(hosen;)-584(if)-500(it)-501(is)-501(repl)1(i-)]TJ -57.0061 -13.5492 Td[(cate)-1(d)1(,)-415(only)]TJ/F48 10.9091 Tf 57.817 0 Td[('BJAC')]TJ/F15 10.9091 Tf 38.7121 0 Td[(or)]TJ/F48 10.9091 Tf 14.0769 0 Td[('SLUDIST')]TJ/F15 10.9091 Tf 55.8938 0 Td[(can)-399(b)-27(e)]TJ -166.4998 -13.5492 Td[(se)-1(lec)-1(t)1(e)-1(d)1(.)]TJ ET -1 0 0 1 945.6337 733.4306 cm +1 0 0 1 941.7187 733.8291 cm q []0 d 0 J @@ -6512,7 +6838,16 @@ q 0.1992 108.3936 l S Q -1 0 0 1 -624.767 -67.746 cm +1 0 0 1 -641.5753 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +641.7746 0.1992 l +S +Q +1 0 0 1 -0.1993 -67.746 cm q []0 d 0 J @@ -6521,11 +6856,11 @@ q 0.1992 67.746 l S Q -1 0 0 1 -320.8667 -665.6846 cm +1 0 0 1 -299.9441 -665.6846 cm BT -/F48 10.9091 Tf 327.0436 723.9462 Td[(mld_coarse_subsolve_)]TJ +/F48 10.9091 Tf 306.121 723.9462 Td[(mld_coarse_subsolve_)]TJ ET -1 0 0 1 447.3661 665.6847 cm +1 0 0 1 426.4435 665.6847 cm q []0 d 0 J @@ -6534,11 +6869,11 @@ q 0.1992 67.746 l S Q -1 0 0 1 -447.3661 -665.6847 cm +1 0 0 1 -426.4435 -665.6847 cm BT -/F48 10.9091 Tf 453.543 723.9462 Td[(character\050len=*\051)]TJ +/F48 10.9091 Tf 432.6204 723.9462 Td[(character\050len=*\051)]TJ ET -1 0 0 1 566.3395 665.6847 cm +1 0 0 1 545.4169 665.6847 cm q []0 d 0 J @@ -6547,11 +6882,11 @@ q 0.1992 67.746 l S Q -1 0 0 1 -566.3395 -665.6847 cm +1 0 0 1 -545.4169 -665.6847 cm BT -/F48 10.9091 Tf 572.5163 723.9462 Td[('ILU')]TJ 0 -13.5492 Td[('MILU')]TJ 0 -13.5492 Td[('ILUT')]TJ 0 -13.5492 Td[('UMF')]TJ 0 -13.5492 Td[('SLU')]TJ +/F48 10.9091 Tf 551.5937 723.9462 Td[('ILU')]TJ 0 -13.5492 Td[('MILU')]TJ 0 -13.5492 Td[('ILUT')]TJ 0 -13.5492 Td[('UMF')]TJ 0 -13.5492 Td[('SLU')]TJ ET -1 0 0 1 651.9956 665.6847 cm +1 0 0 1 648.0806 665.6847 cm q []0 d 0 J @@ -6560,11 +6895,11 @@ q 0.1992 67.746 l S Q -1 0 0 1 -651.9956 -665.6847 cm +1 0 0 1 -648.0806 -665.6847 cm BT -/F48 10.9091 Tf 658.1725 723.9462 Td[('UMF')]TJ +/F48 10.9091 Tf 654.2575 723.9462 Td[('UMF')]TJ ET -1 0 0 1 735.2534 665.6847 cm +1 0 0 1 731.3384 665.6847 cm q []0 d 0 J @@ -6573,11 +6908,11 @@ q 0.1992 67.746 l S Q -1 0 0 1 -735.2534 -665.6847 cm +1 0 0 1 -731.3384 -665.6847 cm BT -/F15 10.9091 Tf 741.4302 723.9462 Td[(Sol)1(v)28(e)-1(r)-242(f)1(or)-243(th)1(e)-243(di)1(agonal)-242(bl)1(o)-28(c)27(ks)-243(of)-242(th)1(e)-243(coars)-1(e)]TJ 0 -13.5492 Td[(matrix)1(,)-610(i)1(n)-554(c)-1(ase)-555(th)1(e)-555(bl)1(o)-28(c)27(k)-554(Jacobi)-554(solv)28(er)]TJ 0 -13.5492 Td[(is)-360(c)27(hosen)-360(as)-361(coarse)-1(st-le)-1(v)28(el)-360(solv)28(e)-1(r)1(:)-498(ILU\050)]TJ/F24 10.9091 Tf 185.6638 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)]TJ -191.1524 -13.5492 Td[(MILU\050)]TJ/F24 10.9091 Tf 33.1819 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)-271(ILU\050)]TJ/F24 10.9091 Tf 33.4162 0 Td[(p;)-167(t)]TJ/F15 10.9091 Tf 14.2765 0 Td[(\051,)-271(LU)-256(fr)1(om)-257(UM)1(FP)84(A)27(CK,)]TJ -86.3632 -13.5492 Td[(LU)-333(from)-333(Su)1(p)-28(erLU,)-333(p)1(lus)-334(t)1(rian)1(gular)-333(solv)28(e.)]TJ +/F15 10.9091 Tf 737.5152 723.9462 Td[(Sol)1(v)28(e)-1(r)-242(f)1(or)-243(th)1(e)-243(di)1(agonal)-242(bl)1(o)-28(c)27(ks)-243(of)-242(th)1(e)-243(coars)-1(e)]TJ 0 -13.5492 Td[(matrix)1(,)-610(i)1(n)-554(c)-1(ase)-555(th)1(e)-555(bl)1(o)-28(c)27(k)-554(Jacobi)-554(solv)28(er)]TJ 0 -13.5492 Td[(is)-360(c)27(hosen)-360(as)-361(coarse)-1(st-le)-1(v)28(el)-360(solv)28(e)-1(r)1(:)-498(ILU\050)]TJ/F24 10.9091 Tf 185.6638 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)]TJ -191.1524 -13.5492 Td[(MILU\050)]TJ/F24 10.9091 Tf 33.1819 0 Td[(p)]TJ/F15 10.9091 Tf 5.4886 0 Td[(\051,)-271(ILU\050)]TJ/F24 10.9091 Tf 33.4162 0 Td[(p;)-167(t)]TJ/F15 10.9091 Tf 14.2765 0 Td[(\051,)-271(LU)-256(fr)1(om)-257(UM)1(FP)84(A)27(CK,)]TJ -86.3632 -13.5492 Td[(LU)-333(from)-333(Su)1(p)-28(erLU,)-333(p)1(lus)-334(t)1(rian)1(gular)-333(solv)28(e.)]TJ ET -1 0 0 1 945.6337 665.6847 cm +1 0 0 1 941.7187 665.6847 cm q []0 d 0 J @@ -6586,7 +6921,16 @@ q 0.1992 67.746 l S Q -1 0 0 1 -624.767 -27.0984 cm +1 0 0 1 -641.5753 -0.3985 cm +q +[]0 d +0 J +0.3985 w +0 0.1992 m +641.7746 0.1992 l +S +Q +1 0 0 1 -0.1993 -27.0984 cm q []0 d 0 J @@ -6595,11 +6939,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -320.8667 -638.5863 cm +1 0 0 1 -299.9441 -638.1878 cm BT -/F48 10.9091 Tf 327.0436 656.2003 Td[(mld_coarse_sweeps_)]TJ +/F48 10.9091 Tf 306.121 655.8018 Td[(mld_coarse_sweeps_)]TJ ET -1 0 0 1 447.3661 638.5863 cm +1 0 0 1 426.4435 638.1878 cm q []0 d 0 J @@ -6608,11 +6952,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -447.3661 -638.5863 cm +1 0 0 1 -426.4435 -638.1878 cm BT -/F48 10.9091 Tf 453.543 656.2003 Td[(integer)]TJ +/F48 10.9091 Tf 432.6204 655.8018 Td[(integer)]TJ ET -1 0 0 1 566.3395 638.5863 cm +1 0 0 1 545.4169 638.1878 cm q []0 d 0 J @@ -6621,11 +6965,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -566.3395 -638.5863 cm +1 0 0 1 -545.4169 -638.1878 cm BT -/F15 10.9091 Tf 572.5163 656.2003 Td[(an)28(y)-2225(in)29(te)-1(ger)]TJ 0 -13.5492 Td[(n)28(u)1(m)27(b)-27(e)-1(r)]TJ/F24 10.9091 Tf 39.7273 0 Td[(>)]TJ/F15 10.9091 Tf 11.5152 0 Td[(0)]TJ +/F15 10.9091 Tf 551.5937 655.8018 Td[(An)28(y)-333(in)29(t.)-333(n)28(um.)]TJ/F24 10.9091 Tf 70.6061 0 Td[(>)]TJ/F15 10.9091 Tf 11.5151 0 Td[(0)]TJ ET -1 0 0 1 651.9956 638.5863 cm +1 0 0 1 648.0806 638.1878 cm q []0 d 0 J @@ -6634,11 +6978,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -651.9956 -638.5863 cm +1 0 0 1 -648.0806 -638.1878 cm BT -/F48 10.9091 Tf 658.1725 656.2003 Td[(4)]TJ +/F48 10.9091 Tf 654.2575 655.8018 Td[(4)]TJ ET -1 0 0 1 735.2534 638.5863 cm +1 0 0 1 731.3384 638.1878 cm q []0 d 0 J @@ -6647,11 +6991,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -735.2534 -638.5863 cm +1 0 0 1 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611.4879 cm +1 0 0 1 426.4435 611.0894 cm q []0 d 0 J @@ -6682,11 +7026,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -447.3661 -611.4879 cm +1 0 0 1 -426.4435 -611.0894 cm BT -/F48 10.9091 Tf 453.543 629.1019 Td[(integer)]TJ +/F48 10.9091 Tf 432.6203 628.7034 Td[(integer)]TJ ET -1 0 0 1 566.3395 611.4879 cm +1 0 0 1 545.4169 611.0894 cm q []0 d 0 J @@ -6695,11 +7039,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -566.3395 -611.4879 cm +1 0 0 1 -545.4169 -611.0894 cm BT -/F15 10.9091 Tf 572.5163 629.1019 Td[(an)28(y)-2225(in)29(te)-1(ger)]TJ 0 -13.5492 Td[(n)28(u)1(m)27(b)-27(e)-1(r)]TJ/F27 10.9091 Tf 39.7273 0 Td[(\025)]TJ/F15 10.9091 Tf 11.5152 0 Td[(0)]TJ +/F15 10.9091 Tf 551.5937 628.7034 Td[(An)28(y)-333(in)29(t.)-333(n)28(um.)]TJ/F27 10.9091 Tf 70.6061 0 Td[(\025)]TJ/F15 10.9091 Tf 11.5151 0 Td[(0)]TJ ET -1 0 0 1 651.9956 611.4879 cm +1 0 0 1 648.0806 611.0894 cm q []0 d 0 J @@ -6708,11 +7052,11 @@ q 0.1992 27.0984 l S Q -1 0 0 1 -651.9956 -611.4879 cm +1 0 0 1 -648.0806 -611.0894 cm BT -/F48 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J @@ -7046,11 +7400,11 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -239.9328 -472.1815 cm +1 0 0 1 -192.1122 -490.9817 cm BT -/F20 10.9091 Tf 243.2783 472.1815 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5678 0 Td[(\051,)-525(dimension\050:\051,)-525(intent\050in\051)]TJ/F15 10.9091 Tf 154.6347 0 Td[(.)]TJ -252.7939 -13.5492 Td[(The)-767(lo)-27(c)-1(al)-766(par)1(t)-767(of)-766(th)1(e)-767(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 164.1766 0 Td[(x)]TJ/F15 10.9091 Tf 6.2348 0 Td[(.)-1744(Note)-767(th)1(at)]TJ/F20 10.9091 Tf 81.5107 0 Td[(typ)51(e)]TJ/F15 10.9091 Tf 28.1399 0 Td[(and)]TJ/F20 10.9091 Tf -280.062 -13.5492 Td[(kind)]TJ +/F20 10.9091 Tf 195.4576 490.9817 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5679 0 Td[(\051,)-525(dimension\050:\051,)-525(intent\050in\051)]TJ/F15 10.9091 Tf 154.6347 0 Td[(.)]TJ -252.794 -13.5492 Td[(The)-767(lo)-27(c)-1(al)-766(par)1(t)-767(of)-766(th)1(e)-767(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 164.1766 0 Td[(x)]TJ/F15 10.9091 Tf 6.2349 0 Td[(.)-1744(Note)-767(th)1(at)]TJ/F20 10.9091 Tf 81.5107 0 Td[(typ)51(e)]TJ/F15 10.9091 Tf 28.1399 0 Td[(and)]TJ/F20 10.9091 Tf -280.0621 -13.5492 Td[(kind)]TJ ET -1 0 0 1 214.4285 445.0831 cm +1 0 0 1 166.6079 463.8833 cm q []0 d 0 J @@ -7059,11 +7413,11 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -214.4285 -445.0831 cm +1 0 0 1 -166.6079 -463.8833 cm BT -/F20 10.9091 Tf 217.774 445.0831 Td[(p)51(ar)51(ameter)]TJ/F15 10.9091 Tf 51.9376 0 Td[(m)27(u)1(s)-1(t)-308(b)-28(e)-309(c)27(h)1(os)-1(en)-309(accordi)1(ng)-309(to)-309(th)1(e)-310(r)1(e)-1(al)1(/c)-1(omplex,)]TJ -76.0247 -13.5492 Td[(single/dou)1(bl)1(e)-334(pr)1(e)-1(cis)-1(i)1(on)-333(v)28(e)-1(r)1(s)-1(ion)-332(of)-333(MLD2P)1(4)-334(u)1(nd)1(e)-1(r)-333(u)1(s)-1(e.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(y)]TJ/F20 10.9091 Tf 45.9708 0 Td[(typ)51(e)]TJ/F48 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ +/F20 10.9091 Tf 169.9533 463.8833 Td[(p)51(ar)51(ameter)]TJ/F15 10.9091 Tf 51.9376 0 Td[(m)27(u)1(s)-1(t)-308(b)-28(e)-309(c)27(h)1(os)-1(en)-309(accordi)1(ng)-309(to)-309(th)1(e)-310(r)1(e)-1(al)1(/c)-1(omplex,)]TJ -76.0247 -13.5492 Td[(single/dou)1(bl)1(e)-334(pr)1(e)-1(cis)-1(i)1(on)-333(v)28(e)-1(r)1(s)-1(ion)-332(of)-333(MLD2P)1(4)-334(u)1(nd)1(e)-1(r)-333(u)1(s)-1(e.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(y)]TJ/F20 10.9091 Tf 45.9708 0 Td[(typ)51(e)]TJ/F48 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ ET -1 0 0 1 239.9328 417.9847 cm +1 0 0 1 192.1122 436.7849 cm q []0 d 0 J @@ -7072,11 +7426,11 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -239.9328 -417.9847 cm +1 0 0 1 -192.1122 -436.7849 cm BT -/F20 10.9091 Tf 243.2783 417.9847 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5678 0 Td[(\051,)-525(dimension\050:\051,)-525(intent\050out\051)]TJ/F15 10.9091 Tf 160.3619 0 Td[(.)]TJ -258.5211 -13.5492 Td[(The)-771(l)1(o)-28(c)-1(al)-770(p)1(art)-770(of)-770(the)-770(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 164.424 0 Td[(y)]TJ/F15 10.9091 Tf 5.7399 0 Td[(.)