diff --git a/docs/pdf/Makefile b/docs/pdf/Makefile index c0e6d85f..2059aeb9 100644 --- a/docs/pdf/Makefile +++ b/docs/pdf/Makefile @@ -84,7 +84,9 @@ # TOPFILE = userguide.tex -SECFILE = title.tex intro.tex methods.tex precs.tex +SECFILE = title.tex abstract.tex overview.tex conventions.tex distribution.tex \ + building.tex gettingstarted.tex highlevelview.tex advanced.tex errors.tex \ + listofroutines.tex bibliography.tex FIGDIR = figures XPDFFLAGS = diff --git a/docs/pdf/abstract.tex b/docs/pdf/abstract.tex new file mode 100644 index 00000000..495d1885 --- /dev/null +++ b/docs/pdf/abstract.tex @@ -0,0 +1,19 @@ +\begin{abstract} +\emph{MLD2P4 (Multi-Level Domain Decomposition Parallel Preconditioners Package based on +PSBLAS}) is a package of parallel algebraic multi-level preconditioners. +It implements various versions of one-level additive and of multi-level additive +and hybrid Schwarz algorithms. In the multi-level case, a purely algebraic approach +is applied to generate coarse-level corrections, so that no geometric background is needed +concerning the matrix to be preconditioned. The matrix is required to be square, real or complex, with a symmetric sparsity pattern \textbf{Non consideriamo anche il caso non simmetrico +con $(A+A^T)/2$?}. + +MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the +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 external libraries such as UMFPACK, SuperLU and SuperLU\_Dist, that +can be exploited in building multi-level preconditioners. +\end{abstract} \ No newline at end of file diff --git a/docs/pdf/advanced.tex b/docs/pdf/advanced.tex new file mode 100644 index 00000000..331a2a27 --- /dev/null +++ b/docs/pdf/advanced.tex @@ -0,0 +1,12 @@ +\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 new file mode 100644 index 00000000..e178fde4 --- /dev/null +++ b/docs/pdf/background.tex @@ -0,0 +1,291 @@ +\section{Multi-level Domain Decomposition Background\label{sec:background}} + +\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'' +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 +into subdomains, e.g. in a PDE discretization, to (approximately) solving the +subproblems corresponding to the subdomains and to building an approximate +solution of the original problem from the local solutions +\cite{Cai_Widlund_92,dd1_94,dd2_96}. + +\emph{Additive Schwarz} preconditioners are DD preconditioners using overlapping +submatrices, i.e.\ with some common rows, to couple the local information +related to the submatrices (see, e.g., \cite{dd2_96}). +The main motivations for choosing Additive Schwarz preconditioners are their +intrinsic parallelism and good \textbf{(dire good e' un po' "`forte"', dato che +subito dopo diciamo che la convergenza dipende dal numero di sottomatrici)} +convergence properties. A drawback of these +preconditioners is that the number of iterations of the preconditioned solvers +generally grows with the number of submatrices. This may be a serious limitation +on parallel computers, since the number of submatrices usually matches the number +of available processors. Optimal convergence rates, i.e.\ iteration numbers +independent of the number of submatrices, can be obtained by correcting the +preconditioner through a suitable approximation of the original linear system +in a coarse space, which globally couples the information related to the single +submatrices. + +\emph{Two-level Schwarz} preconditioners are obtained +by combining basic (one-level) Schwarz preconditioners with coarse-level +corrections. In this context, the one-level preconditioner is often +called smoother. Different two-level preconditioners are obtained by varying the +choice of the smoother, of the coarse-level correction and the +way they are combined \cite{dd2_96}. The same reasoning can be applied starting +from the coarse-level system, i.e.\ a coarse-space correction can be built +from this system, thus obtaining \emph{multi-level} preconditioners. + +It is worth noting that optimal preconditioners do not necessarily correspond +to minimum execution times. Indeed, to obtain effective multilevel preconditioners +a tradeoff between optimality of convergence and the cost of building and applying +the coarse-space corrections must be achieved. The choice of the number of levels, +i.e.\ of the coarse-space corrections, also affects the effectiveness of the +preconditioners. One more goal is to get convergence rates as less sensitive +as possible to variations in the matrix coefficients. + +Two main approaches can be used to build coarse-space corrections. The geometric approach +applies coarsening strategies based on the knowledge of some physical grid associated +to the matrix and requires the user to define grid transfer operators from the fine +to the coarse levels and vice versa. This may result difficult for complex geometries; +furthermore, suitable one-level preconditioners may be required to get efficient +interplay between fine and coarse levels, e.g.\ when matrices with highly varying coefficients +are considered. The algebraic approach builds coarse-space corrections using only matrix +information. It performs a fully automatic coarsening and enforces the interplay between +the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine +interpolation \cite{StubenGMD69_99}. + +MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices +starting from the original matrix. The algebraic approach is based on the \emph{smoothed +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_00}. In the next two subsections we provide +a brief description of the multi-level Schwarz preconditioners and on the smoothed +aggregation technique as implemented in MLD2P4. For further details the user +is referred to \cite{para_04,apnum_07,aaecc_07,dd2_96}. + + +\subsection{Multi-level Schwarz Preconditioners\label{sec:multilevel}} + +The Multilevel preconditioners implemented in MLD2P4 are obtained by combining +Additive Schwarz preconditioners with coarse-space corrections; therefore +we first provide a sketch of the Additive Schwarz preconditioners. + +Given a linear system +\[ Ax=b, \] +where $A=(a_{ij}) \in \Re^{n \times n}$ is a +nonsingular sparse matrix with a symmetric non-zero pattern, +let $G=(W,E)$ be the adjacency graph of $A$, where $W=\{1, 2, \ldots, n\}$ +and $E=\{(i,j) : a_{ij} \neq 0\}$ are the vertex set and the edge set of $G$, +respectively. Two vertices are called adjacent if there is an edge connecting +them. For any integer $\delta > 0$, a $\delta$-overlap +partition of $W$ can be defined recursively as follows. +Given a 0-overlap (or non-overlapping) partition of $W$, +i.e.\ a set of $m$ disjoint nonempty sets $W_i^0 \subset W$ such that +$\cup_{i=1}^m W_i^0 = W$, a $\delta$-overlap +partition of $W$ is obtained by considering the sets +$W_i^\delta \supset W_i^{\delta-1}$, obtained by including the vertices that +are adjacent to any vertex in $W_i^{\delta-1}$. + +Let $n_i^\delta$ be the size of $W_i^\delta$ and $R_i^{\delta} \in +\Re^{n_i^\delta \times n}$ the restriction operator that maps +a vector $v \in \Re^n$ onto the vector $v_i^{\delta} \in \Re^{n_i^\delta}$ +containing the components of $v$ corresponding to the vertices in +$W_i^\delta$. The transpose of $R_i^{\delta}$ is a +prolongation operator from $\Re^{n_i^\delta}$ to $\Re^n$. +The matrix $A_i^\delta=R_i^\delta A (R_i^\delta)^T \in +\Re^{n_i^\delta \times n_i^\delta}$ can be considered +as a restriction of $A$ corresponding to the set $W_i^{\delta}$. + +The \emph{classical one-level AS} preconditioner is defined by +\[ +M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T +(A_i^\delta)^{-1} R_i^{\delta}, +\] +where $A_i^\delta$ is assumed to be nonsingular. Its application +to a vector $v \in \Re^n$ within a Krylov solver requires the following +three steps: +\begin{enumerate} + \item restriction of $v$ as $v_i = R_i^{\delta} v$, $i=1,\ldots,m$; + \item (approximate) solution of the linear systems $A_i^\delta w_i = v_i$, + $i=1,\ldots,m$; + \item prolongation and sum of the $w_i$'s, i.e. $w = \sum_{i=1}^m (R_i^{\delta})^T w_i$. +\end{enumerate} +A variant of the classical AS preconditioner that outperforms it +in terms of both convergence rate and of computation and communication +time on parallel distributed-memory computers is the so-called \emph{Restricted AS +(RAS)} preconditioner~\cite{Cai_Sarkis,Efstathiou_Gander}. It +is obtained by zeroing the components of $w_i$ corresponding to the +overlapping vertices when applying the prolongation. Therefore, +RAS differs from classical AS by the prolongation operator $(R_i^{\delta})^T$, +which is substituted by $(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$, +where $\tilde{R}_i^0$ obtained by zeroing the rows of $R_i^\delta$ +corresponding to the vertices in $W_i^\delta \backslash W_i^0$: +\[ +M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T +(A_i^\delta)^{-1} R_i^{\delta}. +\] +Analogously, the AS variant called \emph{AS with Harmonic extension (ASH)} +is defined by +\[ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T +(A_i^\delta)^{-1} \tilde{R}_i^0. +\] +We note that for $\delta=0$ the three variants of the AS preconditioner are +all equal to the block-Jacobi preconditioner. + +As already observed, the convergence rate of the one-level Schwarz +preconditioned iterative solvers deteriorates as the number $m$ of partitions +of $W$ increases \cite{dd1_94,dd2_96}. To reduce the dependency +of the number of iterations on the degree of parallelism we may +introduce a global coupling among the overlapping partitions by defining +a coarse-space approximation $A_C$ of the matrix $A$. +In a pure algebraic setting, $A_C$ is usually built with +a Galerkin approach. Given a set $W_C$ of \emph{coarse vertices}, +with size $n_C$, and a suitable restriction operator +$R_C \in \Re^{n_C \times n}$, $A_C$ is defined as +\[ +A_C=R_C A R_C^T +\] +and the coarse-level correction matrix to be combined with a generic +one-level AS preconditioner $M_{1L}$ is obtained as +\[ +M_{C}^{-1}= R_C^T A_C^{-1} R_C, +\] +where $A_C$ is assumed to be nonsingular. The application of $M_{C}^{-1}$ +to a vector $v$ corresponds to a restriction, a solution and +a prolongation step; the solution step, involving the matrix $A_C$, +may be carried out also approximately. + +The combination of $M_{C}$ and $M_{1L}$ may be +performed in either an additive or a multiplicative framework. +In the former case, the \emph{two-level additive} Schwarz preconditioner +is obtained: +\[ +M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}. +\] +Applying $M_{2L-A}^{-1}$ to a vector $v$ within a Krylov solver +corresponds to applying $M_{C}^{-1}$ +and $M_{1L}^{-1}$ to $v$ independently and then summing up +the results. + +In the multiplicative case, the combination can be +performed by first applying the smoother $M_{1L}^{-1}$ and then +the coarse-level correction operator $M_{C}^{-1}$: +\[ +\begin{array}{l} +w = M_{1L}^{-1} v, \\ +z = w + M_{C}^{-1} (v-Aw); +\end{array} +\] +this corresponds to the following \emph{two-level hybrid pre-smoothed} +Schwarz preconditioner: +\[ +M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}. +\] +On the other hand, by applying the smoother after the coarse-level correction, +i.e.\ by computing +\[ +\begin{array}{l} +w = M_{C}^{-1} v , \\ +z = w + M_{1L}^{-1} (v-Aw) , +\end{array} +\] +the \emph{two-level hybrid post-smoothed} +Schwarz preconditioner is obtained: +\[ +M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}. +\] +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 +the number of available processors. When the size of the system to be preconditioned +is very large, the use of many proccessors, i.e.