-1755(Note)-771(th)1(at)]TJ/F20 10.9091 Tf 81.717 0 Td[(typ)51(e)]TJ/F15 10.9091 Tf 28.1811 0 Td[(and)]TJ/F20 10.9091 Tf -280.062 -13.5492 Td[(kind)]TJ +/F20 10.9091 Tf 195.4576 436.7849 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5679 0 Td[(\051,)-525(dimension\050:\051,)-525(intent\050out\051)]TJ/F15 10.9091 Tf 160.3619 0 Td[(.)]TJ -258.5212 -13.5492 Td[(The)-771(l)1(o)-28(c)-1(al)-770(p)1(art)-770(of)-770(the)-770(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 164.4241 0 Td[(y)]TJ/F15 10.9091 Tf 5.7399 0 Td[(.)-1755(Note)-771(th)1(at)]TJ/F20 10.9091 Tf 81.717 0 Td[(typ)51(e)]TJ/F15 10.9091 Tf 28.181 0 Td[(and)]TJ/F20 10.9091 Tf -280.062 -13.5492 Td[(kind)]TJ ET -1 0 0 1 214.4285 390.8863 cm +1 0 0 1 166.6079 409.6865 cm q []0 d 0 J @@ -7085,23 +7439,23 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -214.4285 -390.8863 cm +1 0 0 1 -166.6079 -409.6865 cm BT -/F20 10.9091 Tf 217.774 390.8863 Td[(p)51(ar)51(ameter)]TJ/F15 10.9091 Tf 51.9376 0 Td[(m)27(u)1(s)-1(t)-308(b)-28(e)-309(c)27(h)1(os)-1(en)-309(accordi)1(ng)-309(to)-309(th)1(e)-310(r)1(e)-1(al)1(/c)-1(omplex,)]TJ -76.0247 -13.5492 Td[(single/dou)1(bl)1(e)-334(pr)1(e)-1(cis)-1(i)1(on)-333(v)28(e)-1(r)1(s)-1(ion)-332(of)-333(MLD2P)1(4)-334(u)1(nd)1(e)-1(r)-333(u)1(s)-1(e.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(desc_a)-1064(type\050psb_desc_type\051,)-525(intent\050in\051)]TJ/F15 10.9091 Tf 223.5143 0 Td[(.)]TJ -177.5435 -13.5492 Td[(The)-380(com)-1(m)28(un)1(ication)-379(des)-1(crip)1(tor)-379(as)-1(so)-28(ciated)-379(to)-380(th)1(e)-380(m)-1(atr)1(ix)-379(to)-380(b)-27(e)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(d)1(.)]TJ/F48 10.9091 Tf -45.9708 -13.5491 Td[(info)-2114(integer,)-525(intent\050out\051)]TJ/F15 10.9091 Tf 160.515 0 Td[(.)]TJ -114.5442 -13.5492 Td[(Er)1(ror)-307(c)-1(o)-28(d)1(e)-1(.)-435(If)-308(n)1(o)-308(e)-1(r)1(ror,)-313(0)-307(is)-308(re)-1(t)1(urn)1(e)-1(d)1(.)-436(See)-308(Sec)-1(ti)1(on)]TJ +/F20 10.9091 Tf 169.9533 409.6865 Td[(p)51(ar)51(ameter)]TJ/F15 10.9091 Tf 51.9376 0 Td[(m)27(u)1(s)-1(t)-308(b)-28(e)-309(c)27(h)1(os)-1(en)-309(accordi)1(ng)-309(to)-309(th)1(e)-310(r)1(e)-1(al)1(/c)-1(omplex,)]TJ -76.0247 -13.5492 Td[(single/dou)1(bl)1(e)-334(pr)1(e)-1(cis)-1(i)1(on)-333(v)28(e)-1(r)1(s)-1(ion)-332(of)-333(MLD2P)1(4)-334(u)1(nd)1(e)-1(r)-333(u)1(s)-1(e.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(desc_a)-1064(type\050psb_desc_type\051,)-525(intent\050in\051)]TJ/F15 10.9091 Tf 223.5143 0 Td[(.)]TJ -177.5435 -13.5491 Td[(The)-380(com)-1(m)28(un)1(ication)-379(des)-1(crip)1(tor)-379(as)-1(so)-28(ciated)-379(to)-380(th)1(e)-380(m)-1(atr)1(ix)-379(to)-380(b)-27(e)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(d)1(.)]TJ/F48 10.9091 Tf -45.9708 -13.5492 Td[(info)-2114(integer,)-525(intent\050out\051)]TJ/F15 10.9091 Tf 160.515 0 Td[(.)]TJ -114.5442 -13.5492 Td[(Er)1(ror)-307(c)-1(o)-28(d)1(e)-1(.)-435(If)-308(n)1(o)-308(e)-1(r)1(ror,)-313(0)-307(is)-308(re)-1(t)1(urn)1(e)-1(d)1(.)-436(See)-308(Sec)-1(ti)1(on)]TJ ET -1 0 0 1 432.0924 309.5912 cm +1 0 0 1 384.2717 328.3914 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -432.0924 -309.5912 cm +1 0 0 1 -384.2717 -328.3914 cm BT -/F15 10.9091 Tf 432.0924 309.5912 Td[(8)]TJ +/F15 10.9091 Tf 384.2717 328.3914 Td[(8)]TJ ET -1 0 0 1 437.547 309.5912 cm +1 0 0 1 389.7263 328.3914 cm 0 g 0 G -1 0 0 1 -437.547 -309.5912 cm +1 0 0 1 -389.7263 -328.3914 cm BT -/F15 10.9091 Tf 440.9055 309.5912 Td[(for)-307(details.)]TJ/F48 10.9091 Tf -293.1894 -13.5492 Td[(trans)-1589(character\050len=1\051,)-525(optional,)-525(intent\050in\051.)]TJ/F15 10.9091 Tf 45.9708 -13.5492 Td[(If)]TJ/F48 10.9091 Tf 10.0574 0 Td[(trans)]TJ/F15 10.9091 Tf 31.4208 0 Td[(=)]TJ/F48 10.9091 Tf 11.2696 0 Td[('N','n')]TJ/F15 10.9091 Tf 42.8752 0 Td[(th)1(e)-1(n)]TJ/F24 10.9091 Tf 23.9968 0 Td[(op)]TJ/F15 10.9091 Tf 10.7766 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051)-278(=)]TJ/F24 10.9091 Tf 18.7878 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(;)-281(if)]TJ/F48 10.9091 Tf 15.2473 0 Td[(trans)]TJ/F15 10.9091 Tf 31.4208 0 Td[(=)]TJ/F48 10.9091 Tf 11.2696 0 Td[('T','t')]TJ/F15 10.9091 Tf -257.5474 -13.5492 Td[(th)1(e)-1(n)]TJ/F24 10.9091 Tf 23.9616 0 Td[(op)]TJ/F15 10.9091 Tf 10.7765 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051)-278(=)]TJ/F24 10.9091 Tf 18.7878 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(T)]TJ/F15 10.9091 Tf 9.3539 -3.9588 Td[(\050tr)1(ansp)-28(ose)-253(of)]TJ/F24 10.9091 Tf 63.8322 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051;)-279(if)]TJ/F48 10.9091 Tf 19.4309 0 Td[(trans)]TJ/F15 10.9091 Tf 31.3854 0 Td[(=)]TJ/F48 10.9091 Tf 11.2343 0 Td[('C','c')]TJ/F15 10.9091 Tf -257.5473 -13.5492 Td[(th)1(e)-1(n)]TJ/F24 10.9091 Tf 24.8485 0 Td[(op)]TJ/F15 10.9091 Tf 10.7765 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051)-278(=)]TJ/F24 10.9091 Tf 18.7878 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(C)]TJ/F15 10.9091 Tf 10.7332 -3.9588 Td[(\050conj)1(ugate)-334(tr)1(ansp)-28(ose)-334(of)]TJ/F24 10.9091 Tf 115.0001 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051.)]TJ/F48 10.9091 Tf -294.9017 -13.5492 Td[(work)]TJ/F20 10.9091 Tf 45.9708 0 Td[(typ)51(e)]TJ/F48 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ +/F15 10.9091 Tf 393.0848 328.3914 Td[(for)-307(details.)]TJ/F48 10.9091 Tf -293.1894 -13.5492 Td[(trans)-1589(character\050len=1\051,)-525(optional,)-525(intent\050in\051.)]TJ/F15 10.9091 Tf 45.9708 -13.5492 Td[(If)]TJ/F48 10.9091 Tf 10.0575 0 Td[(trans)]TJ/F15 10.9091 Tf 31.4207 0 Td[(=)]TJ/F48 10.9091 Tf 11.2696 0 Td[('N','n')]TJ/F15 10.9091 Tf 42.8752 0 Td[(th)1(e)-1(n)]TJ/F24 10.9091 Tf 23.9969 0 Td[(op)]TJ/F15 10.9091 Tf 10.7765 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9589 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051)-278(=)]TJ/F24 10.9091 Tf 18.7878 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(;)-281(if)]TJ/F48 10.9091 Tf 15.2472 0 Td[(trans)]TJ/F15 10.9091 Tf 31.4208 0 Td[(=)]TJ/F48 10.9091 Tf 11.2696 0 Td[('T','t')]TJ/F15 10.9091 Tf -257.5473 -13.5492 Td[(th)1(e)-1(n)]TJ/F24 10.9091 Tf 23.9615 0 Td[(op)]TJ/F15 10.9091 Tf 10.7765 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051)-278(=)]TJ/F24 10.9091 Tf 18.7878 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(T)]TJ/F15 10.9091 Tf 9.3539 -3.9588 Td[(\050tr)1(ansp)-28(ose)-253(of)]TJ/F24 10.9091 Tf 63.8322 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051;)-279(if)]TJ/F48 10.9091 Tf 19.4309 0 Td[(trans)]TJ/F15 10.9091 Tf 31.3854 0 Td[(=)]TJ/F48 10.9091 Tf 11.2343 0 Td[('C','c')]TJ/F15 10.9091 Tf -257.5473 -13.5492 Td[(th)1(e)-1(n)]TJ/F24 10.9091 Tf 24.8485 0 Td[(op)]TJ/F15 10.9091 Tf 10.7765 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051)-278(=)]TJ/F24 10.9091 Tf 18.7878 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(C)]TJ/F15 10.9091 Tf 10.7332 -3.9588 Td[(\050conj)1(ugate)-334(tr)1(ansp)-28(ose)-334(of)]TJ/F24 10.9091 Tf 115.0001 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F15 10.9091 Tf 4.7323 -3.9588 Td[(\051.)]TJ/F48 10.9091 Tf -294.9017 -13.5492 Td[(work)]TJ/F20 10.9091 Tf 45.9708 0 Td[(typ)51(e)]TJ/F48 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ ET -1 0 0 1 239.9328 241.8453 cm +1 0 0 1 192.1122 260.6455 cm q []0 d 0 J @@ -7110,11 +7464,11 @@ q 3.3455 0.1992 l S Q -1 0 0 1 -239.9328 -241.8453 cm +1 0 0 1 -192.1122 -260.6455 cm BT -/F20 10.9091 Tf 243.2783 241.8453 Td[(p)51(ar)51(ameter)]TJ/F48 10.9091 Tf 48.5678 0 Td[(\051,)-525(dimension\050:\051,)-525(optional,)-525(target)]TJ/F15 10.9091 Tf 188.998 0 Td[(.)]TJ -287.1572 -13.5492 Td[(W)84(orkspace)-1(.)-4466(Its)-1674(s)-1(ize)-1674(s)-1(h)1(ould)-1673(b)-28(e)-1674(at)-1674(leas)-1(t)]TJ/F48 10.9091 Tf 0 -13.5492 Td[(4)-525(*)-525(psb_cd_get_local_cols\050desc_a\051)]TJ/F15 10.9091 Tf 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/Descent -194 -/FontName /TJZGDX+CMR8 +/FontName /UNEXYQ+CMR8 /ItalicAngle 0 /StemV 76 /XHeight 431 /FontBBox [-36 -250 1070 750] /Flags 4 /CharSet (/zero/one/two/equal) -/FontFile 264 0 R +/FontFile 272 0 R >> endobj -514 0 obj +522 0 obj [531 531 531 0 0 0 0 0 0 0 0 0 0 826 ] endobj -513 0 obj << +521 0 obj << /Type /Encoding /Differences [ 0 /.notdef 48/zero/one/two 51/.notdef 61/equal 62/.notdef] >> endobj -261 0 obj << +269 0 obj << /Length1 765 /Length2 745 /Length3 532 @@ -8305,7 +8699,7 @@ stream /ItalicAngle -14.035 def /isFixedPitch false def end readonly def -/FontName /JGTGUW+CMSY8 def +/FontName /ITKOVH+CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -8328,37 +8722,37 @@ currentfile eexec cleartomark endstream endobj -262 0 obj << +270 0 obj << /Type /Font /Subtype /Type1 -/Encoding 515 0 R +/Encoding 523 0 R /FirstChar 0 /LastChar 2 -/Widths 516 0 R -/BaseFont /JGTGUW+CMSY8 -/FontDescriptor 260 0 R +/Widths 524 0 R +/BaseFont /ITKOVH+CMSY8 +/FontDescriptor 268 0 R >> endobj -260 0 obj << +268 0 obj << /Ascent 750 /CapHeight 683 /Descent -194 -/FontName /JGTGUW+CMSY8 +/FontName /ITKOVH+CMSY8 /ItalicAngle -14.035 /StemV 89 /XHeight 431 /FontBBox [-30 -955 1185 779] /Flags 4 /CharSet (/minus/multiply) -/FontFile 261 0 R +/FontFile 269 0 R >> endobj -516 0 obj +524 0 obj [826 0 826 ] endobj -515 0 obj << +523 0 obj << /Type /Encoding /Differences [ 0 /minus 1/.notdef 2/multiply 3/.notdef] >> endobj -258 0 obj << +266 0 obj << /Length1 994 /Length2 4891 /Length3 532 @@ -8378,7 +8772,7 @@ stream /ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /TTYZPK+CMMI8 def +/FontName /OEERJT+CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -8429,37 +8823,37 @@ d cleartomark endstream endobj -259 0 obj << +267 0 obj << /Type /Font /Subtype /Type1 -/Encoding 517 0 R +/Encoding 525 0 R /FirstChar 14 /LastChar 115 -/Widths 518 0 R -/BaseFont /TTYZPK+CMMI8 -/FontDescriptor 257 0 R +/Widths 526 0 R +/BaseFont /OEERJT+CMMI8 +/FontDescriptor 265 0 R >> endobj -257 0 obj << +265 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /TTYZPK+CMMI8 +/FontName /OEERJT+CMMI8 /ItalicAngle -14.04 /StemV 78 /XHeight 431 /FontBBox [-24 -250 1110 750] /Flags 4 /CharSet (/delta/A/C/E/H/L/O/P/R/S/T/i/j/l/m/n/r/s) -/FontFile 258 0 R +/FontFile 266 0 R >> endobj -518 0 obj +526 0 obj [470 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 796 0 757 0 779 0 0 873 0 0 0 723 0 0 806 674 0 800 646 619 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 362 430 0 317 940 645 0 0 0 479 491 ] endobj -517 0 obj << +525 0 obj << /Type /Encoding /Differences [ 0 /.notdef 14/delta 15/.notdef 65/A 66/.notdef 67/C 68/.notdef 69/E 70/.notdef 72/H 73/.notdef 76/L 77/.notdef 79/O/P 81/.notdef 82/R/S/T 85/.notdef 105/i/j 107/.notdef 108/l/m/n 111/.notdef 114/r/s 116/.notdef] >> endobj -205 0 obj << +213 0 obj << /Length1 2088 /Length2 13404 /Length3 532 @@ -8479,7 +8873,7 @@ stream /ItalicAngle 0 def /isFixedPitch true def end readonly def -/FontName /XIRIFK+CMTT10 def +/FontName /WWFPJR+CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -8628,33 +9022,33 @@ I cleartomark endstream endobj -206 0 obj << +214 0 obj << /Type /Font /Subtype /Type1 -/Encoding 519 0 R +/Encoding 527 0 R /FirstChar 33 /LastChar 126 -/Widths 520 0 R -/BaseFont /XIRIFK+CMTT10 -/FontDescriptor 204 0 R +/Widths 528 0 R +/BaseFont /WWFPJR+CMTT10 +/FontDescriptor 212 0 R >> endobj -204 0 obj << +212 0 obj << /Ascent 611 /CapHeight 611 /Descent -222 -/FontName /XIRIFK+CMTT10 +/FontName /WWFPJR+CMTT10 /ItalicAngle 0 /StemV 69 /XHeight 431 /FontBBox [-4 -235 731 800] /Flags 4 /CharSet (/exclam/quotedbl/dollar/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/eight/nine/colon/semicolon/less/equal/greater/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/bracketleft/backslash/bracketright/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/asciitilde) -/FontFile 205 0 R +/FontFile 213 0 R >> endobj -520 0 obj +528 0 obj [525 525 0 525 0 0 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 0 0 525 525 525 525 525 525 525 0 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 0 525 525 525 0 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 0 0 0 525 ] endobj -519 0 obj << +527 0 obj << /Type /Encoding /Differences [ 0 /.notdef 33/exclam/quotedbl 35/.notdef 36/dollar 37/.notdef 39/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five 54/.notdef 56/eight/nine/colon/semicolon/less/equal/greater 63/.notdef 64/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y 90/.notdef 91/bracketleft/backslash/bracketright 94/.notdef 95/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z 123/.notdef 126/asciitilde 127/.notdef] >> endobj @@ -8678,7 +9072,7 @@ stream /ItalicAngle -14.035 def /isFixedPitch false def end readonly def -/FontName /DKDPTY+CMSY10 def +/FontName /KJYDVZ+CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -8726,18 +9120,18 @@ endobj 179 0 obj << /Type /Font /Subtype /Type1 -/Encoding 521 0 R +/Encoding 529 0 R /FirstChar 0 /LastChar 110 -/Widths 522 0 R -/BaseFont /DKDPTY+CMSY10 +/Widths 530 0 R +/BaseFont /KJYDVZ+CMSY10 /FontDescriptor 177 0 R >> endobj 177 0 obj << /Ascent 750 /CapHeight 683 /Descent -194 -/FontName /DKDPTY+CMSY10 +/FontName /KJYDVZ+CMSY10 /ItalicAngle -14.035 /StemV 85 /XHeight 431 @@ -8746,10 +9140,10 @@ endobj /CharSet (/minus/bullet/greaterequal/propersubset/propersuperset/element/negationslash/Rfractur/union/braceleft/braceright/bar/backslash) /FontFile 178 0 R >> endobj -522 0 obj +530 0 obj [778 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 0 0 0 0 778 0 0 0 0 778 778 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 667 0 0 0 0 0 0 0 0 0 722 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 667 0 0 0 0 0 0 0 0 0 0 500 500 0 0 278 0 0 0 500 ] endobj -521 0 obj << +529 0 obj << /Type /Encoding /Differences [ 0 /minus 1/.notdef 15/bullet 16/.notdef 21/greaterequal 22/.notdef 26/propersubset/propersuperset 28/.notdef 50/element 51/.notdef 54/negationslash 55/.notdef 60/Rfractur 61/.notdef 91/union 92/.notdef 102/braceleft/braceright 104/.notdef 106/bar 107/.notdef 110/backslash 111/.notdef] >> endobj @@ -8773,7 +9167,7 @@ stream /ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /GABPLB+CMMI10 def +/FontName /OZMEKO+CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -8861,18 +9255,18 @@ endobj 176 0 obj << /Type /Font /Subtype /Type1 -/Encoding 523 0 R +/Encoding 531 0 R /FirstChar 14 /LastChar 122 -/Widths 524 0 R -/BaseFont /GABPLB+CMMI10 +/Widths 532 0 R +/BaseFont /OZMEKO+CMMI10 /FontDescriptor 174 0 R >> endobj 174 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /GABPLB+CMMI10 +/FontName /OZMEKO+CMMI10 /ItalicAngle -14.04 /StemV 72 /XHeight 431 @@ -8881,18 +9275,18 @@ endobj /CharSet (/delta/theta/omega/period/comma/greater/A/D/E/G/I/M/N/P/R/S/V/W/a/b/e/i/j/l/m/n/o/p/r/s/t/v/w/x/y/z) /FontFile 175 0 R >> endobj -524 0 obj +532 0 obj [444 0 0 0 469 0 0 0 0 0 0 0 0 0 0 0 0 0 0 622 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 278 278 0 0 778 0 0 750 0 0 828 738 0 786 0 440 0 0 0 970 803 0 642 0 759 613 0 0 583 944 0 0 0 0 0 0 0 0 0 529 429 0 0 466 0 0 0 345 412 0 298 878 600 485 503 0 451 469 361 0 485 716 572 490 465 ] endobj -523 0 obj << +531 0 obj << /Type /Encoding /Differences [ 0 /.notdef 14/delta 15/.notdef 18/theta 19/.notdef 33/omega 34/.notdef 58/period/comma 60/.notdef 62/greater 63/.notdef 65/A 66/.notdef 68/D/E 70/.notdef 71/G 72/.notdef 73/I 74/.notdef 77/M/N 79/.notdef 80/P 81/.notdef 82/R/S 84/.notdef 86/V/W 88/.notdef 97/a/b 99/.notdef 101/e 102/.notdef 105/i/j 107/.notdef 108/l/m/n/o/p 113/.notdef 114/r/s/t 117/.notdef 118/v/w/x/y/z 123/.notdef] >> endobj 170 0 obj << -/Length1 1707 -/Length2 13986 +/Length1 1693 +/Length2 13827 /Length3 532 -/Length 16225 +/Length 16052 >> stream %!PS-AdobeFont-1.1: CMTI10 1.00B @@ -8908,7 +9302,7 @@ stream /ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /ECPEGS+CMTI10 def +/FontName /UXLKJT+CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -8939,7 +9333,6 @@ dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put -dup 74 /J put dup 75 /K put dup 76 /L put 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00000 n +0000508794 00000 n +0000508833 00000 n +0000508871 00000 n +0000509069 00000 n trailer << -/Size 548 -/Root 546 0 R -/Info 547 0 R -/ID [ ] +/Size 556 +/Root 554 0 R +/Info 555 0 R +/ID [<5A7B9927D85CB4FE76D0D6A40F0536CB> <5A7B9927D85CB4FE76D0D6A40F0536CB>] >> startxref -500988 +509744 %%EOF diff --git a/mlprec/mld_caggrmap_bld.f90 b/mlprec/mld_caggrmap_bld.f90 index 2ae3110e..0ae7477b 100644 --- a/mlprec/mld_caggrmap_bld.f90 +++ b/mlprec/mld_caggrmap_bld.f90 @@ -47,8 +47,11 @@ ! build the coarse-level matrix. ! ! The aggregation algorithm is a parallel version of that described in -! M. Brezina and P. Vanek, A black-box iterative solver based on a +! * M. Brezina and P. Vanek, A black-box iterative solver based on a ! two-level Schwarz method, Computing, 63 (1999), 233-263. +! * P. Vanek, J. Mandel and M. Brezina, Algebraic Multigrid by Smoothed +! Aggregation for Second and Fourth Order Elliptic Problems, Computing, 56 +! (1996), 179-196. ! For more details see ! P. D'Ambra, D. di Serafino and S. Filippone, On the development of ! PSBLAS-based parallel two-level Schwarz preconditioners, Appl. Num. Math. diff --git a/mlprec/mld_cprecset.f90 b/mlprec/mld_cprecset.f90 index 9ac8c29e..14aeecfd 100644 --- a/mlprec/mld_cprecset.f90 +++ b/mlprec/mld_cprecset.f90 @@ -91,7 +91,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv)) then info = 3111 - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' return endif nlev_ = size(p%baseprecv) @@ -110,7 +110,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then info = 3111 write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' return endif @@ -162,13 +162,16 @@ subroutine mld_cprecseti(p,what,val,info,ilev) if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + case(mld_bjac_) + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_sweeps_) @@ -204,7 +207,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) do ilev_=1,max(1,nlev_-1) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -215,7 +218,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) do ilev_=2,nlev_-1 if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -224,7 +227,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) case(mld_coarse_mat_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - & ': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + & ': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -232,28 +235,29 @@ subroutine mld_cprecseti(p,what,val,info,ilev) case(mld_coarse_solve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ case(mld_bjac_) - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_ilu_n_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_subsolve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return end if @@ -262,7 +266,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) case(mld_coarse_sweeps_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -270,7 +274,7 @@ subroutine mld_cprecseti(p,what,val,info,ilev) case(mld_coarse_fillin_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -355,7 +359,7 @@ subroutine mld_cprecsetc(p,what,string,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -427,7 +431,7 @@ subroutine mld_cprecsetr(p,what,val,info,ilev) end if if (.not.allocated(p%baseprecv)) then - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' info = 3111 return endif @@ -439,7 +443,7 @@ subroutine mld_cprecsetr(p,what,val,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -480,16 +484,24 @@ subroutine mld_cprecsetr(p,what,val,info,ilev) case(mld_fact_thrs_) do ilev_=1,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif p%baseprecv(ilev_)%rprcparm(what) = val end do + case(mld_coarse_fthrs_) + ilev_=nlev_ + if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' + info = -1 + return + endif + p%baseprecv(ilev_)%rprcparm(mld_fact_thrs_) = val case(mld_aggr_damp_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -498,7 +510,7 @@ subroutine mld_cprecsetr(p,what,val,info,ilev) case(mld_aggr_thresh_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif diff --git a/mlprec/mld_daggrmap_bld.f90 b/mlprec/mld_daggrmap_bld.f90 index 90259851..610cd890 100644 --- a/mlprec/mld_daggrmap_bld.f90 +++ b/mlprec/mld_daggrmap_bld.f90 @@ -47,8 +47,11 @@ ! build the coarse-level matrix. ! ! The aggregation algorithm is a parallel version of that described in -! M. Brezina and P. Vanek, A black-box iterative solver based on a +! * M. Brezina and P. Vanek, A black-box iterative solver based on a ! two-level Schwarz method, Computing, 63 (1999), 233-263. +! * P. Vanek, J. Mandel and M. Brezina, Algebraic Multigrid by Smoothed +! Aggregation for Second and Fourth Order Elliptic Problems, Computing, 56 +! (1996), 179-196. ! For more details see ! P. D'Ambra, D. di Serafino and S. Filippone, On the development of ! PSBLAS-based parallel two-level Schwarz preconditioners, Appl. Num. Math. diff --git a/mlprec/mld_dprecset.f90 b/mlprec/mld_dprecset.f90 index a2107360..fc573511 100644 --- a/mlprec/mld_dprecset.f90 +++ b/mlprec/mld_dprecset.f90 @@ -91,7 +91,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv)) then info = 3111 - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' return endif nlev_ = size(p%baseprecv) @@ -110,7 +110,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then info = 3111 write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' return endif @@ -162,13 +162,16 @@ subroutine mld_dprecseti(p,what,val,info,ilev) if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + case(mld_bjac_) + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_sweeps_) @@ -204,7 +207,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) do ilev_=1,max(1,nlev_-1) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -215,7 +218,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) do ilev_=2,nlev_-1 if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -224,7 +227,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) case(mld_coarse_mat_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - & ': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + & ': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -232,28 +235,29 @@ subroutine mld_dprecseti(p,what,val,info,ilev) case(mld_coarse_solve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ case(mld_bjac_) - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_ilu_n_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_subsolve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return end if @@ -262,7 +266,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) case(mld_coarse_sweeps_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -270,7 +274,7 @@ subroutine mld_dprecseti(p,what,val,info,ilev) case(mld_coarse_fillin_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -355,7 +359,7 @@ subroutine mld_dprecsetc(p,what,string,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -426,7 +430,7 @@ subroutine mld_dprecsetr(p,what,val,info,ilev) end if if (.not.allocated(p%baseprecv)) then - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' info = 3111 return endif @@ -438,7 +442,7 @@ subroutine mld_dprecsetr(p,what,val,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -479,7 +483,7 @@ subroutine mld_dprecsetr(p,what,val,info,ilev) case(mld_fact_thrs_) do ilev_=1,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -488,7 +492,7 @@ subroutine mld_dprecsetr(p,what,val,info,ilev) case(mld_coarse_fthrs_) ilev_=nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -496,7 +500,7 @@ subroutine mld_dprecsetr(p,what,val,info,ilev) case(mld_aggr_damp_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -505,7 +509,7 @@ subroutine mld_dprecsetr(p,what,val,info,ilev) case(mld_aggr_thresh_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif diff --git a/mlprec/mld_prec_type.f90 b/mlprec/mld_prec_type.f90 index f815a340..a1d59c91 100644 --- a/mlprec/mld_prec_type.f90 +++ b/mlprec/mld_prec_type.f90 @@ -332,30 +332,30 @@ module mld_prec_type integer, parameter :: mld_smth_avsz_=6, mld_max_avsz_=mld_smth_avsz_ ! - ! Character constants used by mld_file_prec_descr + ! Character constants used by mld_file_prec_descr ! + character(len=19), parameter, private :: & + & smooth_names(1:3)=(/'pre-smoothing ','post-smoothing ',& + & 'pre/post-smoothing'/) character(len=15), parameter, private :: & - & smooth_names(1:3)=(/'Pre-smoothing ','Post-smoothing',& - & 'Smooth both '/) + & aggr_kinds(0:2)=(/'no smoother ','omega smoother',& + & 'bizr. smoother'/) character(len=15), parameter, private :: & - & aggr_kinds(0:2)=(/'No smoother ','Omega smoother',& - & 'Bizr. smoother'/) - character(len=15), parameter, private :: & - & matrix_names(0:1)=(/'Distributed ','Replicated '/) + & matrix_names(0:1)=(/'distributed ','replicated '/) character(len=18), parameter, private :: & - & aggr_names(0:4)=(/'Local aggregation ','Sym. local aggr. ',& - & 'Global aggregation', 'New local aggr. ','New global aggr. '/) + & aggr_names(0:4)=(/'local aggregation ','sym. local aggr. ',& + & 'global aggregation', 'new local aggr. ','new global aggr. '/) character(len=6), parameter, private :: & - & restrict_names(0:4)=(/'None ','Halo ',' ',' ',' '/) + & restrict_names(0:4)=(/'none ','halo ',' ',' ',' '/) character(len=12), parameter, private :: & - & prolong_names(0:3)=(/'None ','Sum ','Average ','Square root'/) + & prolong_names(0:3)=(/'none ','sum ','average ','square root'/) character(len=15), parameter, private :: & - & ml_names(0:3)=(/'None ','Additive ','Multiplicative',& - & 'New ML '/) + & ml_names(0:3)=(/'none ','additive ','multiplicative',& + & 'new ML '/) character(len=15), parameter, private :: & - & fact_names(0:6)=(/'None ','ILU(n) ',& - & 'MILU(n) ','ILU(T) ',& - & 'Sparse SuperLU','UMFPACK Sp. LU',& + & fact_names(0:6)=(/'none ','ILU(n) ',& + & 'MILU(n) ','ILU(t,n) ',& + & 'SuperLU ','UMFPACK LU ',& & 'SuperLU_Dist '/) ! @@ -428,6 +428,8 @@ contains val = psb_sum_ case('AVG') val = psb_avg_ + case('FACT_NONE') + val = mld_f_none_ case('ILU') val = mld_ilu_n_ case('MILU') @@ -462,7 +464,7 @@ contains val = mld_pre_smooth_ case('POST') val = mld_post_smooth_ - case('TWOSIDE','BOTH') + case('TWOSIDE') val = mld_twoside_smooth_ case('NOPREC') val = mld_noprec_ @@ -703,241 +705,431 @@ contains ! Subroutine: mld_file_prec_descr ! Version: real ! - ! This routine prints to a file a description of the preconditioner. + ! This routine prints a description of the preconditioner to the standard + ! output or to a file. It must be called after the preconditioner has been + ! built by mld_precbld. ! ! Arguments: ! p - type(mld_dprec_type), input. ! The preconditioner data structure to be printed out. - ! iout - integer, input. + ! info - integer, output. + ! error code. + ! iout - integer, input, optional. ! The id of the file where the preconditioner description - ! will be printed. + ! will be printed. If iout is not present, then the standard + ! output is condidered. ! - subroutine mld_file_prec_descr(p,iout) + subroutine mld_file_prec_descr(p,info,iout) + use psb_base_mod, only : psb_cd_get_context, psb_info - ! Arguments + ! Arguments type(mld_dprec_type), intent(in) :: p + integer, intent(out) :: info integer, intent(in), optional :: iout - ! Local variables - integer :: ilev + ! Local variables + integer :: ilev, nlev + integer :: ictxt, me, np character(len=20), parameter :: name='mld_file_prec_descr' integer :: iout_ - + + info = 0 if (present(iout)) then iout_ = iout else iout_ = 6 end if + if (iout_ < 0) iout_ = 6 + + if (allocated(p%baseprecv)) then + ictxt = psb_cd_get_context(p%baseprecv(1)%desc_data) + + call psb_info(ictxt,me,np) + + ! + ! The preconditioner description is printed by processor psb_root_. + ! This agrees with the fact that all the parameters defining the + ! preconditioner have the same values on all the procs (this is + ! ensured by mld_precbld). + ! + if (me==psb_root_) then + + write(iout_,*) 'Preconditioner description' + nlev = size(p%baseprecv) + if (nlev >= 1) then - write(iout_,*) 'Preconditioner description' - if (allocated(p%baseprecv)) then - if (size(p%baseprecv)>=1) then - ilev = 1 - write(iout_,*) 'Base preconditioner' - select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) - case(mld_noprec_) - write(iout_,*) 'No preconditioning' - case(mld_diag_) - write(iout_,*) 'Diagonal scaling' - case(mld_bjac_) - write(iout_,*) 'Block Jacobi with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) - case(mld_slu_,mld_umf_,mld_sludist_) - case default - write(iout_,*) 'Should never get here!' - end select - case(mld_as_) - write(iout_,*) 'Additive Schwarz with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) - case(mld_slu_,mld_umf_,mld_sludist_) - case default - write(iout_,*) 'Should never get here!' - end select - write(iout_,*) 'Overlap:',& - & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) - write(iout_,*) 'Restriction: ',& - & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) - write(iout_,*) 'Prolongation: ',& - & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) - end select - end if - if (size(p%baseprecv)>=2) then - do ilev = 2, size(p%baseprecv) - if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then - write(iout_,*) 'Inconsistent MLPREC part!' - return + ! + ! Print description of base preconditioner + ! + + write(iout_,*) ' ' + + if (nlev > 1) then + write(iout_,*) ' Multilevel Schwarz' + write(iout_,*) ' Base preconditioner (smoother) details' endif - - write(iout_,*) 'Multilevel: Level No', ilev - write(iout_,*) 'Multilevel type: ',& - & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) - if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then - write(iout_,*) 'Multilevel aggregation: ', & + + ilev = 1 + select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) + case(mld_noprec_) + write(iout_,*) ' No preconditioning' + case(mld_diag_) + write(iout_,*) ' Diagonal scaling' + case(mld_bjac_) + write(iout_,*) ' Block Jacobi with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + case(mld_ilu_n_,mld_milu_n_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + case(mld_slu_,mld_umf_,mld_sludist_) + case default + write(iout_,*) ' Should never get here!' + end select + case(mld_as_) + write(iout_,*) ' Additive Schwarz with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + case(mld_ilu_n_,mld_milu_n_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + case(mld_slu_,mld_umf_,mld_sludist_) + case default + write(iout_,*) ' Should never get here!' + end select + write(iout_,*) ' Overlap:',& + & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) + write(iout_,*) ' Restriction: ',& + & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) + write(iout_,*) ' Prolongation: ',& + & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) + end select + + end if + + if (nlev > 1) then + + ! + ! Print multilevel details + ! + + write(iout_,*) ' Multilevel details' + + do ilev = 2, nlev + if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then + info = 3111 + write(iout_,*) ' ',name,': error: inconsistent MLPREC part, should call MLD_PRECINIT' + return + endif + end do + + write(iout_,*) ' Number of levels: ',nlev + + ! + ! Currently, all the preconditioner parameters must have the same value at levels + ! 2,...,nlev-1, hence only the values at level 2 are printed + ! + + ilev=2 + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Multilevel type: ',& + & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) + write(iout_,*) ' Smoother position: ',& + & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Aggregation: ', & & aggr_names(p%baseprecv(ilev)%iprcparm(mld_aggr_alg_)) - write(iout_,*) 'Aggregation smoothing: ', & + write(iout_,*) ' Aggregation smoothing: ', & & aggr_kinds(p%baseprecv(ilev)%iprcparm(mld_aggr_kind_)) - write(iout_,*) 'Aggregation threshold: ', & + write(iout_,*) ' Aggregation threshold: ', & & p%baseprecv(ilev)%rprcparm(mld_aggr_thresh_) if (p%baseprecv(ilev)%iprcparm(mld_aggr_kind_) /= mld_no_smooth_) then - write(iout_,*) 'Damping omega: ', & - & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) - write(iout_,*) 'Multilevel smoother position: ',& - & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Damping omega: ', & + & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) + end if + endif + + ! + ! Coarse matrices are different at levels 2,...,nlev-1, hence related + ! info is printed separately + ! + + do ilev = 2, nlev-1 + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Level ',ilev + if (allocated(p%baseprecv(ilev)%nlaggr)) then + write(iout_,*) ' Size of coarse matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if end if - write(iout_,*) 'Coarse matrix: ',& + end do + + ! + ! Print coarsest level details + ! + + ilev = nlev + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + + write(iout_,*) ' Level ',ilev,' (coarsest)' + write(iout_,*) ' Coarsest matrix: ',& & matrix_names(p%baseprecv(ilev)%iprcparm(mld_coarse_mat_)) if (allocated(p%baseprecv(ilev)%nlaggr)) then - write(iout_,*) 'Sizes of aggregates: ', & - & sum( p%baseprecv(ilev)%nlaggr(:)),' : ',p%baseprecv(ilev)%nlaggr(:) + write(iout_,*) ' Size of coarsest matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + if (p%baseprecv(ilev)%iprcparm(mld_coarse_mat_) == mld_distr_mat_ .and. & + & p%baseprecv(ilev)%iprcparm(mld_sub_solve_) /= mld_sludist_) then + write(iout_,*) ' Coarsest matrix solver: block Jacobi with ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + write(iout_,*) ' Number of Jacobi sweeps: ', & + & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + else + write(iout_,*) ' Coarsest matrix solver: ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) end if - write(iout_,*) 'Factorization type: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - write(iout_,*) 'Number of Jacobi sweeps: ', & - & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + end if - end do - end if + end if + + endif else - write(iout_,*) trim(name),': Error: No Base preconditioner available, something is wrong!' + + write(iout_,*) trim(name),': Error: no base preconditioner available, something is wrong!' + info = -2 return endif + end subroutine mld_file_prec_descr - subroutine mld_sfile_prec_descr(p,iout) + subroutine mld_sfile_prec_descr(p,info,iout) + use psb_base_mod, only : psb_cd_get_context, psb_info - ! Arguments + ! Arguments type(mld_sprec_type), intent(in) :: p + integer, intent(out) :: info integer, intent(in), optional :: iout - ! Local variables - integer :: ilev + ! Local variables + integer :: ilev, nlev + integer :: ictxt, me, np character(len=20), parameter :: name='mld_file_prec_descr' integer :: iout_ - + + info = 0 if (present(iout)) then iout_ = iout else iout_ = 6 end if + if (iout_ < 0) iout_ = 6 + + if (allocated(p%baseprecv)) then + ictxt = psb_cd_get_context(p%baseprecv(1)%desc_data) + + call psb_info(ictxt,me,np) + + ! + ! The preconditioner description is printed by processor psb_root_. + ! This agrees with the fact that all the parameters defining the + ! preconditioner have the same values on all the procs (this is + ! ensured by mld_precbld). + ! + if (me==psb_root_) then + + write(iout_,*) 'Preconditioner description' + nlev = size(p%baseprecv) + if (nlev >= 1) then - write(iout_,*) 'Preconditioner description' - if (allocated(p%baseprecv)) then - if (size(p%baseprecv)>=1) then - ilev = 1 - write(iout_,*) 'Base preconditioner' - select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) - case(mld_noprec_) - write(iout_,*) 'No preconditioning' - case(mld_diag_) - write(iout_,*) 'Diagonal scaling' - case(mld_bjac_) - write(iout_,*) 'Block Jacobi with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) - case(mld_slu_,mld_umf_,mld_sludist_) - case default - write(iout_,*) 'Should never get here!' - end select - case(mld_as_) - write(iout_,*) 'Additive Schwarz with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) - case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) - case(mld_slu_,mld_umf_,mld_sludist_) - case default - write(iout_,*) 'Should never get here!' - end select - write(iout_,*) 'Overlap:',& - & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) - write(iout_,*) 'Restriction: ',& - & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) - write(iout_,*) 'Prolongation: ',& - & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) - end select - end if - if (size(p%baseprecv)>=2) then - do ilev = 2, size(p%baseprecv) - if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then - write(iout_,*) 'Inconsistent MLPREC part!' - return + ! + ! Print description of base preconditioner + ! + + write(iout_,*) ' ' + + if (nlev > 1) then + write(iout_,*) ' Multilevel Schwarz' + write(iout_,*) ' Base preconditioner (smoother) details' endif - - write(iout_,*) 'Multilevel: Level No', ilev - write(iout_,*) 'Multilevel type: ',& - & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) - if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then - write(iout_,*) 'Multilevel aggregation: ', & + + ilev = 1 + select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) + case(mld_noprec_) + write(iout_,*) ' No preconditioning' + case(mld_diag_) + write(iout_,*) ' Diagonal scaling' + case(mld_bjac_) + write(iout_,*) ' Block Jacobi with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + case(mld_ilu_n_,mld_milu_n_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + case(mld_slu_,mld_umf_,mld_sludist_) + case default + write(iout_,*) ' Should never get here!' + end select + case(mld_as_) + write(iout_,*) ' Additive Schwarz with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + case(mld_ilu_n_,mld_milu_n_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + case(mld_slu_,mld_umf_,mld_sludist_) + case default + write(iout_,*) ' Should never get here!' + end select + write(iout_,*) ' Overlap:',& + & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) + write(iout_,*) ' Restriction: ',& + & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) + write(iout_,*) ' Prolongation: ',& + & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) + end select + + end if + + if (nlev > 1) then + + ! + ! Print multilevel details + ! + + write(iout_,*) ' Multilevel details' + + do ilev = 2, nlev + if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then + info = 3111 + write(iout_,*) ' ',name,': error: inconsistent MLPREC part, should call MLD_PRECINIT' + return + endif + end do + + write(iout_,*) ' Number of levels: ',nlev + + ! + ! Currently, all the preconditioner parameters must have the same value at levels + ! 