\ of many small submatrices, often +leads to a large coarse-level system, whose solution may be computationally expensive. +On the other hand, the use of few processors often leads to local sumatrices that +are too expensive to be processed on single processors, because of memory and/or +computing requirements. Therefore, it seems natural to use a recursive approach, +in which the coarse-level correction is re-applied starting from the current +coarse-level system. The corresponding preconditioners are called \emph{multi-level}. +One more reason for the multi-level approach is that it may significantly +reduce the computational cost of preconditioning with respect to the two-level case +(see \cite[Chapter 3]{dd2_96}). Additive and hybrid multilevel preconditioners +are obtained as direct extensions of the two-level counterparts. Other combinations +of the smoothers and coarse-level corrections are possible, leading to variants +of the previous algorithms. For a detailed descrition of them, the reader is +referred to \cite[Chapter 3]{dd2_96}. +\textbf{Secondo me qui ci vorrebbe una descrizione algoritmica, a titolo di esempio, +di un precondizionatore multilevel, ad esempio quello ibrido con pre-smoothing, sul tipo +della descrizione in figura 1 della guida di Trilinos ML 4.0. CHE NE PENSATE?} + + +\subsection{Smoothed Aggregation\label{sec:aggregation}} + +To define the restriction operator $R_C$, which is used to compute +the coarse-level matrix $A_C$, MLD2P4 uses the \emph{smoothed aggregation} +algorithm described in \cite{Brezina_Vanek_,Vanek_Mandel_Brezina_}. +The basic idea of this algorithm is to build a coarse set of vertices +$W_C$ by suitably grouping the vertices of $W$ into disjoint subsets +(aggregates), and to define the coarse-to-fine space transfer operator $R_C^T$ by +applying a suitable smoother to a simple piecewise constant +prolongation operator, to improve the quality of the coarse-space correction. + +Three main steps can be identified in the smoothed aggregation procedure: +\begin{itemize} + \item coarsening of the vertex set $W$, to obtain $W_C$; + \item construction of the prolongator $R_C^T$; + \item application of $R_C$ and $R_C^T$ to build $A_C$. +\end{itemize} + +To perform the coarsening step, we have implemented the aggregation algorithm sketched +in \cite{apnum_07}. According to \cite{brezina_vanek}, a modification of this algorithm +has been actually considered, +in which each aggregate $N_r$ is made of vertices of $W$ that are \emph{strongly coupled} +to a certain root vertex $r \in W$, i.e.\ +\[ N_r = \left\{s \in W: |a_{rs}| \geq \theta \sqrt{|a_{rr}a_{ss}|} \right\} \] +for a given $\theta \in [0,1]$. +Since the previous algorithm has a sequential nature, a \emph{decoupled} version of +it has been chosen, where each processor $i$ independently applies the algorithm to +the set of vertices $W_i^0$ assigned to it in the initial data distribution. This +version is embarrassingly parallel, since it does not require any data communication. +On the other hand, it may produce non-uniform aggregates near boundary vertices, +i.e.\ near vertices adjacent to vertices in other processors, and is strongly +dependent on the number of processors and on the initial partitioning of the matrix $A$. +Nevertheless, this algorithm has been chosen for the implementation in MLD2P4, +since it has been shown to produce good results in practice \cite{Tuminaro_Tong_00}. + +The prolongator $P_C=R_C^T$ is built starting from a \emph{tentative prolongator} +$P \in \Re^{n \times n_C}$, defined as +\begin{equation} +P=(p_{ij}), \quad p_{ij}= +\left\{ \begin{array}{ll} +1 & \quad \mbox{if} \; i \in V^j_C \\ +0 & \quad \mbox{otherwise} +\end{array} \right. . +\label{eq:tent_prol} +\end{equation} +$P_C$ is obtained by +applying to $P$ a smoother $S \in \Re^{n \times n}$: +\begin{equation} +P_C = S P, +\label{eq:smoothed_prol} +\end{equation} +in order to remove oscillatory components from the range of the prolongator +and hence to improve the convergence properties of the multi-level +Schwarz method \cite{Brezina_Vanek_,StubenGMD69_99}. +A simple choice for $S$ is the damped Jacobi smoother: +\begin{equation} +S = I - \omega D^{-1} A , +\label{eq:jac_smoother} +\end{equation} +where the value of $\omega$ can be chosen +using some estimate of the spectral radius of $D^{-1}A$ \cite{Brezina_Vanek}. +\textbf{Cenno al filtering di $A$ nello smoothing, dicendo che pero' non e' stato +implementato?} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "userguide" +%%% End: diff --git a/docs/pdf/bibliography.tex b/docs/pdf/bibliography.tex new file mode 100644 index 00000000..16a778c7 --- /dev/null +++ b/docs/pdf/bibliography.tex @@ -0,0 +1,152 @@ +\begin{thebibliography}{99} + +% +\bibitem{PARA04FOREST} +Bella, G., Filippone, S., De Maio, A., Testa, M.: +A Simulation Model for Forest Fires. +In: Dongarra, J., Madsen, K., Wasniewski, J. (eds.): +Proceedings of PARA~04 Workshop on State of the Art +in Scientific Computing. Lecture Notes in Computer Science, 3732. Berlin: +Springer, 2005 +% +\bibitem{aaecc_07} A. Buttari, D. di Serafino, P. D'Ambra, S. Filippone,\newblock +2LEV-D2P4: a package of high-performance preconditioners,\newblock +Applicable Algebra in Engineering, Communications and Computing, +Volume 18, Number 3, May, 2007, pp. 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\newblock +On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners +\newblock +Applied Numerical Mathematics, Elsevier Science, +Volume 57, Issues 11-12, November-December 2007, Pages 1181-1196. +%published online 3 February 2007, {\tt +% http://dx.doi.org/10.1016/j.apnum.2007.01.006} + +%% \bibitem{DOUGLAS} +%% R.E.~Bank and C.C.~Douglas, +%% {\em SMMP: Sparse Matrix Multiplication Package}, +%% Advances in Computational Mathematics, 1993, 1, 127-137. +%% (See also {\tt http://www.mgnet.org/~douglas/ccd-codes.html}) +% +% +\bibitem{para_04} +A.~Buttari, P.~D'Ambra, D.~di Serafino and S.~Filippone, +{\em Extending PSBLAS to Build Parallel Schwarz Preconditioners}, +in , J.~Dongarra, K.~Madsen, J.~Wasniewski, editors, +Proceedings of PARA~04 Workshop on State of the Art +in Scientific Computing, pp.~593--602, Lecture Notes in Computer Science, +Springer, 2005. +% +%% \bibitem{CAI_SAAD} +%% X.~C.~Cai and Y.~Saad, +%% {\em Overlapping Domain Decomposition Algorithms for General Sparse Matrices}, +%% Numerical Linear Algebra with Applications, 3(3), pp.~221--237, 1996. +%% % +%% \bibitem{CAI_SARKIS} +%% X.C.~Cai and M.~Sarkis, +%% {\em A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems}, +%% SIAM Journal on Scientific Computing, 21(2), pp.~792--797, 1999. +% +\bibitem{Cai_Widlund_92} +X.C.~Cai and O.~B.~Widlund, +{\em Domain Decomposition Algorithms for Indefinite Elliptic Problems}, +SIAM Journal on Scientific and Statistical Computing, 13(1), pp.~243--258, 1992. +% +\bibitem{dd1_94} +T.~Chan and T.~Mathew, +{\em Domain Decomposition Algorithms}, +in A.~Iserles, editor, Acta Numerica 1994, pp.~61--143, 1994. +Cambridge University Press. +%% % +%% \bibitem{UMFPACK} +%% T.A.~Davis, +%% {\em Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal +%% Method with a Column Pre-ordering Strategy}, +%% ACM Transactions on Mathematical Software, 30, pp.~196--199, 2004. +%% (See also {\tt http://www.cise.ufl.edu/~davis/}) +%% % +%% \bibitem{SUPERLU} +%% J.W.~Demmel, S.C.~Eisenstat, J.R.~Gilbert, X.S.~Li and J.W.H.~Liu, +%% A supernodal approach to sparse partial pivoting, +%% SIAM Journal on Matrix Analysis and Applications, 20(3), pp.~720--755, 1999. +% +\bibitem{BLACS} +J.~J.~Dongarra and R.~C.~Whaley, +{\em A User's Guide to the BLACS v.~1.1}, +Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of +Tennessee, March 1995 (updated May 1997). +% +\bibitem{sblas_97} +I.~Duff, M.~Marrone, G.~Radicati and C.~Vittoli, +{\em Level 3 Basic Linear Algebra Subprograms for Sparse Matrices: +a User Level Interface}, +ACM Transactions on Mathematical Software, 23(3), pp.~379--401, 1997. +% +\bibitem{sblas_02} +I.~Duff, M.~Heroux and R.~Pozo, +{\em An Overview of the Sparse Basic Linear +Algebra Subprograms: the New Standard from the BLAS Technical Forum}, +ACM Transactions on Mathematical Software, 28(2), pp.~239--267, 2002. +% +\bibitem{psblas_00} +S.~Filippone and M.~Colajanni, +{\em PSBLAS: A Library for Parallel Linear Algebra +Computation on Sparse Matrices}, +\newblock +ACM Transactions on Mathematical Software, 26(4), pp.~527--550, 2000. +% +\bibitem{KIVA3PSBLAS} +S.~Filippone, P.~D'Ambra, M.~Colajanni, +{\em Using a Parallel Library of Sparse Linear Algebra in a Fluid Dynamics +Applications Code on Linux Clusters}, +in G.~Joubert, A.~Murli, F.~Peters, M.~Vanneschi, editors, +Parallel Computing - Advances \& Current Issues, +pp.~441--448, Imperial College Press, 2002. +% +\bibitem{METIS} +Karypis, G. and Kumar, V., +{\em {METIS}: Unstructured Graph Partitioning and Sparse Matrix + Ordering System}. +Minneapolis, MN 55455: University of Minnesota, Department of + Computer Science, 1995. +Internet Address: {\verb|http://www.cs.umn.edu/~karypis|}. +\bibitem{BLAS1} +Lawson, C., Hanson, R., Kincaid, D. and Krogh, F., + Basic {L}inear {A}lgebra {S}ubprograms for {F}ortran usage, +{ACM Trans. Math. Softw.} vol.~{5}, 38--329, 1979. + +\bibitem{machiels} +{Machiels, L. and Deville, M.} +{\em Fortran 90: An entry to object-oriented programming for the solution + of partial differential equations.} +{ACM Trans. Math. Softw.} vol.~{23}, 32--49. +\bibitem{metcalf} +{Metcalf, M., Reid, J. and Cohen, M.} +{\em Fortran 95/2003 explained.} +{Oxford University Press}, 2004. + +\bibitem{dd2_96} +B.~Smith, P.~Bjorstad and W.~Gropp, +{\em Domain Decomposition: Parallel Multilevel Methods for Elliptic +Partial Differential Equations}, +Cambridge University Press, 1996. + +\bibitem{MPI1} +M.~Snir, S.~Otto, S.~Huss-Lederman, D.~Walker and J.~Dongarra, +{\em MPI: The Complete Reference. Volume 1 - The MPI Core}, second edition, +MIT Press, 1998. +% +\bibitem{BREZINA_VANEK} +M.~Brezina and P.~Van{\v e}k, +{\em A Black-Box Iterative Solver Based on a Two-Level Schwarz Method}, +Computing, 1999, 63, 233-263. +% +% +\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. +% + +\end{thebibliography} \ No newline at end of file diff --git a/docs/pdf/building.tex b/docs/pdf/building.tex new file mode 100644 index 00000000..e6acbbf4 --- /dev/null +++ b/docs/pdf/building.tex @@ -0,0 +1,7 @@ +\section{Configuring and Building MLD2P4\label{sec:configuring}} + - uso di GNU autoconf e automake \\ + - software di base necessario (MPI, BLACS, BLAS, PSBLAS - specificare versioni) \\ + - software opzionale (UMFPACK, SuperLU, SuperLUdist - specificare versioni e opzioni di configure) \\ + - sistemi operativi e compilatori su cui MLD2P4 e' stato costruito con successo \\ + - sono previste opzioni di configurazione per il debugging o per il profiling? \\ + - albero delle directory \\ diff --git a/docs/pdf/conventions.tex b/docs/pdf/conventions.tex new file mode 100644 index 00000000..39c91f22 --- /dev/null +++ b/docs/pdf/conventions.tex @@ -0,0 +1,6 @@ +\section{Notational Conventions\label{sec:conventions}} + - caratteri tipografici usati nella guida (vedi guida ML recente e guida Aztec) \\ + - convenzioni sui nomi di routine (differenza tra high-level e medium-level), + strutture dati,\\ + moduli, costanti, etc. (vedi guida psblas) \\ + - versione reale e complessa\\ \ No newline at end of file diff --git a/docs/pdf/distribution.tex b/docs/pdf/distribution.tex new file mode 100644 index 00000000..3d8f75e9 --- /dev/null +++ b/docs/pdf/distribution.tex @@ -0,0 +1,41 @@ +\section{Code Distribution\label{sec:distribution}} + +The MLD2P4 is freely distributable under the following copyright +terms: +\begin{verbatim} + MLD2P4 version 1.0 +MultiLevel Domain Decomposition Parallel Preconditioners Package + based on PSBLAS (Parallel Sparse BLAS version 2.3) + +(C) Copyright 2008 + + Salvatore Filippone University of Rome Tor Vergata + Alfredo Buttari University of Rome Tor Vergata + Pasqua D'Ambra ICAR-CNR, Naples + Daniela di Serafino Second University of Naples + + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions +are met: + 1. Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + 2. Redistributions in binary form must reproduce the above copyright + notice, this list of conditions, and the following disclaimer in the + documentation and/or other materials provided with the distribution. + 3. The name of the MLD2P4 group or the names of its contributors may + not be used to endorse or promote products derived from this + software without specific written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR +PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS +BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR +CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF +SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN +CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) +ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE +POSSIBILITY OF SUCH DAMAGE. +\end{verbatim} diff --git a/docs/pdf/errors.tex b/docs/pdf/errors.tex new file mode 100644 index 00000000..2ba4f60b --- /dev/null +++ b/docs/pdf/errors.tex @@ -0,0 +1,9 @@ +\section{Error Handling}\label{sec:errors} + +Error handling + - Breve descrizione con rinvio alla guida di PSBLAS + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "userguide" +%%% End: diff --git a/docs/pdf/gettingstarted.tex b/docs/pdf/gettingstarted.tex new file mode 100644 index 00000000..92c81021 --- /dev/null +++ b/docs/pdf/gettingstarted.tex @@ -0,0 +1,224 @@ +\section{Getting Started\label{sec:started}} + +We describe the basics for building and applying MLD2P4 one-level and multi-level +Schwarz preconditioners with the Krylov solvers included in PSBLAS \cite{}. +The following five steps are required: +\begin{enumerate} +\item \emph{Allocate and initialize the preconditioner data structure, according to + a preconditioner type chosen by the user}. This is performed by the routine + \verb|mld_precinit|, which also sets a default preconditioner for each preconditioner + type selected by the user. The default preconditioner associated to each preconditioner + type is listed in Table~\ref{tab:precinit}; the string used by \verb|mld_precinit| + to identify each preconditioner type is also given. The preconditioner data structure is + the derived data type \verb|mld_prec_type|, which is accessed to the user only + through the MLD2P4 routines. +\item \emph{Choose a specific variant of the selected preconditioner type, by setting + the preconditioner parameters.} This is performed by the routine \verb|mld_precset|. + A few examples concerning the use of \verb|mld_precset| are given in + Sections~\ref{sec:example1} and \ref{sec:example1}; a complete list of all the + preconditioner parameters and their allowed values is provided in + Section~\ref{sec:highlevel}. +\item \emph{Build the preconditioner for a given matrix.} This is performed by + the routine \verb|mld_precbld|. +\item \emph{Apply the preconditioner at each iteration of a Krylov solver.} + This is performed by the routine \verb|mld_precaply|. When using the PSBLAS Krylov solvers, + this step is completely transparent to the user, since \verb|mld_precaply| is called + by the PSBLAS routine implementing the Krylov solver (\verb|psb_krylov|). +\item \emph{Deallocate the preconditioner data structure}. This is performed by + the routine \verb|mld_precfree|. This step is complementary to step 1 and should + be performed when the preconditioner is no more used. +\end{enumerate} +A detailed description of the above routines is given in Section~\ref{sec:highlevel}. + +Note that the Fortran 95 module \verb|mld_prec_mod| must be used in the program +calling the MLD2P4 routines. Furthermore, to apply MLD2P4 with the Krylov solvers +from PSBLAS, the module \verb|psb_krylov_mod| must be used too. + +Two simple example programs showing the (basic) use of MLD2P4 are reported in +Section~\ref{sec:examples}. + +\begin{table}[th] +{ +\begin{center} +\begin{tabular}{|l|l|p{6.7cm}|} +\hline +Type & String & Default preconditioner \\ \hline +No preconditioner &'NOPREC'& (Considered only to use the PSBLAS + Krylov solvers with no preconditioner.) \\ +Diagonal & 'DIAG' & --- \\ +Block Jacobi & 'BJAC' & ILU(0) on the local blocks.\\ +Additive Schwarz & 'AS' & Restricted Additive Schwarz (RAS), + with overlap 1 and ILU(0) on the local blocks. \\ +Multilevel &'ML' & Multi-level hybrid preconditioner (additive on the + same level and multiplicative through the levels), + with post-smoothing only. Number of levels: 2; + post-smoother: block-Jacobi preconditioner, with ILU(0) + on the local blocks; coarsest matrix: distributed among the + processors; corase-level solver: 4 sweeps of the + block-Jacobi solver, with ILU(0) on the blocks. \\ +\hline +\end{tabular} +\end{center} +} +\caption{Preconditioner types and default choices.\label{tab:precinit}} +\end{table} + +\subsection{Examples\label{sec:examples}} + +The simple code reported below shows how to set and apply the MLD2P4 default multi-level +preconditioned, i.e.\ the two-level hybrid post-smoothed Schwarz preconditioner, using block-Jacobi with ILU(0) on the blocks as basic preconditioner, +a coarse matrix distributed among the processors, and four block-Jacobi sweeps with ILU(0) on the blocks as approximate coarse-level solver. The choice of this preconditioner is made +by simply specifying \verb|'ML'| as second argument of \verb|mld_precinit| +(a call to \verb|mld_precset| is not needed). +The preconditioner is applied within the BiCGSTAB solver provided by PSBLAS. + +The part of the code concerning the +reading and assembling of the sparse matrix and the right-hand side vector, performed +through the PSBLAS routines for sparse matrix and vector management, is not reported +here for brevity. Other statements concerning the use of PSBLAS are neglected too. +The complete code can be found in the example program file \verb|example_2lev_default.f90| +in the directory \textbf{XXXXXX (SPECIFICARE).} Note that the modules \verb|psb_base_mod| +and \verb|psb_util_mod| at the beginning of the code are required by PSBLAS. +For details on the use of the PSBLAS routines, see the PSBLAS User's Guide \cite{}. + +\begin{verbatim} + use psb_base_mod + use psb_util_mod + use mld_prec_mod + use psb_krylov_mod +... ... +! +! sparse matrix + type(psb_dspmat_type) :: A +! sparse matrix descriptor + type(psb_desc_type) :: DESC_A +! preconditioner + type(mld_prec_type) :: PRE +... ... +! +! initialize the parallel environment + call psb_init(ictxt) + call psb_info(ictxt,iam,np) +... ... +! +! read and assemble the matrix A and the right-hand +! side b using PSBLAS routines for sparse matrix / +! vector management +... ... +! +! initialize the default multi-level preconditioner +! (two-level hybrid post-smoothed Schwarz) + call mld_precinit(PRE,'ML',info) +! +! build the preconditioner + call psb_precbld(A,PRE,DESC_A,info) +! +! set the solver parameters and the initial guess + ... ... +! +! solve Ax=b with preconditioned BiCGSTAB + call psb_krylov('BICGSTAB',A,PRE,b,x,tol,DESC_A,info) + ... ... +! +! cleanup the preconditioner + call mld_precfree(PRE,info) +! +! cleanup other data structures + ... ... +! +! exit the parallel environment + call psb_exit(ictxt) + stop +\end{verbatim} + + +\textbf{MODIFICARE TUTTA LA PARTE CHE SEGUE:\\ +- solo istruzioni diverse dall'esempio precedente (essenzialmente il setting del precondizionatore, magari con piu' chiamate a precset;\\ +- lasciare l'osservazione sulla specifica esplicita del numero di livelli;\\ +- rimandare al paragrafo successivo per una decrizione accurata di tutti i parametri;\\ +- lasciare l'osservazione sui vecchi utenti di PSBLAS.}\\ + +In the following we describe the general procedure for setting and building one of the MLD2P4 preconditioners. +The user has first to prepare the preconditioner data structure by using the routine \verb|mld_precinit|. Input parameters +for this routine include a string parameter, needed to define the preconditioner type, and an optional integer parameter +specifying the number of the levels in the case of a multi-level preconditioner. +Note that if the optional parameter is not present and a multi-level preconditioner has been chosen, +a two-level preconditioner is set. On the other hand, the integer parameter is ignored if the type of the preconditioner is not multilevel. +In Table \ref{tab:precinit} we report both the possible choices for the preconditioner type +and the related default preconditioners. + + +The user of MLD2P4 may set a lot of parameters for one-level and multi-level Schwarz, in order +to define a different preconditioner than that of default choices. The parameters +can be set through the routine \verb|mld_precset|. The APIs of \verb|mld_precinit| and \verb|mld_precset| as well as the complete +list of the parameters that can be set with the corresponding allowed values are reported in Section \ref{sec:highlevel}. In the following a simple code +for a three-level hybrid post-smoothed Schwarz preconditioner, using RAS with overlap 1 as local preconditioner, +with ILU(0) on the local blocks, a distributed coarse matrix, four block-Jacobi sweeps with the UMFPACK LU +factorization on the blocks as coarse-matrix solver, is reported. Note that for the multi-level preconditioners, the levels are numbered in increasing +order starting from the finest one, i.e. level 1 is the finest level. +For more details, see the test program \verb|example2.f90| in xxxx(directory dei test).\\[0.5cm] + +\begin{verbatim} + use psb_base_mod + use psb_util_mod + use mld_prec_mod + use psb_krylov_mod +... ... +! +! sparse matrix + type(psb_dspmat_type) :: A +! sparse matrix descriptor + type(psb_desc_type) :: DESC_A +! preconditioner data + type(mld_dprec_type) :: PRE +... ... +! +! initialization of the parallel environment + + call psb_init(ictxt) + call psb_info(ictxt,iam,np) +... ... +! read and assemble the matrix A and the right-hand +! side vector b using PSBLAS routines for sparse +! matrix/vector management +... ... +! prepare the three-level hybrid post-smoothed Schwarz +! using RAS with overlap 1 as local preconditioner +! + call mld_precinit(PRE,'ML',info,nlev=3) + call mld_precset(PRE,mld_n_ovr_,novr=1,info,ilev=1) + call mld_precset(PRE,mld_sub_restr_,psb_halo_,info,ilev=1) +NOTA: e' PROPRIO BRUTTO "PSB_HALO_", BISOGNEREBBE AVERE COSTANTI CHE HANNO IL PREFISSO MLD! +! +! build preconditioner + call psb_precbld(A,PRE,DESC_A,info) +! +! set solver parameters and initial guess + ... ... +! solve Ax=b with preconditioned BiCGSTAB + + call psb_krylov('BICGSTAB',A,PRE,b,x,tol,DESC_A,info) + ... ... +! +! cleanup storage and exit +! + call mld_precfree(PRE,info) +! + call psb_gefree(b,DESC_A,info) + call psb_gefree(x,DESC_A,info) + call psb_spfree(A,DESC_A,info) + call psb_cdfree(DESC_A,info) +! + call psb_exit(ictxt) + stop + +\end{verbatim} + +{\bf Remark for users with PSBLAS-based legacy codes:} when MLD2P4 is installed, a PSBLAS user, with a PSBLAS-based legacy code +calling base preconditioners included in PSBLAS (NOPREC, DIAG and BJAC), is able to use the same preconditioners without changes to the code, if she/he +includes in her/his program the file \verb|psb_prec_mod|. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "userguide" +%%% End: diff --git a/docs/pdf/highlevelview.tex b/docs/pdf/highlevelview.tex new file mode 100644 index 00000000..99ed7af1 --- /dev/null +++ b/docs/pdf/highlevelview.tex @@ -0,0 +1,279 @@ +\section{High-Level User Interface\label{sec:highlevel}} + +At the upper layer of MLD2P4, five black-box routines encapsulate all the functionalities for the construction +and the application of any of the multi-level preconditioners. +In the following we give the details of the above routines. Note that for each routine are available four +different versions depending on involved data types: Real-Single/Double Precision, Complex-Single/Double Precision. + +\subsection{Preconditioner Setup and Building}\label{sec:setup} + +The setup of a MLD2P4 preconditioner is obtained by using the \verb|mld_precinit| routine, which +allocates and initializes the preconditioner data structure. +The API of this routine as well as the description of the arguments is reported in Fig.~\ref{fig:prcinit}. +Note that the allowed values for the \verb|ptype| argument are reported in Table~\ref{tab:precinit} (Sec. \ref{sec:started}). +% +\begin{figure}[h] +\begin{center} +{\small +\begin{verbatim} +mld_precinit(p,ptype,info,nlev) + +Arguments: + p type(mld_dprec_type), input/output. + The preconditioner data structure. + ptype character, input. The type of preconditioner. + info integer, output. Error code. + nlev integer, optional, input. + The number of levels of the multilevel preconditioner. + If nlev is not present and ptype=`ML'/`ml', + then nlev=2 is assumed. + Otherwise, nlev is ignored. +\end{verbatim} +} +\end{center} +\caption{API of the routine for preconditioner allocation and inizialization.\label{fig:prcinit}} +\end{figure} +% +% +\begin{figure}[h] +\begin{center} +{\small +\begin{verbatim} +mld_precfree(p,info) + +Arguments: + p - type(mld_dprec_type), input/output. + The preconditioner data structure to be deallocated. + info - integer, output. + Error code. +\end{verbatim} +} +\end{center} +\caption{API of the routine for preconditioner deallocation.\label{fig:prcfree}} +\end{figure} + +A twin routine for deallocation of the preconditioner data structure is the \verb|mld_precfree| routine, whose API is reported in +Fig.~\ref{fig:prcfree}. +As mentioned in Section~\ref{sec:multilevel}, a multi-level preconditioner is a combination +of coarse-level corrections and one-level preconditioner (or smoothers). +Different combinations of these components together with different type of one-level preconditioner +as well as different algorithms to build and apply coarse-level corrections allow to the user of defining different multi-level +preconditioners. +The user of MLD2P4 may specify the type of multi-level framework (additive or multiplicative), details on the +aggregation algorithm, details on the type and the way for applying the one-level preconditioner +(as pre-smoother, post-smoother or both), the coarsest matrix storage +(distributed or replicated), the type of the solver to be employed at the coarsest level +and related details, by setting some parameters through the routine \verb|mld_precset| (see Section~\ref{sec:list}). +The API of this routine is reported in Fig.~\ref{fig:prcset}. +% +\begin{figure}[h] +\begin{center} +{\small +\begin{verbatim} +mld_precset(p,what,val,info,ilev) + + Arguments: + p - type(mld_dprec_type), input/output. + The preconditioner data structure. + what - integer, input. + The number identifying the parameter to be set. + A mnemonic constant has been associated to each of these + numbers. + val - integer/character, input. + The value of the parameter to be set. + info - integer, output. + Error code. + ilev - integer, optional, input. + For the multilevel preconditioner, the level at which the + preconditioner parameter has to be set. + If nlev is not present, the parameter identified by 'what' + is set at all the appropriate levels. +\end{verbatim} +} +\end{center} +\caption{API of the routine for preconditioner setup.\label{fig:prcset}} +\end{figure} +% +Finally, to build a preconditioner, according to the requirements made trough the routines \verb|mld_precinit| and \verb|mld_precset|, +a user of MLD2P4 have to call the \verb|prec_build| routine, whose API is reported in Figure~\ref{fig:prcbld}. +% +\begin{figure}[h] +\begin{center} +{\small +\begin{verbatim} +mld_precbld(a,desc_a,prec,info) + + Arguments: + a - type(psb_dspmat_type). + The sparse matrix structure containing the local part of the + matrix to be preconditioned. + desc_a - type(psb_desc_type), input. + The communication descriptor of a. + p - type(mld_dprec_type), input/output. + The preconditioner data structure containing the local part + of the preconditioner to be built. + info - integer, output. + Error code. +\end{verbatim} +} +\end{center} +\caption{API of the routine for preconditioner building.\label{fig:prcbld}} +\end{figure} + +\subsubsection{List of the preconditioner parameters\label{sec:list}} + +In the following we report the list of possible parameters to be set through the \verb|mld_precset| routine, +in order to choose the type of multi-level preconditioner. The parameters are classified depending on their scope. +Note that for character data both uppercase and lowercase strings are allowed. +\begin{table}[h] +{\small \label{tab:prec_type} +\begin{tabular}{ll} +Parameter (\verb|what|) & Allowed values ( \verb|val|)\\ +\verb|mld_ml_type_| & 'ADD', 'MULT'\\ + & Define the type of multi-level preconditioner.\\ +\verb|mld_prec_type_| & 'DIAG', 'BJAC', 'AS' \\ + & Define the smoother at a certain level.\\ +\verb|mld_smooth_pos_| & 'PRE', 'POST', 'BOTH'\\ + & Define the way to apply the smoother.\\ +\end{tabular} +\caption{Parameters for preconditioner type.} +} +\end{table} + +In order to build a coarse matrix from a fine one, this version of MLD2P4 implements the +smoothed aggregation algorithm described in Section~\ref{sec:aggregation}. However, since for nonsymmetric problems the +application of a correct smoothed procedure is yet an open problem~\cite{lin}, the user +may also choose to apply a nonsmoothed aggregation technique, where the prolongator operator from +the coarse to fine-space vertices is the simple piecewice constant interpolation +(the tentative prolongator) operator defined in Section~\ref{sec:aggregation}. +The coarsening scheme takes into account possible anisotropic features of the problems, by using +a threshold level to be used for dropping matrix coefficients during the process. +The parallel implementation of the coarsening algorithm is based on a decoupled approach, where each process applies the coarsening scheme +to its own local data. The uncoupled scheme can be applied to the matrix $A+A^T$, in the case of matrices with nonsymmetric sparsity pattern. +In the Table \ref{tab:aggr_type} we list the parameters that the user can specify for the aggregation algorithm. +\begin{table}[h] +{\small \label{tab:aggr_type} +\begin{tabular}{ll} +Parameter & Allowed values \\ +(\verb|what|) & ( \verb|val|)\\ +\verb|mld_aggr_alg_| & 'DEC', 'SYMDEC'\\ + & Define the aggregation scheme\\ + & Now, only decoupled aggregation is available \\ + & (if 'SYMDEC' is set, the symmetric part of the matrix is considered)\\ +\verb|mld_aggr_kind_| & 'SMOOTH', 'RAW'\\ + & Define the type of aggregation technique (smoothed or nonsmoothed).\\ +\verb|mld_aggr_thresh_| & Dropping threshold in aggregation.\\ + & Default 0.0\\ +\verb|mld_aggr_eig_| & NON E' DEFINITA LA STRINGA CORRISPONDENTE a mldmaxnorm\\ + & Define the algorithm to evaluate the maximum eigenvalue\\ + & of $D^{-1}A$ for smoothed aggregation. Now only the A-norm of the\\ + & matrix is available.\\ +\end{tabular} +\caption{Parameters for aggregation type.} +} +\end{table} + +Some options are available for the system involving the coarsest matrix. +Indeed, this matrix can be replicated or distributed among the processors. +In the former case, various versions of incomplete LU (ILU) factorizations of the +coarsest matrix are available in order to solve the coarsest system. +In the current version of MLD2P4, the following factorizations are available~\cite{saad}: +\begin{description} +\item[ILU(k):] ILU factorization with fill-in level $k$; +\item[MILU(k):] modified ILU factorization with fill-in level $k$; +\item[ILU(k,t):] ILU with threshold $t$ and $k$ additional entries in each row of the L and U factors with respect to the initial sparsity pattern. +\end{description} +Furthermore, interfaces to UMFPACK~\cite{UMFPACK}, version 4.4, and to SuperLU package~\cite{SUPERLU}, version 3.0, have been also available to deal +with the coarsest system, when the coarsest matrix is replicated among the processors. +On the other hand, to solve the coarsest-level system when the coarsest matrix is distributed, +a block-Jacobi routine has been developed. It uses the different versions of ILU or the LU +factorization on the coarse matrix diagonal blocks held by the processors. In the case of +distributed coarsest matrix is also available an interface to SupeLU$\_$dist~\cite{SUPERLUDIST}, version 2.0, for distributed +sparse factorization and solve. +See the Table \ref{tab:coarse_mat} for details. +\begin{table}[h] +{\small \label{tab:coarse_mat} +\begin{tabular}{ll} +Parameter & Allowed values\\ +( \verb|what|) & ( \verb|val|)\\ +\verb|mld_coarse_mat_| & 'DISTR', 'REPL' \\ + & Coarse Matrix: distributed or replicated \\ +\verb|mld_coarse_solve_| & 'ILU', 'MILU', 'ILUT', 'SLU', 'UMF', SLUDIST', BJAC????\\ + & Available Coarse solver.\\ + & Only SLUDIST e BJAC can be used when coarse matrix is distributed\\ +\verb|mld_coarse_BJAC_sweeps_| & (NON VA BENE mldcoarsesweeps) number of Block-Jacobi sweeps when BJAC is used as coarsest solver\\ +\verb|mld_coarse_fill_in_| & level of fill-in in MILU and ILU factorization\\ + & E IL THRESHOLD PER ILUT? \\ +\end{tabular} +\caption{Parameters for coarsest matrix solver.} +} +\end{table} + +When a Schwarz algorithm is considered as smoother at a certain level or as one-level preconditioner, the user may set many parameters +in order to choose the type of additive Schwarz version (AS,RAS,ASH), the number of overlaps as well as the local solver. +All the parameters are reported in Table \ref{tab:schwarz_type}. +\begin{table}[h] +{\small \label{tab:schwarz_type} +\begin{tabular}{ll} +Parameter & Allowed values\\ +(\verb|what|) & (\verb|val|)\\ +\verb|mld_n_ovr_| & Number of overlaps \\ +\verb|mld_sub_restr_| & 'HALO', 'NONE'\\ +\verb|mld_sub_prol_| & 'SUM', 'NONE'\\ +\verb|mld_sub_solve_| & 'ILU', 'MILU', 'ILUT', 'SLU', 'UMF'\\ +\verb|mld_sub_ren_| & MANCANO LE STRINGHE\\ +\verb|mld_sub_fill_in_| & level of fill-in in local diagonal blocks, when ILU-type factorizations are used\\ +\end{tabular} +\caption{Parameters for Schwarz smoother/preconditioner type.} +} +\end{table} +Its worth noting that, the classical AS method corresponds to the couple of values 'HALO' and 'SUM' of the argument \verb|val|, +for the values \verb|mld_sub_restr_| and \verb|mld_sub_prol_| of the argument \verb|what|, respectively. While, the RAS method corresponds to +the couple of values 'NONE' and 'SUM' and ASH method corresponds to the couple of values 'HALO' and 'NONE'. + +\subsection{Preconditioner Application} \label{sec:application} + +Once the preconditioner has been built, it may be applied at each iteration +of a Krylov solver by calling the routine \verb|mld_precaply| (CAMBIARE NOME ROUTINE NEL SOFTWARE EVITANDO L'UNDERSCORE), +whose API is shown in Figure~\ref{fig:prcaply}. +This routine computes $y = op(M^{-1})\, x$, where $M$ is the previously built +preconditioner, stored in the \verb|prec| data structure, and $op$ +denotes the matrix itself or its transpose, according to the value of \verb|trans|. +Note that this routine is called within the PSBLAS-based Krylov solver available in the PSBLAS library (see the PSBLAS User's Guide for details), +therefore, the use of this routine is generally transparent to the MLD2P4 user. +% +\begin{figure}[h] +\begin{center} +{\small +\begin{verbatim} + mld_precaply(prec,x,y,desc_data,info,trans,work) + + Arguments: + prec - type(mld_dprec_type), input. + The preconditioner data structure containing the local part + of the preconditioner to be applied. + x - real(psb_dpk_), dimension(:), input. + The local part of the vector X in Y := op(M^(-1)) * X. + y - real(psb_dpk_), dimension(:), output. + The local part of the vector Y in Y := op(M^(-1)) * X. + desc_data - type(psb_desc_type), input. + The communication descriptor associated to the matrix to be + preconditioned. + info - integer, output. + Error code. + trans - character(len=1), optional. + If trans='N','n' then op(M^(-1)) = M^(-1); + if trans='T','t' then op(M^(-1)) = M^(-T) (transpose of M^(-1)). + work - real(psb_dpk_), dimension (:), optional, target. + Workspace. Its size must be at + least 4*psb_cd_get_local_cols(desc_data). +\end{verbatim} +} +\end{center} +\caption{API of the routine for preconditioner application.\label{fig:prcaply}} +\end{figure} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "userguide" +%%% End: diff --git a/docs/pdf/listofroutines.tex b/docs/pdf/listofroutines.tex new file mode 100644 index 00000000..0829ef60 --- /dev/null +++ b/docs/pdf/listofroutines.tex @@ -0,0 +1,10 @@ +\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/overview.tex b/docs/pdf/overview.tex new file mode 100644 index 00000000..eac4bac9 --- /dev/null +++ b/docs/pdf/overview.tex @@ -0,0 +1,62 @@ +\section{General Overview\label{sec:overview}} + +The \emph{Multi-Level Domain Decomposition Parallel Preconditioners Package based on +PSBLAS (MLD2P4}) provides various versions of multi-level Schwarz preconditioners~\cite{DD2}, +to be used in the iterative solutions of sparse linear systems $Ax=b$, where +$A$ is a square, real or complex, sparse matrix with a symmetric sparsity pattern. +\textbf{Ma non abbiamo detto che, se il pattern di sparista' non e' simmetrico, +lavoriamo su $(A+A^T)/2$? Ma questo vale solo per l'aggregazione? Dovremmo fare +qualcosa di consistente anche con 1-lev Schwarz.} +Both additive and hybrid preconditioners, i.e.\ multiplicative among the levels +and additive inside a level, are implemented; the basic additive Schwarz preconditioners +are obtained by considering only one level. A purely algebraic approach is used to +generate a sequence of coarse-level corrections to a basic preconditioner, without +explicitly using any information on the geometry of the original problem (e.g.\ the +discretization of a PDE). The smoothed aggregation technique is applied +as algebraic coarsening strategy~\cite{}. + +The package is written in Fortran~95, using object-oriented techniques, +and is based on a distributed-memory parallel programming paradigm. \textbf{SALVATORE, +potresti aggiungere due righe sulla scelta del Fortran 95 e sul semplice interfacciamento +con i legacy codes, senza ripetere quello che e' detto sotto sulla scelta di PSBLAS?} +Single and double precision implementations of MLD2P4 are available for both the +real and the complex case, that can be used through a single interface. +\textbf{SALVATORE, funziona tutto?} + +MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners +in the context of the PSBLAS (Parallel Sparse BLAS) computational framework~\cite{}. +PSBLAS is a library originally developed to address the parallel implementation of +iterative solvers for sparse linear system, by providing basic linear algebra +operators and data management facilities for distributed sparse matrices; it +also includes parallel Krylov solvers, built on the top of the basic PSBLAS kernels. +The preconditioners available in MLD2P4 can be used with these Krylov solvers. +The choice of PSBLAS has been mainly motivated by the need of having +a portable and efficient software infrastructure implementing ``de facto'' standard +parallel sparse linear algebra kernels, to pursue goals such as performance, +portability, modularity ed extensibility in the development of the preconditioner +package. On the other hand, the implementation of MLD2P4 has led to some +revisions and extentions of the PSBLAS kernels, leading to the +recent PSBLAS 2.0 version~\cite{}. The inter-process comunication required +by MLD2P4 is encapsulated into the PSBLAS routines, except few cases where +MPI~\cite{} is explicitly called. Therefore, MLD2P4 can be run on any parallel +machine where PSBLAS and MPI implementations are available. + +MLD2P4 has a layered and modular software architecture where three main layers can be identified. The lower layer consists of the PSBLAS kernels, the middle one implements +the construction and application phases of the preconditioners, and the upper one +provides a uniform and easy-to-use interface to all the preconditioners. +This architecture allows for different levels of use of the package: +few black-box routines at the upper level allow non-expert users to easily +build any preconditioner available in MLD2P4 and to apply it within a PSBLAS Krylov solver. +On the other hand, the routines of the middle and lower layer can be used and extended +by expert users to build new versions of multi-level Schwarz preconditioners.\\ + +\textbf{Organizzazione della guida:\\ +dire che per il momento non +forniamo anche la documentazione del middle layer, ma lo faremo in seguito\\} + +\textbf{Evidenziare le parole chiave che caratterizzano il nostro package} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "userguide" +%%% End: diff --git a/docs/pdf/userguide.tex b/docs/pdf/userguide.tex index 0ff357a7..3ac58b0c 100644 --- a/docs/pdf/userguide.tex +++ b/docs/pdf/userguide.tex @@ -1,4 +1,4 @@ -\documentclass[10pt,a4paper,twoside]{article} +\documentclass[11pt,a4paper,twoside]{article} \usepackage{pstricks} \usepackage{fancybox} \usepackage{amsfonts} @@ -22,17 +22,17 @@ \pdfoutput=1 \relax \pdfcompresslevel=0 %-- 0 = none, 9 = best -\pdfinfo{ %-- Info dictionary of PDF output /Author (Alfredo Buttari) - /Title ( MultiLevel Domain Decomposition Parallel Preconditioners Package - based on PSBLAS V. 1.0) - /Subject ( MultiLevel Domain Decomposition Parallel Preconditioners - Package) - /Keywords (Computer Science Linear Algebra Fluid Dynamics Parallel Linux MPI PSBLAS Iterative Solvers Preconditioners) - /Creator (pdfLaTeX) - /Producer ($Id: userguide.tex 1978 2007-10-19 14:51:12Z sfilippo $) +\pdfinfo{ %-- Info dictionary of PDF output /Author (PD, DdS, SF) + /Title (MultiLevel Domain Decomposition Parallel Preconditioners Package + based on PSBLAS, V. 1.0) + /Subject (MultiLevel Domain Decomposition Parallel Preconditioners Package) + /Keywords (Parallel Numerical Software, Algebraic Multilevel Preconditioners, Sparse Iterative Solvers, PSBLAS, MPI) + /Creator (pdfLaTeX) + /Producer ($Id: userguide.tex 2008-04-08 Pasqua D'Ambra, Daniela di Serafino, + Salvatore Filippone$) } -\pdfcatalog{ %-- Catalog dictionary of PDF output. - /URI (http://ce.uniroma2.it/psblas) +\pdfcatalog{ %-- Catalog dictionary of PDF output. +% /URI (http://ce.uniroma2.it/psblas) } \newcounter{subroutine}[subsection] @@ -78,175 +78,43 @@ \begin{document} \include{title} +%\cleardoublepage +\clearpage +\ \\ +\thispagestyle{empty} +\clearpage + +\pagenumbering{roman} % Roman numbering +\setcounter{page}{1} % Abstract start on page i + +\include{abstract} \cleardoublepage + \begingroup \renewcommand*{\thepage}{toc} - \pagenumbering{roman} % Roman numbering - \setcounter{page}{1} % Abstract start on page ii + %\pagenumbering{roman} % Roman numbering + %\setcounter{page}{1} % Abstract start on page ii \tableofcontents \endgroup - \cleardoublepage \pagenumbering{arabic} % Arabic numbering \setcounter{page}{1} % Chapters start on page 1 -\include{intro} -\include{precs} -\include{methods} +\include{overview} +\include{conventions} +\include{distribution} +\include{building} +\include{background} +\include{gettingstarted} +\include{highlevelview} +\include{advanced} +\include{errors} +\include{listofroutines} \cleardoublepage -\begin{thebibliography}{99} - -\bibitem{PARA04FOREST} -G.~Bella, S.~Filippone, A.~De Maio and M.~Testa, -{\em A Simulation Model for Forest Fires}, -in J.~Dongarra, K.~Madsen, J.~Wasniewski, editors, -Proceedings of PARA~04 Workshop on State of the Art -in Scientific Computing, pp.~546--553, Lecture Notes in Computer Science, -Springer, 2005. -\bibitem{2007d} A. Buttari, D. di Serafino, P. D'Ambra, S. Filippone,\newblock -2LEV-D2P4: a package of high-performance preconditioners,\newblock -Applicable Algebra in Engineering, Communications and Computing, -Volume 18, Number 3, May, 2007, pp. 223-239 -%Published online: 13 February 2007, {\tt http://dx.doi.org/10.1007/s00200-007-0035-z} -% -\bibitem{2007c} P. D'Ambra, S. Filippone, D. Di Serafino\newblock -On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners -\newblock -Applied Numerical Mathematics, Elsevier Science, -Volume 57, Issues 11-12, November-December 2007, Pages 1181-1196. -%published online 3 February 2007, {\tt -% http://dx.doi.org/10.1016/j.apnum.2007.01.006} - -\bibitem{BLAS2} - Dongarra, J. J., DuCroz, J., Hammarling, S. and Hanson, R., -An Extended Set of {F}ortran {B}asic {L}inear {A}lgebra {S}ubprograms, -{ACM Trans. Math. Softw.} vol.~{14}, 1--17, 1988. -\bibitem{BLAS3} - Dongarra, J., DuCroz, J., Hammarling, S. and Duff, I., -A Set of level 3 Basic Linear Algebra Subprograms, -{ACM Trans. Math. Softw.} vol.~{16}, 1--17, 1990. -%% \bibitem{DOUGLAS} -%% R.E.~Bank and C.C.~Douglas, -%% {\em SMMP: Sparse Matrix Multiplication Package}, -%% Advances in Computational Mathematics, 1993, 1, 127-137. -%% (See also {\tt http://www.mgnet.org/~douglas/ccd-codes.html}) -% -% -%% \bibitem{PARA04} -%% A.~Buttari, P.~D'Ambra, D.~di Serafino and S.~Filippone, -%% {\em Extending PSBLAS to Build Parallel Schwarz Preconditioners}, -%% in , J.~Dongarra, K.~Madsen, J.~Wasniewski, editors, -%% Proceedings of PARA~04 Workshop on State of the Art -%% in Scientific Computing, pp.~593--602, Lecture Notes in Computer Science, -%% Springer, 2005. -% -%% \bibitem{CAI_SAAD} -%% X.~C.~Cai and Y.~Saad, -%% {\em Overlapping Domain Decomposition Algorithms for General Sparse Matrices}, -%% Numerical Linear Algebra with Applications, 3(3), pp.~221--237, 1996. -%% % -%% \bibitem{CAI_SARKIS} -%% X.C.~Cai and M.~Sarkis, -%% {\em A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems}, -%% SIAM Journal on Scientific Computing, 21(2), pp.~792--797, 1999. -% -%% \bibitem{CAI_WIDLUND} -%% X.C.~Cai and O.~B.~Widlund, -%% {\em Domain Decomposition Algorithms for Indefinite Elliptic Problems}, -%% SIAM Journal on Scientific and Statistical Computing, 13(1), pp.~243--258, 1992. -% -%% \bibitem{DD1} -%% T.~Chan and T.~Mathew, -%% {\em Domain Decomposition Algorithms}, -%% in A.~Iserles, editor, Acta Numerica 1994, pp.~61--143, 1994. -%% Cambridge University Press. -%% % -%% \bibitem{APNUM06} -%% P.~D'Ambra, D.~di Serafino and S.~Filippone, -%% On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners, -%% Applied Numerical Mathematics, to appear, 2007. -% -%% \bibitem{UMFPACK} -%% T.A.~Davis, -%% {\em Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal -%% Method with a Column Pre-ordering Strategy}, -%% ACM Transactions on Mathematical Software, 30, pp.~196--199, 2004. -%% (See also {\tt http://www.cise.ufl.edu/~davis/}) -%% % -%% \bibitem{SUPERLU} -%% J.W.~Demmel, S.C.~Eisenstat, J.R.~Gilbert, X.S.~Li and J.W.H.~Liu, -%% A supernodal approach to sparse partial pivoting, -%% SIAM Journal on Matrix Analysis and Applications, 20(3), pp.~720--755, 1999. -% -\bibitem{BLACS} -J.~J.~Dongarra and R.~C.~Whaley, -{\em A User's Guide to the BLACS v.