2,...,nlev-1, hence only the values at level 2 are printed + ! + + ilev=2 + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Multilevel type: ',& + & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) + write(iout_,*) ' Smoother position: ',& + & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Aggregation: ', & & aggr_names(p%baseprecv(ilev)%iprcparm(mld_aggr_alg_)) - write(iout_,*) 'Aggregation smoothing: ', & + write(iout_,*) ' Aggregation smoothing: ', & & aggr_kinds(p%baseprecv(ilev)%iprcparm(mld_aggr_kind_)) - write(iout_,*) 'Aggregation threshold: ', & + write(iout_,*) ' Aggregation threshold: ', & & p%baseprecv(ilev)%rprcparm(mld_aggr_thresh_) if (p%baseprecv(ilev)%iprcparm(mld_aggr_kind_) /= mld_no_smooth_) then - write(iout_,*) 'Damping omega: ', & - & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) - write(iout_,*) 'Multilevel smoother position: ',& - & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Damping omega: ', & + & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) end if - write(iout_,*) 'Coarse matrix: ',& + endif + + ! + ! Coarse matrices are different at levels 2,...,nlev-1, hence related + ! info is printed separately + ! + + do ilev = 2, nlev-1 + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Level ',ilev + if (allocated(p%baseprecv(ilev)%nlaggr)) then + write(iout_,*) ' Size of coarse matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + end if + end do + + ! + ! Print coarsest level details + ! + + ilev = nlev + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + + write(iout_,*) ' Level ',ilev,' (coarsest)' + write(iout_,*) ' Coarsest matrix: ',& & matrix_names(p%baseprecv(ilev)%iprcparm(mld_coarse_mat_)) if (allocated(p%baseprecv(ilev)%nlaggr)) then - write(iout_,*) 'Sizes of aggregates: ', & - & sum( p%baseprecv(ilev)%nlaggr(:)),' : ',p%baseprecv(ilev)%nlaggr(:) + write(iout_,*) ' Size of coarsest matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + if (p%baseprecv(ilev)%iprcparm(mld_coarse_mat_) == mld_distr_mat_ .and. & + & p%baseprecv(ilev)%iprcparm(mld_sub_solve_) /= mld_sludist_) then + write(iout_,*) ' Coarsest matrix solver: block Jacobi with ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + write(iout_,*) ' Number of Jacobi sweeps: ', & + & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + else + write(iout_,*) ' Coarsest matrix solver: ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) end if - write(iout_,*) 'Factorization type: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - write(iout_,*) 'Number of Jacobi sweeps: ', & - & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + end if - end do - end if + end if + + endif else - write(iout_,*) trim(name),': Error: No Base preconditioner available, something is wrong!' + + write(iout_,*) trim(name),': Error: no base preconditioner available, something is wrong!' + info = -2 return endif + end subroutine mld_sfile_prec_descr + function mld_prec_short_descr(p) type(mld_dprec_type), intent(in) :: p character(len=20) :: mld_prec_short_descr @@ -958,231 +1150,417 @@ contains ! The id of the file where the preconditioner description ! will be printed. ! - subroutine mld_zfile_prec_descr(p,iout) + subroutine mld_zfile_prec_descr(p,info,iout) + use psb_base_mod, only : psb_cd_get_context, psb_info - ! Arguments + ! Arguments type(mld_zprec_type), intent(in) :: p + integer, intent(out) :: info integer, intent(in), optional :: iout - ! Local variables - integer :: ilev + ! Local variables + integer :: ilev, nlev + integer :: ictxt, me, np character(len=20), parameter :: name='mld_file_prec_descr' integer :: iout_ - + + info = 0 if (present(iout)) then iout_ = iout else iout_ = 6 end if + if (iout_ < 0) iout_ = 6 + + if (allocated(p%baseprecv)) then + ictxt = psb_cd_get_context(p%baseprecv(1)%desc_data) + + call psb_info(ictxt,me,np) + + ! + ! The preconditioner description is printed by processor psb_root_. + ! This agrees with the fact that all the parameters defining the + ! preconditioner have the same values on all the procs (this is + ! ensured by mld_precbld). + ! + if (me==psb_root_) then + + write(iout_,*) 'Preconditioner description' + nlev = size(p%baseprecv) + if (nlev >= 1) then + + ! + ! Print description of base preconditioner + ! + + write(iout_,*) ' ' - write(iout_,*) 'Preconditioner description' - if (allocated(p%baseprecv)) then - if (size(p%baseprecv)>=1) then - write(iout_,*) 'Base preconditioner' - ilev=1 - select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) - case(mld_noprec_) - write(iout_,*) 'No preconditioning' - case(mld_diag_) - write(iout_,*) 'Diagonal scaling' - case(mld_bjac_) - write(iout_,*) 'Block Jacobi with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + if (nlev > 1) then + write(iout_,*) ' Multilevel Schwarz' + write(iout_,*) ' Base preconditioner (smoother) details' + endif + + ilev = 1 + select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) + case(mld_noprec_) + write(iout_,*) ' No preconditioning' + case(mld_diag_) + write(iout_,*) ' Diagonal scaling' + case(mld_bjac_) + write(iout_,*) ' Block Jacobi with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - case(mld_as_) - write(iout_,*) 'Additive Schwarz with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + case(mld_as_) + write(iout_,*) ' Additive Schwarz with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - write(iout_,*) 'Overlap:',& - & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) - write(iout_,*) 'Restriction: ',& - & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) - write(iout_,*) 'Prolongation: ',& - & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) - end select - end if - if (size(p%baseprecv)>=2) then - do ilev = 2, size(p%baseprecv) - if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then - write(iout_,*) 'Inconsistent MLPREC part!' - return - endif - - write(iout_,*) 'Multilevel: Level No', ilev - write(iout_,*) 'Multilevel type: ',& - & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) - if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then - write(iout_,*) 'Multilevel aggregation: ', & + write(iout_,*) ' Overlap:',& + & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) + write(iout_,*) ' Restriction: ',& + & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) + write(iout_,*) ' Prolongation: ',& + & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) + end select + + end if + + if (nlev > 1) then + + ! + ! Print multilevel details + ! + + write(iout_,*) ' Multilevel details' + + do ilev = 2, nlev + if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then + info = 3111 + write(iout_,*) ' ',name,': error: inconsistent MLPREC part, should call MLD_PRECINIT' + return + endif + end do + + write(iout_,*) ' Number of levels: ',nlev + + ! + ! Currently, all the preconditioner parameters must have the same value at levels + ! 2,...,nlev-1, hence only the values at level 2 are printed + ! + + ilev=2 + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Multilevel type: ',& + & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) + write(iout_,*) ' Smoother position: ',& + & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Aggregation: ', & & aggr_names(p%baseprecv(ilev)%iprcparm(mld_aggr_alg_)) - write(iout_,*) 'Aggregation smoothing: ', & + write(iout_,*) ' Aggregation smoothing: ', & & aggr_kinds(p%baseprecv(ilev)%iprcparm(mld_aggr_kind_)) - write(iout_,*) 'Aggregation threshold: ', & + write(iout_,*) ' Aggregation threshold: ', & & p%baseprecv(ilev)%rprcparm(mld_aggr_thresh_) if (p%baseprecv(ilev)%iprcparm(mld_aggr_kind_) /= mld_no_smooth_) then - write(iout_,*) 'Smoothing omega: ', & - & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) - write(iout_,*) 'Smoothing position: ',& - & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Damping omega: ', & + & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) end if - write(iout_,*) 'Coarse matrix: ',& + endif + + ! + ! Coarse matrices are different at levels 2,...,nlev-1, hence related + ! info is printed separately + ! + + do ilev = 2, nlev-1 + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Level ',ilev + if (allocated(p%baseprecv(ilev)%nlaggr)) then + write(iout_,*) ' Size of coarse matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + end if + end do + + ! + ! Print coarsest level details + ! + + ilev = nlev + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + + write(iout_,*) ' Level ',ilev,' (coarsest)' + write(iout_,*) ' Coarsest matrix: ',& & matrix_names(p%baseprecv(ilev)%iprcparm(mld_coarse_mat_)) if (allocated(p%baseprecv(ilev)%nlaggr)) then - write(iout_,*) 'Aggregation sizes: ', & - & sum( p%baseprecv(ilev)%nlaggr(:)),' : ',p%baseprecv(ilev)%nlaggr(:) + write(iout_,*) ' Size of coarsest matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + if (p%baseprecv(ilev)%iprcparm(mld_coarse_mat_) == mld_distr_mat_ .and. & + & p%baseprecv(ilev)%iprcparm(mld_sub_solve_) /= mld_sludist_) then + write(iout_,*) ' Coarsest matrix solver: block Jacobi with ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + write(iout_,*) ' Number of Jacobi sweeps: ', & + & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + else + write(iout_,*) ' Coarsest matrix solver: ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) end if - write(iout_,*) 'Factorization type: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - write(iout_,*) 'Number of Jacobi sweeps: ', & - & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + end if - end do - end if + end if + + endif else - write(iout_,*) trim(name),': Error: No Base preconditioner available, something is wrong!' + + write(iout_,*) trim(name),': Error: no base preconditioner