~1.1}, -Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of -Tennessee, March 1995 (updated May 1997). -% -\bibitem{sblas97} -I.~Duff, M.~Marrone, G.~Radicati and C.~Vittoli, -{\em Level 3 Basic Linear Algebra Subprograms for Sparse Matrices: -a User Level Interface}, -ACM Transactions on Mathematical Software, 23(3), pp.~379--401, 1997. -% -\bibitem{sblas02} -I.~Duff, M.~Heroux and R.~Pozo, -{\em An Overview of the Sparse Basic Linear -Algebra Subprograms: the New Standard from the BLAS Technical Forum}, -ACM Transactions on Mathematical Software, 28(2), pp.~239--267, 2002. -\bibitem{PSBLAS} -S.~Filippone and M.~Colajanni, -{\em PSBLAS: A Library for Parallel Linear Algebra -Computation on Sparse Matrices}, -\newblock -ACM Transactions on Mathematical Software, 26(4), pp.~527--550, 2000. -% -\bibitem{KIVA3PSBLAS} -S.~Filippone, P.~D'Ambra, M.~Colajanni, -{\em Using a Parallel Library of Sparse Linear Algebra in a Fluid Dynamics -Applications Code on Linux Clusters}, -in G.~Joubert, A.~Murli, F.~Peters, M.~Vanneschi, editors, -Parallel Computing - Advances \& Current Issues, -pp.~441--448, Imperial College Press, 2002. -% -\bibitem{METIS} -Karypis, G. and Kumar, V., -{\em {METIS}: Unstructured Graph Partitioning and Sparse Matrix - Ordering System}. -Minneapolis, MN 55455: University of Minnesota, Department of - Computer Science, 1995. -Internet Address: {\verb|http://www.cs.umn.edu/~karypis|}. -\bibitem{BLAS1} -Lawson, C., Hanson, R., Kincaid, D. and Krogh, F., - Basic {L}inear {A}lgebra {S}ubprograms for {F}ortran usage, -{ACM Trans. Math. Softw.} vol.~{5}, 38--329, 1979. - -\bibitem{machiels} -{Machiels, L. and Deville, M.} -{\em Fortran 90: An entry to object-oriented programming for the solution - of partial differential equations.} -{ACM Trans. Math. Softw.} vol.~{23}, 32--49. -\bibitem{metcalf} -{Metcalf, M., Reid, J. and Cohen, M.} -{\em Fortran 95/2003 explained.} -{Oxford University Press}, 2004. -% -%% \bibitem{DD2} -%% B.~Smith, P.~Bjorstad and W.~Gropp, -%% {\em Domain Decomposition: Parallel Multilevel Methods for Elliptic -%% Partial Differential Equations}, -%% Cambridge University Press, 1996. -% -\bibitem{MPI1} -M.~Snir, S.~Otto, S.~Huss-Lederman, D.~Walker and J.~Dongarra, -{\em MPI: The Complete Reference. Volume 1 - The MPI Core}, second edition, -MIT Press, 1998. -% -\end{thebibliography} +\include{bibliography} \end{document} %%% Local Variables: diff --git a/docs/userguide.pdf b/docs/userguide.pdf index 8d64e33a..07569223 100644 --- a/docs/userguide.pdf +++ b/docs/userguide.pdf @@ -9,74 +9,92 @@ endobj << /S /GoTo /D (section.1) >> endobj 12 0 obj -(1 Introduction) +(1 General Overview) endobj 13 0 obj -<< /S /GoTo /D (subsection.1.1) >> +<< /S /GoTo /D (section.2) >> endobj 16 0 obj -(1.1 Programming model) +(2 Notational Conventions) endobj 17 0 obj -<< /S /GoTo /D (section.2) >> +<< /S /GoTo /D (section.3) >> endobj 20 0 obj -(2 Preconditioner routines) +(3 Code Distribution) endobj 21 0 obj -<< /S /GoTo /D (section*.2) >> +<< /S /GoTo /D (section.4) >> endobj 24 0 obj -(mld\137precinit) +(4 Configuring and Building MLD2P4) endobj 25 0 obj -<< /S /GoTo /D (section*.5) >> +<< /S /GoTo /D (section.5) >> endobj 28 0 obj -(mld\137precset) +(5 Getting Started) endobj 29 0 obj -<< /S 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-11.9552 Td[(In)28(ten)28(t:)]TJ/F29 9.9626 Tf 33.4856 0 Td[(in)]TJ/F8 9.9626 Tf 9.5475 0 Td[(.)]TJ -43.0331 -11.9552 Td[(Sp)-27(ec)-1(i\014)1(e)-1(d)-332(as)-1(:)-444(a)-333(s)-1(tr)1(uctured)-333(dat)1(a)-334(of)-333(t)28(yp)-27(e)]TJ +/F15 10.9091 Tf 253.5385 365.0222 Td[(di)1(s)-1(t)-327([)]TJ/F39 10.9091 Tf 24.2415 0 Td[(?)]TJ/F15 10.9091 Tf 5.9242 0 Td[(],)-328(v)28(e)-1(r)1(s)-1(ion)-327(2.0,)-328(for)-327(distri)1(bu)1(te)-1(d)-327(s)-1(p)1(arse)]TJ -189.7864 -13.5492 Td[(factori)1(z)-1(ati)1(on)-333(and)-333(solv)28(e.)-445(S)1(e)-1(e)-333(the)-334(T)83(ab)1(le)]TJ ET -1 0 0 1 346.5264 456.5647 cm +1 0 0 1 277.6452 351.473 cm 0 0 1 rg 0 0 1 RG -1 0 0 1 -346.5264 -456.5647 cm +1 0 0 1 -277.6452 -351.473 cm BT -/F34 9.9626 Tf 346.5264 456.5647 Td[(psb)]TJ +/F15 10.9091 Tf 277.6452 351.473 Td[(6.1.1)]TJ ET -1 0 0 1 362.8451 456.5647 cm -q -[]0 d -0 J -0.3985 w -0 0.1992 m -3.1382 0.1992 l -S -Q -1 0 0 1 -362.8451 -456.5647 cm +1 0 0 1 300.0695 351.473 cm +0 g 0 G +1 0 0 1 -300.0695 -351.473 cm BT -/F34 9.9626 Tf 365.9833 456.5647 Td[(desc)]TJ +/F15 10.9091 Tf 303.7059 351.473 Td[(for)-333(d)1(e)-1(t)1(ails.)]TJ ET -1 0 0 1 387.5322 456.5647 cm -q -[]0 d -0 J -0.3985 w -0 0.1992 m -3.1382 0.1992 l -S -Q -1 0 0 1 -387.5322 -456.5647 cm +1 0 0 1 93.9178 339.4341 cm +0 g 0 G +1 0 0 1 -93.9178 -339.4341 cm BT -/F34 9.9626 Tf 390.6705 456.5647 Td[(type)]TJ +/F15 9.9626 Tf 99.8954 331.0655 Td[(P)28(ar)1(am)-1(ete)-1(r)-8727(Allo)28(w)28(e)-1(d)-333(v)56(alu)1(e)-1(s)]TJ 0 -11.9552 Td[(\050)]TJ/F52 9.9626 Tf 7.1952 0 Td[(what)]TJ/F15 9.9626 Tf 20.9214 0 Td[(\051)-10064(\050)]TJ/F52 9.9626 Tf 111.3317 0 Td[(val)]TJ/F15 9.9626 Tf 15.691 0 Td[(\051)]TJ/F52 9.9626 Tf -155.1393 -11.9551 Td[(mld_coarse_mat_)]TJ/F15 9.9626 Tf 132.253 0 Td[('DIS)1(T)-1(R')1(,)-334(')1(REPL')]TJ 0 -11.9552 Td[(Coarse)-334(Matr)1(ix:)-444(di)1(s)-1(tr)1(ibu)1(te)-1(d)-333(or)-333(r)1(e)-1(p)1(licate)-1(d)]TJ/F52 9.9626 Tf -132.253 -11.9552 Td[(mld_coarse_solve_)]TJ/F15 9.9626 Tf 132.253 0 Td[('ILU',)-333('M)1(ILU',)-333('ILUT',)-333('SLU')1(,)-334(')1(UMF',)-333(S)1(LUDIST',)-333(BJA)28(C)-1(????)]TJ 0 -11.9551 Td[(Av)56(ailab)1(le)-334(Coarse)-334(solv)28(e)-1(r)1(.)]TJ 0 -11.9552 Td[(On)1(ly)-333(SLUDIS)1(T)-334(e)-334(BJA)28(C)-334(can)-333(b)-27(e)-334(use)-1(d)-333(wh)1(e)-1(n)-333(coarse)-334(matrix)-333(is)-333(distri)1(but)1(e)-1(d)]TJ/F52 9.9626 Tf -132.253 -11.9552 Td[(mld_coarse_BJAC_sweeps_)]TJ/F15 9.9626 Tf 132.253 0 Td[(\050NON)-333(V)111(A)-333(B)-1(E)1(NE)-333(mldcoarse)-1(sw)27(ee)-1(p)1(s)-1(\051)-333(n)28(u)1(m)27(b)-27(er)-333(of)-333(B)-1(l)1(o)-28(c)27(k-Jacobi)-333(sw)27(ee)-1(p)1(s)-334(when)-333(B)-1(J)1(A)27(C)-333(is)-334(u)1(s)-1(ed)-333(as)-334(coars)-1(est)-334(solv)28(er)]TJ/F52 9.9626 Tf -132.253 -11.9551 Td[(mld_coarse_fill_in_)]TJ/F15 9.9626 Tf 132.253 0 Td[(lev)28(e)-1(l)-333(of)-333(\014)1(ll-in)-333(in)-332(MILU)-334(an)1(d)-333(ILU)-334(f)1(ac)-1(t)1(orization)]TJ 0 -11.9552 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BT -/F8 9.9626 Tf 171.1837 630.4128 Td[(M.)-443(S)1(nir)1(,)-471(S)1(.)-443(Otto,)-470(S.)-443(Huss-)-1(Lederman,)-470(D.)-443(W)83(al)1(k)28(e)-1(r)-443(an)1(d)-443(J.)-443(Don)1(garra,)]TJ/F19 9.9626 Tf 300.6449 0 Td[(MPI:)]TJ -300.6449 -11.9552 Td[(The)-365(C)1(o)-1(m)1(plete)-365(R)50(ef)1(er)51(enc)51(e.)-365(V)76(ol)1(ume)-365(1)-365(-)-365(The)-365(MPI)-365(Cor)52(e)]TJ/F8 9.9626 Tf 228.8028 0 Td[(,)-343(s)-1(ec)-1(on)1(d)-341(e)-1(d)1(ition)1(,)-343(MIT)]TJ -228.8028 -11.9551 Td[(Pr)1(e)-1(ss)-1(,)-333(1998.)]TJ +/F15 10.9091 Tf 164.1628 293.6754 Td[(P)84(.)-336(V)83(an)28(\024)472(ek,)-337(J.)-336(M)1(andel)-336(and)-336(M)1(.)-336(B)-1(r)1(e)-1(zina,)]TJ/F41 10.9091 Tf 181.6433 0 Td[(A)25(l)1(gebr)51(aic)-360(Mul)1(tigri)1(d)-361(by)-360(Smo)51(oth)-1(e)52(d)]TJ -181.6433 -13.5492 Td[(A)51(ggr)51(e)51(gation)-407(for)-407(S)-1(e)52(c)51(on)-1(d)-407(an)-1(d)-407(F)77(ourth)-407(Or)51(der)-407(El)-51(l)1(iptic)-407(P)-1(r)52(oblems)]TJ/F15 10.9091 Tf 286.4035 0 Td[(,)-401(Compu)1(t-)]TJ -286.4035 -13.5492 Td[(in)1(g,)-333(1996,)-334(56,)-333(179-196.)]TJ ET -1 0 0 1 150.7049 90.4377 cm +1 0 0 1 141.7385 91.633 cm 0 g 0 G -1 0 0 1 -150.7049 -90.4377 cm +1 0 0 1 -141.7385 -91.633 cm BT -/F8 9.9626 Tf 317.5791 90.4377 Td[(16)]TJ +/F15 10.9091 Tf 315.6115 91.633 Td[(22)]TJ ET -1 0 0 1 494.416 90.4377 cm +1 0 0 1 500.3935 91.633 cm 0 g 0 G endstream endobj -216 0 obj << +271 0 obj << /Type /Page -/Contents 217 0 R -/Resources 215 0 R +/Contents 272 0 R +/Resources 270 0 R /MediaBox [0 0 595.2756 841.8898] -/Parent 214 0 R +/Parent 250 0 R >> endobj -218 0 obj << -/D [216 0 R /XYZ 150.7049 740.9981 null] +273 0 obj << +/D [271 0 R /XYZ 141.7385 740.0018 null] >> endobj -219 0 obj << -/D [216 0 R /XYZ 150.7049 716.0915 null] +274 0 obj << +/D [271 0 R /XYZ 141.7385 715.0952 null] >> endobj -220 0 obj << -/D [216 0 R /XYZ 150.7049 676.2963 null] +275 0 obj << +/D [271 0 R /XYZ 141.7385 643.3351 null] >> endobj -221 0 obj << -/D [216 0 R /XYZ 150.7049 644.4158 null] +276 0 obj << +/D [271 0 R /XYZ 141.7385 579.8689 null] >> endobj -215 0 obj << -/Font << /F8 66 0 R /F19 96 0 R >> +277 0 obj << +/D [271 0 R /XYZ 141.7385 530.558 null] +>> endobj +278 0 obj << +/D [271 0 R /XYZ 141.7385 480.944 null] +>> endobj +279 0 obj << +/D [271 0 R /XYZ 141.7385 444.8793 null] +>> endobj +280 0 obj << +/D [271 0 R /XYZ 141.7385 395.2653 null] +>> endobj +281 0 obj << +/D [271 0 R /XYZ 141.7385 345.6513 null] +>> endobj +282 0 obj << +/D [271 0 R /XYZ 141.7385 309.5866 null] +>> endobj +270 0 obj << +/Font << /F15 78 0 R /F41 90 0 R /F52 143 0 R >> /ProcSet [ /PDF /Text ] >> endobj -155 0 obj -[50 0 R /Fit] -endobj -154 0 obj -[50 0 R /Fit] -endobj -128 0 obj -[50 0 R /Fit] -endobj -179 0 obj << -/Length1 744 -/Length2 1109 +235 0 obj << +/Length1 735 +/Length2 1050 /Length3 532 -/Length 2385 +/Length 2317 >> stream -%!PS-AdobeFont-1.1: CMMI7 1.100 -%%CreationDate: 1996 Jul 23 07:53:53 +%!PS-AdobeFont-1.1: CMR8 1.0 +%%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin -/version (1.100) readonly def +/version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def -/FullName (CMMI7) readonly def +/FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def -/ItalicAngle -14.04 def +/ItalicAngle 0 def /isFixedPitch false def end readonly def -/FontName /BVYXMW+CMMI7 def +/FontName /YTJMMN+CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for -dup 105 /i put +dup 49 /one put readonly def -/FontBBox{0 -250 1171 750}readonly def +/FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec -oc;j~EЪ)s̾;.;rTejiK/df5A|{S/ )Sc\^ȟmp+#vL17~k d# -]LeVߐGoo٥\k 9Mv92#:iQ?.n>%jzGm >NjS])7uևUQh4)-×p#z{8YаGP9 )"Pƣ3ГzYYZgp_PC߬CܔDC'<|(Rl/IxXMYzntԟlAv)|*iQP3|TxL" ʃ?Aw·pҋ:r)6,N{?JoA剳Js(0T9$ն''0j%&4LwQcKHpY Bֶ6$bZXZmX>B#9xҥ6IUJeG#݆>@P1o(ߋUcCо"[D.)>;@LyB Gp^yc96d{f'(~z[ܸQRn|48!΍eϟg'_$R;pb>زX _|M%r8U{^^1="\[dZB6CoK8/Gƪ86g84ʹLeoI4&>NU[pf[ž@vam(8a.Qr T+IFdj=aY@O)UQ4c,>inY( (o -V!gJb`B*g9!+spҞ @1YaČd_*\hIլ-$u0000000000000000000000000000000000000000000000000000000000000000 +oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou\@[6]nhmlhaH+4/?3&n=a6E#|~.ԅˡw,"rg[eHi>u +Wת>~ӖѿJmdvA +[4|ܾ.Vz_1. Ff |X9^Rw] +ۊ31S\DKZW` P^3 eێ 8`2?l;Ȋ!eXxh +Df@=$*IBXԏ!Wx-b1 C; p}hQi=HXbgbmǣR,>_Q! +Z؞Ar]< Qjt8?B,mN3v լǵ+K6 t{AJcQ%r?v60RmXsŤ+pǸi:Us)kPW\~; 7T.N }.&:D~zp0C_HOzCByAOve1k`G|f28'j0&®uk6sw, I5&{oP}p2d +CA!8Z⓪٦ǎaBLeb>HC;*B|^um%;J|uwcўПQ:F.vÕ3d^~ =Ov$LQ=^vSOE;{L|,7No]dFjVېm즀Z{sl> endobj -178 0 obj << +234 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /BVYXMW+CMMI7 -/ItalicAngle -14.04 -/StemV 81 +/FontName /YTJMMN+CMR8 +/ItalicAngle 0 +/StemV 76 /XHeight 431 -/FontBBox [0 -250 1171 750] +/FontBBox [-36 -250 1070 750] /Flags 4 -/CharSet (/i) -/FontFile 179 0 R +/CharSet (/one) +/FontFile 235 0 R >> endobj -223 0 obj -[404 ] +284 0 obj +[531 ] endobj -222 0 obj << +283 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 105/i 106/.notdef] +/Differences [ 0 /.notdef 49/one 50/.notdef] >> endobj -125 0 obj << -/Length1 915 -/Length2 2883 +232 0 obj << +/Length1 745 +/Length2 580 /Length3 532 -/Length 4330 +/Length 1857 >> stream -%!PS-AdobeFont-1.1: CMR8 1.0 -%%CreationDate: 1991 Aug 20 16:39:40 +%!PS-AdobeFont-1.1: CMSY8 1.0 +%%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def -/FullName (CMR8) readonly def +/FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def -/ItalicAngle 0 def +/ItalicAngle -14.035 def /isFixedPitch false def end readonly def -/FontName /OELLFO+CMR8 def +/FontName /DHYKSS+CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for -dup 45 /hyphen put -dup 84 /T put -dup 97 /a put -dup 99 /c put -dup 101 /e put -dup 103 /g put -dup 104 /h put -dup 105 /i put -dup 110 /n put -dup 114 /r put -dup 115 /s put -dup 116 /t put -dup 118 /v put +dup 0 /minus put readonly def -/FontBBox{-36 -250 1070 750}readonly def +/FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec -oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou\@[6]nhmlhaH+4/?3&n=a6E#|~.