available, something is wrong!' + info = -2 return endif + end subroutine mld_zfile_prec_descr - subroutine mld_cfile_prec_descr(p,iout) + subroutine mld_cfile_prec_descr(p,info,iout) - ! Arguments + use psb_base_mod, only : psb_cd_get_context, psb_info + + ! Arguments type(mld_cprec_type), intent(in) :: p + integer, intent(out) :: info integer, intent(in), optional :: iout - ! Local variables - integer :: ilev + ! Local variables + integer :: ilev, nlev + integer :: ictxt, me, np character(len=20), parameter :: name='mld_file_prec_descr' integer :: iout_ - + + info = 0 if (present(iout)) then iout_ = iout else iout_ = 6 end if + if (iout_ < 0) iout_ = 6 + + if (allocated(p%baseprecv)) then + ictxt = psb_cd_get_context(p%baseprecv(1)%desc_data) + + call psb_info(ictxt,me,np) + + ! + ! The preconditioner description is printed by processor psb_root_. + ! This agrees with the fact that all the parameters defining the + ! preconditioner have the same values on all the procs (this is + ! ensured by mld_precbld). + ! + if (me==psb_root_) then + + write(iout_,*) 'Preconditioner description' + nlev = size(p%baseprecv) + if (nlev >= 1) then + + ! + ! Print description of base preconditioner + ! + + write(iout_,*) ' ' - write(iout_,*) 'Preconditioner description' - if (allocated(p%baseprecv)) then - if (size(p%baseprecv)>=1) then - write(iout_,*) 'Base preconditioner' - ilev=1 - select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) - case(mld_noprec_) - write(iout_,*) 'No preconditioning' - case(mld_diag_) - write(iout_,*) 'Diagonal scaling' - case(mld_bjac_) - write(iout_,*) 'Block Jacobi with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + if (nlev > 1) then + write(iout_,*) ' Multilevel Schwarz' + write(iout_,*) ' Base preconditioner (smoother) details' + endif + + ilev = 1 + select case(p%baseprecv(ilev)%iprcparm(mld_smoother_type_)) + case(mld_noprec_) + write(iout_,*) ' No preconditioning' + case(mld_diag_) + write(iout_,*) ' Diagonal scaling' + case(mld_bjac_) + write(iout_,*) ' Block Jacobi with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - case(mld_as_) - write(iout_,*) 'Additive Schwarz with: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + case(mld_as_) + write(iout_,*) ' Additive Schwarz with ',& + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - write(iout_,*) 'Overlap:',& - & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) - write(iout_,*) 'Restriction: ',& - & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) - write(iout_,*) 'Prolongation: ',& - & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) - end select - end if - if (size(p%baseprecv)>=2) then - do ilev = 2, size(p%baseprecv) - if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then - write(iout_,*) 'Inconsistent MLPREC part!' - return - endif - - write(iout_,*) 'Multilevel: Level No', ilev - write(iout_,*) 'Multilevel type: ',& - & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) - if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then - write(iout_,*) 'Multilevel aggregation: ', & + write(iout_,*) ' Overlap:',& + & p%baseprecv(ilev)%iprcparm(mld_sub_ovr_) + write(iout_,*) ' Restriction: ',& + & restrict_names(p%baseprecv(ilev)%iprcparm(mld_sub_restr_)) + write(iout_,*) ' Prolongation: ',& + & prolong_names(p%baseprecv(ilev)%iprcparm(mld_sub_prol_)) + end select + + end if + + if (nlev > 1) then + + ! + ! Print multilevel details + ! + + write(iout_,*) ' Multilevel details' + + do ilev = 2, nlev + if (.not.allocated(p%baseprecv(ilev)%iprcparm)) then + info = 3111 + write(iout_,*) ' ',name,': error: inconsistent MLPREC part, should call MLD_PRECINIT' + return + endif + end do + + write(iout_,*) ' Number of levels: ',nlev + + ! + ! Currently, all the preconditioner parameters must have the same value at levels + ! 2,...,nlev-1, hence only the values at level 2 are printed + ! + + ilev=2 + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Multilevel type: ',& + & ml_names(p%baseprecv(ilev)%iprcparm(mld_ml_type_)) + write(iout_,*) ' Smoother position: ',& + & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Aggregation: ', & & aggr_names(p%baseprecv(ilev)%iprcparm(mld_aggr_alg_)) - write(iout_,*) 'Aggregation smoothing: ', & + write(iout_,*) ' Aggregation smoothing: ', & & aggr_kinds(p%baseprecv(ilev)%iprcparm(mld_aggr_kind_)) - write(iout_,*) 'Aggregation threshold: ', & + write(iout_,*) ' Aggregation threshold: ', & & p%baseprecv(ilev)%rprcparm(mld_aggr_thresh_) if (p%baseprecv(ilev)%iprcparm(mld_aggr_kind_) /= mld_no_smooth_) then - write(iout_,*) 'Smoothing omega: ', & - & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) - write(iout_,*) 'Smoothing position: ',& - & smooth_names(p%baseprecv(ilev)%iprcparm(mld_smoother_pos_)) + write(iout_,*) ' Damping omega: ', & + & p%baseprecv(ilev)%rprcparm(mld_aggr_damp_) end if - write(iout_,*) 'Coarse matrix: ',& + endif + + ! + ! Coarse matrices are different at levels 2,...,nlev-1, hence related + ! info is printed separately + ! + + do ilev = 2, nlev-1 + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + write(iout_,*) ' Level ',ilev + if (allocated(p%baseprecv(ilev)%nlaggr)) then + write(iout_,*) ' Size of coarse matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + end if + end do + + ! + ! Print coarsest level details + ! + + ilev = nlev + + if (p%baseprecv(ilev)%iprcparm(mld_ml_type_)>mld_no_ml_) then + + write(iout_,*) ' Level ',ilev,' (coarsest)' + write(iout_,*) ' Coarsest matrix: ',& & matrix_names(p%baseprecv(ilev)%iprcparm(mld_coarse_mat_)) if (allocated(p%baseprecv(ilev)%nlaggr)) then - write(iout_,*) 'Aggregation sizes: ', & - & sum( p%baseprecv(ilev)%nlaggr(:)),' : ',p%baseprecv(ilev)%nlaggr(:) + write(iout_,*) ' Size of coarsest matrix: ', & + & sum( p%baseprecv(ilev)%nlaggr(:)) + write(iout_,*) ' Sizes of aggregates: ', & + & p%baseprecv(ilev)%nlaggr(:) + end if + if (p%baseprecv(ilev)%iprcparm(mld_coarse_mat_) == mld_distr_mat_ .and. & + & p%baseprecv(ilev)%iprcparm(mld_sub_solve_) /= mld_sludist_) then + write(iout_,*) ' Coarsest matrix solver: block Jacobi with ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) + write(iout_,*) ' Number of Jacobi sweeps: ', & + & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + else + write(iout_,*) ' Coarsest matrix solver: ', & + & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) end if - write(iout_,*) 'Factorization type: ',& - & fact_names(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) select case(p%baseprecv(ilev)%iprcparm(mld_sub_solve_)) case(mld_ilu_n_,mld_milu_n_) - write(iout_,*) 'Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) - case(mld_ilu_t_) - write(iout_,*) 'Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + case(mld_ilu_t_) + write(iout_,*) ' Fill level:',p%baseprecv(ilev)%iprcparm(mld_sub_fillin_) + write(iout_,*) ' Fill threshold :',p%baseprecv(ilev)%rprcparm(mld_fact_thrs_) case(mld_slu_,mld_umf_,mld_sludist_) case default - write(iout_,*) 'Should never get here!' + write(iout_,*) ' Should never get here!' end select - write(iout_,*) 'Number of Jacobi sweeps: ', & - & (p%baseprecv(ilev)%iprcparm(mld_smoother_sweeps_)) + end if - end do - end if + end if + + endif else - write(iout_,*) trim(name),': Error: No Base preconditioner available, something is wrong!' + + write(iout_,*) trim(name),': Error: no base preconditioner available, something is wrong!' + info = -2 return endif + end subroutine mld_cfile_prec_descr + function mld_zprec_short_descr(p) type(mld_zprec_type), intent(in) :: p character(len=20) :: mld_zprec_short_descr diff --git a/mlprec/mld_saggrmap_bld.f90 b/mlprec/mld_saggrmap_bld.f90 index a0884896..2fbbe5e8 100644 --- a/mlprec/mld_saggrmap_bld.f90 +++ b/mlprec/mld_saggrmap_bld.f90 @@ -47,8 +47,11 @@ ! build the coarse-level matrix. ! ! The aggregation algorithm is a parallel version of that described in -! M. Brezina and P. Vanek, A black-box iterative solver based on a +! * M. Brezina and P. Vanek, A black-box iterative solver based on a ! two-level Schwarz method, Computing, 63 (1999), 233-263. +! * P. Vanek, J. Mandel and M. Brezina, Algebraic Multigrid by Smoothed +! Aggregation for Second and Fourth Order Elliptic Problems, Computing, 56 +! (1996), 179-196. ! For more details see ! P. D'Ambra, D. di Serafino and S. Filippone, On the development of ! PSBLAS-based parallel two-level Schwarz preconditioners, Appl. Num. Math. diff --git a/mlprec/mld_sprecset.f90 b/mlprec/mld_sprecset.f90 index a7dc898a..b551acba 100644 --- a/mlprec/mld_sprecset.f90 +++ b/mlprec/mld_sprecset.f90 @@ -91,7 +91,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv)) then info = 3111 - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' return endif nlev_ = size(p%baseprecv) @@ -110,7 +110,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then info = 3111 write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' return endif @@ -162,13 +162,16 @@ subroutine mld_sprecseti(p,what,val,info,ilev) if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + case(mld_bjac_) + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_sweeps_) @@ -204,7 +207,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) do ilev_=1,max(1,nlev_-1) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -215,7 +218,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) do ilev_=2,nlev_-1 if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -224,7 +227,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) case(mld_coarse_mat_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - & ': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + & ': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -232,28 +235,29 @@ subroutine