ԅˡw,"rg[eHi>u -Wת>~ӖѿJmdvA -[4|ܾ.Vz_1. Ff |X9^Rw] -ۊ31S\DKZW` P^3 eێ 8`2?l;Ȋ!eXxh -Df@=$*IBXԏ!Wx-b1 C; p}hQi=HXbgbmǣR,>_Q! -Z؞Ar]< Qjt8?B,mN3v լǵ+K6 t{AJcQ%r?v60RmXsŤ+pǸi:Us)kPW\~; 7T.N }.&:D~zp0C_HOzCByAOve1k`G|f28'j0&®uk6sw, I5&{oP}p2d -CA!8Z⓪٦ǎaBLebG\[F_}+w!JLz>:<,?˲s =,Fj#V?" y(=D{qPƷC6w"B ҥ!aN arE[##oS)MUL*sI8<0_?k`ˠE女 vFyx8Ԑ%6lF"aJ^wtBy͖gV~J 2g/dƦ+]:KU¡S8D( $"ۀ"36y0%b9y]5 q4aB>I~]\q޼sSWP4k[A6GXzBA74eUa t$.WPH*SFI[U7X; -B~,A6`)M> !]}Ȗ =-2#Nmh,0\)E52 8S+\%jQρyJ>R4*׿z6zuF+OfRɻy3FJ":>T@3T?߄jم -w{]5ρcP`%h,P$s'6HܚFLQnf{۫ -'%lU[Ode@Etl>OgGQZb D,DsǴ#r %j!u]} ӆ}!Ʉ ]jy+)P['%t=1:՘o#\>ڃ -yQu3u\:Yڶ3Et>Gb%^6X? :6"Y iKΡv1b \Wh~oC>̎EػffPf#48#|Nǿaԫ#O=QiP d=אkŷЗF-1".V!svyȣLcpDn<g8$kxl!凌eb:#܏4^B)@?~nb*>RlSY4~dz?S*$:R_C^%f?,:yji~DPDl> `&յCǗn$MFħucKkD&ʼa@ӹ2`,QuBpAdk*3ԓuɬ$ꦋU`]|J"4=iM_XGKs[#i[^]+GsN6;>5XMqA^gw!Pc-SH.t#@zA] +:w+bo9cϲ Σ_@C&вk45(~kY|ة +V+R`Fg> endobj -124 0 obj << -/Ascent 694 +231 0 obj << +/Ascent 750 /CapHeight 683 /Descent -194 -/FontName /OELLFO+CMR8 -/ItalicAngle 0 -/StemV 76 +/FontName /DHYKSS+CMSY8 +/ItalicAngle -14.035 +/StemV 89 /XHeight 431 -/FontBBox [-36 -250 1070 750] +/FontBBox [-30 -955 1185 779] /Flags 4 -/CharSet (/hyphen/T/a/c/e/g/h/i/n/r/s/t/v) -/FontFile 125 0 R +/CharSet (/minus) +/FontFile 232 0 R >> endobj -225 0 obj -[354 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 0 767 0 0 0 0 0 0 0 0 0 0 0 0 531 0 472 0 472 0 531 590 295 0 0 0 0 590 0 0 0 414 419 413 0 561 ] +286 0 obj +[826 ] endobj -224 0 obj << +285 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 45/hyphen 46/.notdef 84/T 85/.notdef 97/a 98/.notdef 99/c 100/.notdef 101/e 102/.notdef 103/g/h/i 106/.notdef 110/n 111/.notdef 114/r/s/t 117/.notdef 118/v 119/.notdef] +/Differences [ 0 /minus 1/.notdef] >> endobj -121 0 obj << +219 0 obj << /Length1 735 -/Length2 1052 +/Length2 1053 /Length3 532 -/Length 2319 +/Length 2320 >> stream -%!PS-AdobeFont-1.1: CMR6 1.0 -%%CreationDate: 1991 Aug 20 16:39:02 +%!PS-AdobeFont-1.1: CMR7 1.0 +%%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def -/FullName (CMR6) readonly def +/FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def -/FontName /ICOPRL+CMR6 def +/FontName /PXXUZA+CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -3522,13 +2918,16 @@ end readonly def 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def -/FontBBox{-20 -250 1193 750}readonly def +/FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec -oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou\@[6]nhmlhaH+4/?3&n=a6E#|~.ԅˡ}_B$~\|"4Pxҍ>P% ~ߏ4q.C3s蛼q翈by?Z72z6LpHC1D"28s B -~ OPQ -O\O}';Pǂ;߉~ I;^קf5)P)6Ö9}ln~E1 k N|dEmϫNO3h>vM83M- ۃB3?(qJ9k7I%֧N93πh.ÉV6k#8"++=fw6Naʹn'`goNq#./eJ[$2E;ԣ *z͕Qt1kBa\ye`90000000000000000000000000000000000000000000000000000000000000000 +oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou\ʻl24`}IRB s\pzEGAۋa4_p0uAEG)Q|8B*dGWj'0Vix#(=@'qIT5i +(Ѵt` ?E)4E^yҴF+$8&G"Yl?eYjt+TI2! nXKt`˱~h]{>Qaiσ\mFU#Gvh2un8W;R4Ej1B4b ;b9y[lV:P񕈇9g`\J~ O>%V0 ݬ;cJԺjYrt6yql4Qo٫~o_=6xqw,",kϷo +#vI7_ fz!y(3)5*кYb#aM?3G%wG*M"/z%+x DxYWh&D6/.qC2byU\7LǫAב: +aW0Wcaqj@jn5pRH\`J+#\ȕTMxR0EZD(64EJ^a^M:X#;A{ M +=u0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 @@ -3539,75 +2938,69 @@ currentfile eexec cleartomark endstream endobj -122 0 obj << +220 0 obj << /Type /Font /Subtype /Type1 -/Encoding 226 0 R +/Encoding 287 0 R /FirstChar 49 /LastChar 49 -/Widths 227 0 R -/BaseFont /ICOPRL+CMR6 -/FontDescriptor 120 0 R +/Widths 288 0 R +/BaseFont /PXXUZA+CMR7 +/FontDescriptor 218 0 R >> endobj -120 0 obj << +218 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /ICOPRL+CMR6 +/FontName /PXXUZA+CMR7 /ItalicAngle 0 -/StemV 83 +/StemV 79 /XHeight 431 -/FontBBox [-20 -250 1193 750] +/FontBBox [-27 -250 1122 750] /Flags 4 /CharSet (/one) -/FontFile 121 0 R +/FontFile 219 0 R >> endobj -227 0 obj -[611 ] +288 0 obj +[569 ] endobj -226 0 obj << +287 0 obj << /Type /Encoding /Differences [ 0 /.notdef 49/one 50/.notdef] >> endobj -118 0 obj << -/Length1 751 -/Length2 1203 +216 0 obj << +/Length1 745 +/Length2 581 /Length3 532 -/Length 2486 +/Length 1858 >> stream -%!PS-AdobeFont-1.1: CMR7 1.0 -%%CreationDate: 1991 Aug 20 16:39:21 +%!PS-AdobeFont-1.1: CMSY7 1.0 +%%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def -/FullName (CMR7) readonly def +/FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def -/ItalicAngle 0 def +/ItalicAngle -14.035 def /isFixedPitch false def end readonly def -/FontName /MGZAWE+CMR7 def +/FontName /MOOCOC+CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for -dup 49 /one put -dup 50 /two put +dup 0 /minus put readonly def -/FontBBox{-27 -250 1122 750}readonly def +/FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec -oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou\ʻl24`}IRB s\pzEGAۋa4_p0uAEG)Q|8B*dGWj'0Vix#(=@'qIT5i -(Ѵt` ?E)4E^yҴF+$8&G"Yl?eYjt+TI2! nXKt`˱~h]{>Qaiσ\mFU#Gvh2un8W;R4Ej1B4b ;b9y[lV:P񕈇9g`\J~ O>%V0 ݬ;cJԺjYrt6yql4Qo٫~o_=6xqw,",kϷo -#vI7_ fz!y(3)5*кYb#aM?3G%wG*L4}. f0Fp)_@ڴ -.E z~Fi_F $oG^b=09u61%7ZKUe.E'fe 0cGy\ɑk+zz;U5{CNjvcr0&=Xw;7/LJgz}8̚ug@{.'b[>PP.DW,={ #VX6nzT`qN&=`~+GH c%=TF{v/҄fwsj j-${fwkxze4G$JB%p"&g4BW[x-n)ǡ~('?CE-QeSO9sY=`lz%i$/neu*TVyBܡ mM/|8ySx{+˄sߨ39Ӟ:QXE':c *ȝ?/vwC'kH0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 @@ -3618,41 +3011,41 @@ _2 cleartomark endstream endobj -119 0 obj << +217 0 obj << /Type /Font /Subtype /Type1 -/Encoding 228 0 R -/FirstChar 49 -/LastChar 50 -/Widths 229 0 R -/BaseFont /MGZAWE+CMR7 -/FontDescriptor 117 0 R +/Encoding 289 0 R +/FirstChar 0 +/LastChar 0 +/Widths 290 0 R +/BaseFont /MOOCOC+CMSY7 +/FontDescriptor 215 0 R >> endobj -117 0 obj << -/Ascent 694 +215 0 obj << +/Ascent 750 /CapHeight 683 /Descent -194 -/FontName /MGZAWE+CMR7 -/ItalicAngle 0 -/StemV 79 +/FontName /MOOCOC+CMSY7 +/ItalicAngle -14.035 +/StemV 93 /XHeight 431 -/FontBBox [-27 -250 1122 750] +/FontBBox [-15 -951 1252 782] /Flags 4 -/CharSet (/one/two) -/FontFile 118 0 R +/CharSet (/minus) +/FontFile 216 0 R >> endobj -229 0 obj -[569 569 ] +290 0 obj +[893 ] endobj -228 0 obj << +289 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 49/one/two 51/.notdef] +/Differences [ 0 /minus 1/.notdef] >> endobj -107 0 obj << -/Length1 1125 -/Length2 4973 +142 0 obj << +/Length1 1907 +/Length2 11959 /Length3 532 -/Length 6630 +/Length 14398 >> stream %!PS-AdobeFont-1.1: CMTT10 1.00B @@ -3668,21 +3061,66 @@ stream /ItalicAngle 0 def /isFixedPitch true def end readonly def -/FontName /VCYLXU+CMTT10 def +/FontName /BRVQRR+CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for +dup 33 /exclam put +dup 34 /quotedbl put +dup 39 /quoteright put +dup 40 /parenleft put +dup 41 /parenright put +dup 42 /asterisk put +dup 44 /comma put +dup 45 /hyphen put dup 46 /period put dup 47 /slash put +dup 48 /zero put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 52 /four put +dup 56 /eight put +dup 57 /nine put dup 58 /colon put +dup 59 /semicolon put +dup 61 /equal put +dup 65 /A put +dup 66 /B put +dup 67 /C put +dup 68 /D put +dup 69 /E put +dup 70 /F put +dup 71 /G put +dup 72 /H put +dup 73 /I put +dup 74 /J put +dup 76 /L put +dup 77 /M put +dup 78 /N put +dup 79 /O put +dup 80 /P put +dup 81 /Q put +dup 82 /R put +dup 83 /S put +dup 84 /T put +dup 85 /U put +dup 86 /V put +dup 87 /W put +dup 88 /X put +dup 89 /Y put +dup 94 /asciicircum put dup 95 /underscore put +dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put +dup 102 /f put +dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put @@ -3691,42 +3129,140 @@ dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put +dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put +dup 120 /x put dup 121 /y put +dup 122 /z put dup 126 /asciitilde put readonly def -/FontBBox{-4 -235 731 800}readonly def +/FontBBox{-4 -235 731 800}readonly def +currentdict end +currentfile eexec +oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou_c2Bطj=-8\Dg݌] /%bԺnٻڿSy b*L(9sWF R:EMksH02E?Oe+Z'zK΃оif <,EDNZ|J#BW3ϗMp(%5%~(5Bk}Aj c'S-8*!iy$G.w g J0nr +;଍ Kr^ox7p4= CyO#y[49'n*xx/J}o֯p6Oޏe}hà1W6Ϋ%(i]Ii +텴""b3=5sJzbJ>[Wе H 2"ێ0Oy9.t:=$EǪ ]to`% +'EKKfrRK^y`vO^v~ZwR iNMW3HSp+T,q!s0(ع;U+3"J8q3dJ`77+IuؿhH<=!'T́2.r% +v y*S\L$r)_`5_O㛘H&ESq=* 4vDg%+POS6ՕlQz0ȝ/A+fA!PͪzVe.&(՛BEGI n6^0QMvn$1S (DMѰgifPjs-(2'`wzAN&> _tOz`Mu~o$'6L[ݽz> X 7 ٣"c8t$BjgOHtMI̓E);s:(cܶ- ucӁ žBMQ)+\ V0'އB5[|0c 5MV"v-2gM +vסpeS duu+;<nL +.>3e}TQ l:"Pq%/k|3m ACs/8;\ V0$ILu1[&:E nM¯˧srW)3"G9=4Z&񮣡Ѱ^P>ז姜aw5{ladjEd7,O_Ұ2OeJejF8+WТz"ްcUc6&B >)LPs"@yֹU[gULe${ʇZ:]gM7`Ҷ=v!VXptȢof-S0n DJre8FvX1 +&3hsrpGOER!; xqm&\d=:R.D>}ZG_ziP wQsvΙ30kND#C?X* XH nUw}6uHF /-ml3A/Y /bz>;6w,K僜!d%oo_ykȗQEd.Ck4 FgȜJ9f.FE"\0L?̷(RtT^Y}d.!.!X 4 GJ|/wǜV 4uL#1rrŕLVdEO,y:HZ-|V,Nǀ_ea5_m +%þjB)oɢ롊/yػnhb8sI`DDq)]p8bo( jocb,o]I`e`Ob QC](Q8]q ``-)i`5AX n ۣ>j!H앑lf `}x75ư_@u!Y]x^snxшu-fU$--N +h}ByF+<]0 87rì{v1:c/ :gnjA#UV|Ǥ½TY\Kdh;d"kPaoۺ%;dVt[@ZgCdwC;ڏis~?;4tFӾ,ЛGK.I$ HbʣTN( < GՁv,&mEy3=?ݜJ8'u`BsݴmW>V0چɸҏR1քE0} +j" R)5nߺ{zGdCwnͩ\k7],+"g׍5OT@:67U |+GMTZd|3n5qԶ->5oEVŏVM=ߠ}YTnp95_d:h1R QʬNGPmcRE /]Ճtv1$]+|LA +!4,#wVs[ pHr,;V)0YXd13MquRX(3s\ R۫6v4ԭ3 +$ xPPҎ.Dsu[$fc;M5vWz/'65h ^iO1vQx dy;K.Dʐ+RWX i<4/"β`f^Ż$ g&:t[\Fw^ŎX[4WhФvح_3{S0A%i'jQ{:IT8x=77fk]Q,42i:k6jSHA61Mw^O< \ƌL9M,=ZAM@.X,r q-<.Zy\ JIaO狤ij ?NH3kz7v%S0r/+5f)5FJ!^[ x[\D$Yf|z&+z!sށ~lqF  +J0pVthU;c>QV Z /݊@i=LMɖOz+^3 }[.z? +GT%X > +C}6zdq΋ժ"qPq:"B@BJeqAo؜8Me?#+z@$&_*4ʜI s"'ӛZ{BG'rLke~Gб ><H('$+X^T:Jm"?61-i};^B,u)Whc7e^q3 T.%c0 U7gL.n[݀tD>o^,+GT^]}jg@2q)"fJ[Y ߉ez.@<7D?rE_H61pH +,.{aRh oБ |X/[ uþKĚ%.|s0soӚs9/ 2/;~|1̪>@|`ࠫiRgnUmr%ɹ&7.w}3PFk|DL 2xfjJ k6+UTRa MܭznG./+EwjL^qKFmb׆]p?&u^6G` @kTy^.50?Fuy}5 vا\l @hx-TaT`1 #x+' 2cTS.7D`VKcAĂW}+Qr*eLџLy0?`Va 9[5V0ü~i U5y˫ؼC|kTMM.?-l8MbP-rn24,+*gx-O'4]@9sv0Uy|21>Y02Ua?+pBly*r)ֿ M!b_=p瑝G^3'[w': _RExM[qoSZ^o.Iy 禡=7 ᕺO=tQ[?q=ݯ.4oM<0\L#/0N41saS`(@iR5D;sMM32騵|ȆA<9+mu_z,ϕ 霶5D*<ŪDl. a0{+v>,_>6.11DG,-t:6s#҈蛖Xv b/Gv83C^@__^UyAkMmyni4opN!<#8BO!VDY:1~DAA "p H,IiBJ=JE(tEMJ j|n>+gCї-z{I*-U528}D \q*5 FQz܁=R .!zo?!ˈAVGv]|& ё>5~JA6•spHMͭU<{~;Q噛h·Qfzf>iϛ +$crK.TTJ5p7Ou1cBKBP?[7]DہX`MNSBC|5qs u j[<@+YZ/ }^௑Xqm4f  z2uГ+jl%4t(\ppoze/Afv jsRG7rk0z$:U|CwxZ{ea,~!}SسIx-M$)r{|]B+Kg;畊0حK~A}wlFC_mHޣn;[BBp(ա%KǶn&$^쐲%fH֙5£kd={UKhϞcw! 0`}WqkʓH,{.o.؇˦v{zi=Rۙa! $%M=h ]C8߲ !RXB^N:ˈ`-2I,.uN*scE{ O"̳ 9_pYi c x[fNV_A:/%a Vu``M0RvzҮډM[ tMyJ.YkȷnS⡴g$W$瞧A贐ѝ~ewXSQuCqWݞUqAhLQh< +?N!mC&ѤԖ1sv]BdfDp2XDD{q- el)WO y~_d,=2Wsj59 +U^q }IMO`t[nnMr)r:0,cy (i5aG0s?B;wwIRq躜yAZ䡏=bil=i*CLgk턭TWسweO4vBo U[ Ep~. Ձ/ʟ5gR&E ǟ4EBN3&e$Y/`=|9FN2 +8H#6:bz^e^\OxnY#$~s6qlAth +9M{"a^PI1ɧXZ=D,O:I1@52jPetyGCj '^`㕯BQnd#b7E!qy!1n9WM/C_t3ɕ[^ Љ.] I;biT -%F;{C$ v+tlecT'[wYg +$:3;-!g.?0mĩc| (ٽQ(.sҍe|O2cn*Cv%;ljwI9be5T47"o*@\_*=\ȘD:kOݱpCh_m&p6 l"S 5M>;)š^jΥB 3]JZ)|''@s]`le4eB!k'UZxMĭo߼0N5-?fn]G5^i] + +1]33 +)ђ][zO5+,7b.?͟ejaQmA!jotʜVAvvƌpqݪf#Qyw4zn2J$l֛ppv=}PUc?~Yoٙ GD ۝X:Z{'ȢK6ևCs;ƍD%0TMEsD"qbOcpCjI#gNNN>=@1{`ggk{H_HQ*Ǐ9j}/eOFkлڑ`$NԧEF^dL! I[uTkeyXI$~[)19@˽Sh{*DclW3s JwGEvĻ!rQjJ !*%k8nEe*rH; 姂'BHUf;%tO_[+-D$znUdhbfSw2꩖2!+v%Nb8\.YUq\1|@ vR܉=KnwH>+O&@8MܧHĞ H.j=p% e/s]㦦ZI흗i.I$ +k%Q<ň|\ Hs heN%*ȾH>}܄4+)a,ܙpǼfOᦱ-WYp!p6(G&nQ#mp.c4Z#b$0X-06ܲb  Zn2(^.<ӳѽ3q=PGNܔ.e7fJNZMR +KUCh1(2Q9D9 ¬m!:˥Qy U\5|Cdq~֭Mmf~J]͢~,هA\#%SglQ՜b;(9YzvHy @8G\>I`e`C6:n&4K~SiÉ{j2.ġOzw t'A. a8CpXDa*] +a{sX^צzk +΀&`;~F#? Fu57uS["Vj0,3 XI$P8^nM=q(79M=bth'W-_}y+ cu66q7L:〻ek|z2Uz,F+ӡcbl2WnQL bNl86WhqF (n| +/.PtD#LqRXfg=ڶr_G<%CVM5c:VIH*6ؒ lhu~7˨}gƷ  tn])ދfKDw[>#|~oZm,97\0ðG7ާޘ2q [O*$XtY ۯ2-z됼n M7A.9mD}y֤`C9.|nN"H/_sT 婺 + 9Bsd*aU9O49^xLKط>;N*Dad{ 4`<r S z-2+y3i?OsM!