mld_sprecseti(p,what,val,info,ilev) case(mld_coarse_solve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ case(mld_bjac_) - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_ilu_n_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_subsolve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return end if @@ -262,7 +266,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) case(mld_coarse_sweeps_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -270,7 +274,7 @@ subroutine mld_sprecseti(p,what,val,info,ilev) case(mld_coarse_fillin_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -355,7 +359,7 @@ subroutine mld_sprecsetc(p,what,string,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -426,7 +430,7 @@ subroutine mld_sprecsetr(p,what,val,info,ilev) end if if (.not.allocated(p%baseprecv)) then - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' info = 3111 return endif @@ -438,7 +442,7 @@ subroutine mld_sprecsetr(p,what,val,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -479,16 +483,24 @@ subroutine mld_sprecsetr(p,what,val,info,ilev) case(mld_fact_thrs_) do ilev_=1,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif p%baseprecv(ilev_)%rprcparm(what) = val end do + case(mld_coarse_fthrs_) + ilev_=nlev_ + if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' + info = -1 + return + endif + p%baseprecv(ilev_)%rprcparm(mld_fact_thrs_) = val case(mld_aggr_damp_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -497,7 +509,7 @@ subroutine mld_sprecsetr(p,what,val,info,ilev) case(mld_aggr_thresh_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif diff --git a/mlprec/mld_zaggrmap_bld.f90 b/mlprec/mld_zaggrmap_bld.f90 index 295b59a2..0232aafb 100644 --- a/mlprec/mld_zaggrmap_bld.f90 +++ b/mlprec/mld_zaggrmap_bld.f90 @@ -47,8 +47,11 @@ ! build the coarse-level matrix. ! ! The aggregation algorithm is a parallel version of that described in -! M. Brezina and P. Vanek, A black-box iterative solver based on a +! * M. Brezina and P. Vanek, A black-box iterative solver based on a ! two-level Schwarz method, Computing, 63 (1999), 233-263. +! * P. Vanek, J. Mandel and M. Brezina, Algebraic Multigrid by Smoothed +! Aggregation for Second and Fourth Order Elliptic Problems, Computing, 56 +! (1996), 179-196. ! For more details see ! P. D'Ambra, D. di Serafino and S. Filippone, On the development of ! PSBLAS-based parallel two-level Schwarz preconditioners, Appl. Num. Math. diff --git a/mlprec/mld_zprecset.f90 b/mlprec/mld_zprecset.f90 index 2d58862d..33422255 100644 --- a/mlprec/mld_zprecset.f90 +++ b/mlprec/mld_zprecset.f90 @@ -91,7 +91,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv)) then info = 3111 - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' return endif nlev_ = size(p%baseprecv) @@ -110,7 +110,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then info = 3111 write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' return endif @@ -162,13 +162,16 @@ subroutine mld_zprecseti(p,what,val,info,ilev) if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + case(mld_bjac_) + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_sweeps_) @@ -204,7 +207,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) do ilev_=1,max(1,nlev_-1) if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -215,7 +218,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) do ilev_=2,nlev_-1 if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -224,7 +227,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) case(mld_coarse_mat_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - & ': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + & ': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -232,28 +235,29 @@ subroutine mld_zprecseti(p,what,val,info,ilev) case(mld_coarse_solve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif if (nlev_ > 1) then p%baseprecv(nlev_)%iprcparm(mld_coarse_solve_) = val - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == 0) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = val p%baseprecv(nlev_)%iprcparm(mld_smoother_type_) = mld_bjac_ + p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ select case (val) case(mld_umf_, mld_slu_) p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_repl_mat_ case(mld_bjac_) - p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_ilu_n_ - case default - p%baseprecv(nlev_)%iprcparm(mld_coarse_mat_) = mld_distr_mat_ + if (p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) == mld_bjac_) & + & p%baseprecv(nlev_)%iprcparm(mld_sub_solve_) = mld_umf_ end select endif case(mld_coarse_subsolve_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return end if @@ -262,7 +266,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) case(mld_coarse_sweeps_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -270,7 +274,7 @@ subroutine mld_zprecseti(p,what,val,info,ilev) case(mld_coarse_fillin_) if (.not.allocated(p%baseprecv(nlev_)%iprcparm)) then write(0,*) name,& - &': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + &': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -355,7 +359,7 @@ subroutine mld_zprecsetc(p,what,string,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%iprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -427,7 +431,7 @@ subroutine mld_zprecsetr(p,what,val,info,ilev) end if if (.not.allocated(p%baseprecv)) then - write(0,*) name,': Error: Uninitialized preconditioner, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner, should call MLD_PRECINIT' info = 3111 return endif @@ -439,7 +443,7 @@ subroutine mld_zprecsetr(p,what,val,info,ilev) return endif if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = 3111 return endif @@ -480,16 +484,24 @@ subroutine mld_zprecsetr(p,what,val,info,ilev) case(mld_fact_thrs_) do ilev_=1,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif p%baseprecv(ilev_)%rprcparm(what) = val end do + case(mld_coarse_fthrs_) + ilev_=nlev_ + if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' + info = -1 + return + endif + p%baseprecv(ilev_)%rprcparm(mld_fact_thrs_) = val case(mld_aggr_damp_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif @@ -498,7 +510,7 @@ subroutine mld_zprecsetr(p,what,val,info,ilev) case(mld_aggr_thresh_) do ilev_=2,nlev_ if (.not.allocated(p%baseprecv(ilev_)%rprcparm)) then - write(0,*) name,': Error: Uninitialized preconditioner component, should call MLD_PRECINIT' + write(0,*) name,': Error: uninitialized preconditioner component, should call MLD_PRECINIT' info = -1 return endif diff --git a/test/fileread/cf_sample.f90 b/test/fileread/cf_sample.f90 index 429dd1ee..7da658f5 100644 --- a/test/fileread/cf_sample.f90 +++ b/test/fileread/cf_sample.f90 @@ -289,7 +289,7 @@ program cf_sample call psb_sum(ictxt,descsize) call psb_sum(ictxt,precsize) if (iam==psb_root_) then - call mld_precdescr(prec) + call mld_precdescr(prec,info) write(*,'("Matrix: ",a)')mtrx_file write(*,'("Computed solution on ",i8," processors")')np write(*,'("Iterations to convergence : ",i6)')iter diff --git a/test/fileread/df_sample.f90 b/test/fileread/df_sample.f90 index e8b53160..1896a423 100644 --- a/test/fileread/df_sample.f90 +++ b/test/fileread/df_sample.f90 @@ -289,7 +289,7 @@ program df_sample call psb_sum(ictxt,descsize) call psb_sum(ictxt,precsize) if (iam==psb_root_) then - call mld_precdescr(prec) + call mld_precdescr(prec,info) write(*,'("Matrix: ",a)')mtrx_file write(*,'("Computed solution on ",i8," processors")')np write(*,'("Iterations to convergence : ",i6)')iter diff --git a/test/fileread/sf_sample.f90 b/test/fileread/sf_sample.f90 index 5d255e07..9a8c572c 100644 --- a/test/fileread/sf_sample.f90 +++ b/test/fileread/sf_sample.f90 @@ -289,7 +289,7 @@ program sf_sample call psb_sum(ictxt,descsize) call psb_sum(ictxt,precsize) if (iam==psb_root_) then - call mld_precdescr(prec) + call mld_precdescr(prec,info) write(*,'("Matrix: ",a)')mtrx_file write(*,'("Computed solution on ",i8," processors")')np write(*,'("Iterations to convergence : ",i6)')iter diff --git a/test/fileread/zf_sample.f90 b/test/fileread/zf_sample.f90 index 4eb452b7..db9e4a43 100644 --- a/test/fileread/zf_sample.f90 +++ b/test/fileread/zf_sample.f90 @@ -289,7 +289,7 @@ program zf_sample call psb_sum(ictxt,descsize) call psb_sum(ictxt,precsize) if (iam==psb_root_) then - call mld_precdescr(prec) + call mld_precdescr(prec,info) write(*,'("Matrix: ",a)')mtrx_file write(*,'("Computed solution on ",i8," processors")')np write(*,'("Iterations to convergence : ",i6)')iter diff --git a/test/pargen/Makefile b/test/pargen/Makefile index 12ab63a7..98300a56 100644 --- a/test/pargen/Makefile +++ b/test/pargen/Makefile @@ -9,6 +9,7 @@ FINCLUDES=$(FMFLAG). $(FMFLAG)$(MLDLIBDIR) $(FMFLAG)$(PSBDIR) $(FIFLAG). EXEDIR=./runs +all: ppde spde ppde: ppde.o data_input.o $(F90LINK) ppde.o data_input.o -o ppde $(MLD_LIB) $(PSBLAS_LIB) $(LDLIBS) diff --git a/test/pargen/ppde.f90 b/test/pargen/ppde.f90 index 5a74c271..139d520d 100644 --- a/test/pargen/ppde.f90 +++ b/test/pargen/ppde.f90 @@ -222,7 +222,7 @@ program ppde call psb_amx(ictxt,tprec) if (iam == psb_root_) write(*,'("Preconditioner time : ",es10.4)')tprec - if (iam == psb_root_) call mld_precdescr(prec) + if (iam == psb_root_) call mld_precdescr(prec,info) if (iam == psb_root_) write(*,'(" ")') ! diff --git a/test/pargen/spde.f90 b/test/pargen/spde.f90 index 110cda67..55309e6c 100644 --- a/test/pargen/spde.f90 +++ b/test/pargen/spde.f90 @@ -222,7 +222,7 @@ program spde call psb_amx(ictxt,tprec) if (iam == psb_root_) write(*,'("Preconditioner time : ",es10.4)')tprec - if (iam == psb_root_) call mld_precdescr(prec) + if (iam == psb_root_) call mld_precdescr(prec,info) if (iam == psb_root_) write(*,'(" ")') !