-cU KNa%Kr@S'{҂Tko]WxR*v9q_V6UkTE KW 8Nu$~#+JYlsJ<4;`[DgVsY%L*}{.bt؇bI95o-[jC; Py/_R[n>36{'@#<)=8jAyl%=!Ŵ'Q2Zr659WdYM7/NqL R8XN&jOiV>AoK4;p .Xq(?rBB/OU!| ,gȸfuR]od;?}6A OxP:%]|Tvw+DUcI!@zZKrtJZQ>R)*vpZz>'..I] R67U_qm|diE,רm"z Aen/mOosiF:yxF^(f Q=`ADEtq p\4/JILyX+{3!>4EnICr'^w.+Cv;i`kssdNNzOE*~7L jJ;-i+Y-y1է#(A!P܈Ƶs*y!8hlUOc0տ=>3b&wi+`4OAbaH4Δx-Veиi@hslq +jq^6ϩ~ӦD¯N1)#:A%|z1i|[8>ptO=&D7YVT]c,hD9m 'Nf5ms˨e dM@olqfOIXpeYmTAKjjrPm l~`B?%̶( +I4_"n3hNCt){q_ҍż)6WTtڿ +o1ұs.9h7/$\u,WH%$Cos;d}Sv*:Fub -|Br;'mdedIWz9P\S8b~Xؖڍڏ8+?58. 1ÎWErutM>ô* ,*ex9۞dNĕ :١U{TgrP Q/[9@~qIbon8[o\W%A`[mjOVC\T=2z-h5 [/:@ [u/#ژ? HLa ;r>?,?+4ާ.nV`q{cYvwIeإKnj0B*nc+LjaC0ՎIb>gpqĸ$w, '[/M޶!6Ȯ"5H{@feqDXTY$J + 憈eWS> endobj +141 0 obj << +/Ascent 611 +/CapHeight 611 +/Descent -222 +/FontName /BRVQRR+CMTT10 +/ItalicAngle 0 +/StemV 69 +/XHeight 431 +/FontBBox [-4 -235 731 800] +/Flags 4 +/CharSet (/exclam/quotedbl/quoteright/parenleft/parenright/asterisk/comma/hyphen/period/slash/zero/one/two/three/four/eight/nine/colon/semicolon/equal/A/B/C/D/E/F/G/H/I/J/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/asciitilde) +/FontFile 142 0 R +>> endobj +292 0 obj +[525 525 0 0 0 0 525 525 525 525 0 525 525 525 525 525 525 525 525 525 0 0 0 525 525 525 525 0 525 0 0 0 525 525 525 525 525 525 525 525 525 525 0 525 525 525 525 525 525 525 525 525 525 525 525 525 525 0 0 0 0 525 525 525 525 525 525 525 525 525 525 525 525 0 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 0 0 0 525 ] +endobj +291 0 obj << +/Type /Encoding +/Differences [ 0 /.notdef 33/exclam/quotedbl 35/.notdef 39/quoteright/parenleft/parenright/asterisk 43/.notdef 44/comma/hyphen/period/slash/zero/one/two/three/four 53/.notdef 56/eight/nine/colon/semicolon 60/.notdef 61/equal 62/.notdef 65/A/B/C/D/E/F/G/H/I/J 75/.notdef 76/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y 90/.notdef 94/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i 106/.notdef 107/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z 123/.notdef 126/asciitilde 127/.notdef] +>> endobj +127 0 obj << +/Length1 745 +/Length2 1134 +/Length3 532 +/Length 2411 +>> +stream +%!PS-AdobeFont-1.1: CMMI8 1.100 +%%CreationDate: 1996 Jul 23 07:53:54 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /VMTMFX+CMMI8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 84 /T put +readonly def +/FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec -oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j+ou_c2Bطj=-8\Dg݌] /%bԺnٻڿSy b*L(9sWF R:EMksH02E?Oe+Z'zK΃оif <,EDNZ|J#BW3ϗMp(%5%~(5Bk}Aj c'S-8*!iy$G.w g J0nr -;଍ Kr^ox7p4= CyO#y[49'n*xx/J}o֯p6Oޏe}hà1W6Ϋ%(i]Ii -텴""b3=5sJzbJ>[Wе H 2"ێ0Oy9.t:=$EǪ ]to`% -'EKKfrRK^y`vO^v~ZwR iNMW3HSp+T,q!s0(ع;U+3"J8q3dJ`77+IuؿhH<=!'T́2.r% -v y*S\L$r)Xm!UgWyD3y#<+iYPo+e=a.q%B )z=v"1); t -x<b?a,~JDgpt1UsOi1y -E|Kڤڵ8 ar|z(B]KpWSF ~p}+__EqfEXXE\:~*Ys}G[`Рb`4mWUи+ᰁKVdNzqm=7zj -h6 1 xoMgR2, ns4 BtߚC2x`ZBq2avSfW)G e?wb+M4BW6)iO.]%`N[0J;}]_a&Tn`[kFB|xpB4:Ye Xba521gcUbBq cۺkiYQEϲb^Ṋg X컄 `Ĥ1&DΨя:x7fH!ɫ2nU.o {yku p$ibzԶ.^DIБ-4gȲrS#iQ|-X j3bSc+ZiGq0eΪ[egWճic?Us2oi6fu5- ݆zmSArf?$O(ϐ$X[zmhv6}|nP+QcR -:-X2ڡ厫내EXP <ANqe -)V}=XfZ˥|cJCQئ]jS!i @6L-a?{Ʒ~zwLMsImh%k|K@‰E8)Mwl -E"rX{v-r/?IbL]a4ĺfCi6z#qi~%'m:iXRTQt]#h &oh0PD6X6Ө‘^c7h^9nƑpu>&. -Ū;ZƟo崖gH&ё2LlD!uTQ O&*zg)rI` p 25qF 3<\I|2|oa._AS<7h%CiZTۃC/}m8?!0^ݘFvY/Ӧ+&˄F[X AO`9uC/9!ɒ &k-]5 _:vcm5xCZD.,-F EY+28d rDNoQmg%"ڠ¡B(Ÿ\lGEtOocs.nq[}`Q4.#E.}sTU׋(Ud!>f*%tQ~z QcV5\J/M>lHD*v kf`BZzd)<إJU z=x(|#$n} ]*Kc|~̆YqG{|~Zw!6-C.!Bޖ: H~-]DCM;zrIH+c)2*G9DBR4x 6n= DvTs( 6evSuN^oN+MK|vhQ7lXCVO Yj9`idpȼ_*ٽN/g[i@2] ҧU}4KU(ԸPfKb@T_ ##.*H7@^?H=̪j\H5\D6<ҳt{JN(8Y@5G{*vce-ⲶFi՚{Wh}p1CjNTl6Y_ĪM8<)|QRtы~_ddYu3w#Qkld -՗:JI˖&Q)K[u#1fM,@YX譻)-~6 Wmk8<[ōZ)@c8k% w8ZDb Y8.o1Du\7N5^ >6ieUmV[T%^pH?58pކG/9hȕi&OiLY@  'J@ozlSAGX-N`hB(}aX;O -gɭI!k$p'  -iot% T-=%`؂v1zCZXNh)_ d.y__ĺW-fK;LʌcHO y8R[kk 'x*] #Ow)JuVEH{dQ_ YSs iv3Fyk܈&sI)"{ -2 g6<!{A٢џ|"ErK[}`Vh4D ؕ;˻%~6p p4MgCUG]n]KskEHzZ?YIwk;+>d ,MvW$5u/E2 vH|6Už/z)gVsK[4ğ}T'oKkKlMiQ?\G MqPi6 -mrgºC!P/ZNl ӻ YX}`:!nE !lBK*n;fʠh{zS*slvg?pdj=8'(Z\0vӜZ#C] hʊ !-~/T B(i9gԴ.+#FYqSԷ,s >bUIܪک -!D~$oD'MOFBA^guNZGrFan ]_C+E -\2ȗ\t¬#u3m҇j3G({ <ֿwnU;w ǒ0000000000000000000000000000000000000000000000000000000000000000 +oc;j~EЪ)s̾;.;rTejiK/df5A|{S/ )Sc\^ȟmp+#vL17~k d# +]LeVߐGoo٥\k 9M֨[G(aܘ|RP6n=: b9s2m4{~CD%xSd,&jA x"ПmѬ,+;ŝboB|ծTboI*VDZBJϟ2a{Y0Q(/e@AGƀ挄п]w}ga8.ݲ cS,t^Ujq.or!N]@j +Gē75uʑm$KWNWVF2P,KP˻ 62abw;SBֳ{ u&DChNe^L2Ib^ǞY62"CSl5X!okR6D UjT|S{G3:6A| ٺJ 33kLKEnMpٽqlHZoxV KË=]L+e4fv:*ys%kٞ((~~{O{m_NPJ> endobj -106 0 obj << -/Ascent 611 -/CapHeight 611 -/Descent -222 -/FontName /VCYLXU+CMTT10 -/ItalicAngle 0 -/StemV 69 +126 0 obj << +/Ascent 694 +/CapHeight 683 +/Descent -194 +/FontName /VMTMFX+CMMI8 +/ItalicAngle -14.04 +/StemV 78 /XHeight 431 -/FontBBox [-4 -235 731 800] +/FontBBox [-24 -250 1110 750] /Flags 4 -/CharSet (/period/slash/colon/underscore/a/b/c/d/e/h/i/k/l/m/n/o/p/r/s/t/u/v/w/y/asciitilde) -/FontFile 107 0 R +/CharSet (/T) +/FontFile 127 0 R >> endobj -231 0 obj -[525 525 0 0 0 0 0 0 0 0 0 0 525 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 0 525 0 525 525 525 525 525 0 0 525 525 0 525 525 525 525 525 525 0 525 525 525 525 525 525 0 525 0 0 0 0 525 ] +294 0 obj +[619 ] endobj -230 0 obj << +293 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 46/period/slash 48/.notdef 58/colon 59/.notdef 95/underscore 96/.notdef 97/a/b/c/d/e 102/.notdef 104/h/i 106/.notdef 107/k/l/m/n/o/p 113/.notdef 114/r/s/t/u/v/w 120/.notdef 121/y 122/.notdef 126/asciitilde 127/.notdef] +/Differences [ 0 /.notdef 84/T 85/.notdef] >> endobj -104 0 obj << -/Length1 1038 -/Length2 4549 +95 0 obj << +/Length1 743 +/Length2 1156 /Length3 532 -/Length 6119 +/Length 2431 >> stream -%!PS-AdobeFont-1.1: CMMI10 1.100 -%%CreationDate: 1996 Jul 23 07:53:57 +%!PS-AdobeFont-1.1: CMMI7 1.100 +%%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def -/FullName (CMMI10) readonly def +/FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /RFRKVT+CMMI10 def +/FontName /GPIOOC+CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for -dup 28 /tau put -dup 59 /comma put -dup 60 /less put -dup 62 /greater put -dup 65 /A put -dup 73 /I put -dup 76 /L put -dup 85 /U put -dup 97 /a put -dup 98 /b put -dup 99 /c put -dup 101 /e put -dup 105 /i put -dup 109 /m put -dup 112 /p put -dup 114 /r put -dup 115 /s put -dup 116 /t put -dup 120 /x put -dup 121 /y put +dup 84 /T put readonly def -/FontBBox{-32 -250 1048 750}readonly def +/FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec oc;j~EЪ)s̾;.;rTejiK/df5A|{S/ )Sc\^ȟmp+#vL17~k d# -]LeVߐGoo٥\k 9Mՙ= 0)xf -1kX9JS:6V =!X>KM)9! I}HWIUcl=GeZm>nQk2=1p%R`%$]c{!W+8?v(+)0ӈ!YahtA?H @m?|M X9Հ֨#2 -:m,MZ+ǒ\Ex9Yn<Ŀ2^FC[`T+ȬD]ǸFF$oB!>;A\T hxh|5:6;ϖUǷ6~k.T*{8K`y8'dT|-QkKmy9 91|Afr (K*)%_ʬtP`.rY?nrn{v2O_5;\g>R{4_ -: 9pEz~Gu E[n窙i9ヽu#=X&>@Q-2fT{0lY+%dm%dBag(MInǹpFt%V/~@6!tL:nS&Il*rH7d?ܡPZ=uۻ3MfM[S`fvuԳ3Ttw~^QbamM`c4Z7ʺ!@ s9G!t!-_a@Tp[SͰ5s|DmCB@uDT6QU)x2.5BEU;ɉjK#vB -e x9H}#;% u/^7Yȭ[SdtNGbBpV'@!zudL&>te?Bj%1O HKEO 53AOPpg fdxȿ)K VqnpQ, xX C ] ᣛ6;@%A3D8pT#!FD -kYDA|ۊ]Xj,GRohvgN.)9 )* -t״nߦ&_k䑏eKv%D/5Wa_eǡ̦j &":.9dolc:^_>4/MC -s'|CYȖD -_s^>@dc2W \zMvG:QOmGX~Ն]L 3SK |s 24+V\ -~bŌ m~4~Gȩ -'|1Փ~ pQOp?NCL[@'`-pq}1+0U?bSpT ~к̑K**qU{ -_[`w F6&!';uHDn7EXخԘ`U'0ߛUY^=s?t)up"dd!.;d52~ OĮdf?E+d2m~qfR$$*O@dA:7] ,FuT:,qIUDx -G^_NwÅɼ; 0Vt -vOWY2 b<o[-_hᆽR (A:gOK~xm@vʙϳS"p -m? ! 6ɤlQa` ǰ\$)o鸷[BƉM'N_sK1K K THCYViFum NvrƒȪLl-w1l,N xE{w$ T M1>cnۢ[Ar쿎'Қ]`%v; >'/C&%/M*pxވqUu/ƂQ*DeY{42c\7}$9I-ow5a!<\`tH%g>l!+1sGy+1jp9o ktHh2쁆tC~[ uT\R$@Ce]"2Gļjɶg~GhŦ'3ÍM^eI=O τM%ƍxj jCz#Aˊ3B^n>A41 4Ĕ}4b-p\\Lb 0R΃}܍[}LI3 ȨI<8WۭE5zç̦Pb>p'mK Lzr?EQ7f{E6W Aa?:+| -*:l0000>ΡJʴhWP)[\J0l%/41G,pD5%:?h0,dR91ɩ6Y/T7vFI@E0['vB.b_ Nn?'INTZsE7Mc?Lxvu{SM^ZrJ>STqR5YO~fpk}nc(Ծ)~}I؞7F9y{ZL;f= :(u H)(sS)(GDy[oV>{k"\YwU>E $;ҷb{$*@䉌+̹ ҀVUɀM6?;B+=(ɜb혧<*M!bã 3jtgUUűoMf9nxnҝffC9nbg;#7Y?f7*X-Ju⡘®8eG_lb[51(T^\{Ϟ2f$?8'N)@uȉMW#3n}b <˶`&HN3n.c41h r.yMTiʼFӗ셼qIMTA0aZ 1:QxCM5!Vd*muBy()oHHPsqzr\Qj;!dcr**D7BXWϦ8<5.fs@[B! ?f_xx -/.VSM1F|S5Rq|z5=ܒ/_3RB1Xqrȫ\O8H~8=4a d)n\)ʜ>vɕtXI {#s*NQ?otI/*=WLy&")%jzGm >NjS])7uևUQh4)-×p#z{8YаGP9 )"Pƣ3ГzYYZgp_PC߬CܔDC'<|(Rl/IxXMYzntԟlAv)|*iQP3|TxL" ʃ?Aw·pҋ:r)6,N{?JoA剳Js(0T9$ն''0j%&4LwQcKHpY Bֶ6$bZXZmX>B#9xҥ6IUJeG#݆>@P1o(ߋUcCо"[D.)>;@LyB Gp^yc96dFlVQ罭TID,eΏaXWŬq*@ +i }Ͱ4vfFʧ{ m?IF2tÕFGȐ=Nk{ULg0a܌dYbĽDT}_j.?ZI8$Wɘ<1;n,}Hι#cLוr6m|6)VrK䋪 ,tz1=ھE{z[#t;u/<(m sﶍ𻁧C۟HhfsהZ5qr_<*kwu2qy(: p~2<epQ@;-hٶu\Al=0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 @@ -3846,76 +3347,92 @@ V cleartomark endstream endobj -105 0 obj << +96 0 obj << /Type /Font /Subtype /Type1 -/Encoding 232 0 R -/FirstChar 28 -/LastChar 121 -/Widths 233 0 R -/BaseFont /RFRKVT+CMMI10 -/FontDescriptor 103 0 R +/Encoding 295 0 R +/FirstChar 84 +/LastChar 84 +/Widths 296 0 R +/BaseFont /GPIOOC+CMMI7 +/FontDescriptor 94 0 R >> endobj -103 0 obj << +94 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /RFRKVT+CMMI10 +/FontName /GPIOOC+CMMI7 /ItalicAngle -14.04 -/StemV 72 +/StemV 81 /XHeight 431 -/FontBBox [-32 -250 1048 750] +/FontBBox [0 -250 1171 750] /Flags 4 -/CharSet (/tau/comma/less/greater/A/I/L/U/a/b/c/e/i/m/p/r/s/t/x/y) -/FontFile 104 0 R +/CharSet (/T) +/FontFile 95 0 R >> endobj -233 0 obj -[437 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 0 278 778 0 778 0 0 750 0 0 0 0 0 0 0 440 0 0 681 0 0 0 0 0 0 0 0 683 0 0 0 0 0 0 0 0 0 0 0 529 429 433 0 466 0 0 0 345 0 0 0 878 0 0 503 0 451 469 361 0 0 0 572 490 ] +296 0 obj +[675 ] endobj -232 0 obj << +295 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 28/tau 29/.notdef 59/comma/less 61/.notdef 62/greater 63/.notdef 65/A 66/.notdef 73/I 74/.notdef 76/L 77/.notdef 85/U 86/.notdef 97/a/b/c 100/.notdef 101/e 102/.notdef 105/i 106/.notdef 109/m 110/.notdef 112/p 113/.notdef 114/r/s/t 117/.notdef 120/x/y 122/.notdef] +/Differences [ 0 /.notdef 84/T 85/.notdef] >> endobj -101 0 obj << -/Length1 812 -/Length2 902 +92 0 obj << +/Length1 898 +/Length2 3076 /Length3 532 -/Length 2246 +/Length 4506 >> stream -%!PS-AdobeFont-1.1: CMSY10 1.0 -%%CreationDate: 1991 Aug 15 07:20:57 +%!PS-AdobeFont-1.1: CMMI10 1.100 +%%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin -/version (1.0) readonly def +/version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def -/FullName (CMSY10) readonly def +/FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def -/ItalicAngle -14.035 def +/ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /GDGSMG+CMSY10 def +/FontName /DPJNGB+CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for -dup 0 /minus put -dup 15 /bullet put -dup 21 /greaterequal put -dup 107 /bardbl put +dup 61 /slash put +dup 65 /A put +dup 68 /D put +dup 77 /M put +dup 98 /b put +dup 107 /k put +dup 111 /o put +dup 112 /p put +dup 116 /t put +dup 120 /x put +dup 121 /y put readonly def -/FontBBox{-29 -960 1116 775}readonly def +/FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec -oc;j~EЪ/ ȭX~id}S5Q!gtⵎkJc;rN^X5.Sy +'IqV:r㚉#,# dBZ *R*"7٨y=cLIPsF'f> ba -]fv+QAwdO[x"%Sx~{p҈덡|O -BÄ/GL3h+Ng03jU1~akDzq=U}.KY碌 ֻ1?C N2Muh/4Gm |Q] -OaFދp0z -1ɛQx%:.~e`Q} t-;CH']0NbܒN4٭ -So~.,ԼʴQCMbΞ0+G7<E9˦c/H˥A.gM:^ӽÑp 8nTGU H|P A_HMNi#aj{of凣+z2(Wq^""XG0$BX;[ϸ 2W'^&* Ō@Y`)zJr8kgi˰v/bL(:dZ\h&I% 5̗n$=ɶHA^twwߔ*xnhߩX"CEX/˷f\M5>;γ}P&·ՇPͽcMqݏA;~0es8-Nqt6 7₇2AVkbELyc?RZgz֮ ef0000000000000000000000000000000000000000000000000000000000000000 +oc;j~EЪ)s̾;.;rTejiK/df5A|{S/ )Sc\^ȟmp+#vL17~k d# +]LeVߐGoo٥\k 9Mՙ= 0)xf +1kX9JS:6V =!X>KM)9! I}HWIUcl=GeZm>nQk2=1p%R`%$]c{!W+8?v(+)0ӈ!YahtA?H @m?|M X9Հ֨#2 +:m,MZ+ǒ\Ex9Yn<Ŀ2^FC[`T+ȬD]ǸFF$oB!>;A\T hxh|5:6;ϖUǷ6~k.T*{8K`y8'dT|-QkKmy9 91|Afr (K*)%_ʬtP`.rY?nrn{v2O_5;\g>R{4_ +: 9pEz~Gu E[n窙i9ヽu#=X&>@Q-2fT{0lY+%dm%dBag(MInǹpFt%V/~@6!tL:nS&Il*rH7d?ܡPZ=uۻ3MfM[S`fu1O/C0˕*5jp7䕌=)8 +-n#]=ǃctǓ!Rpqxe7<' D1' |OMMYޝi eFOm(pNA_|=i-H[c@?n+]LM4@M c|1Q>N~q}c^ߣo <}g +V<eҲF_m4۸۞&na1N\ Ғ~z64L3Q_ E.9!D"^ii}B.΅ߟMZgF1]0ۊ,qtƏcf P.DQ#is~Ar}Z4`}.,&*%*X>vAŝlݵAGR1$WWbV&G 4M{G_1zB{% ) 9_dk,_jW7[ÉϵSA1q/0*вLM.}x`y8ި燺35˅?cɮ + c:釬Թ+[fN#pȞ\1Y>+V1"rF IcB~{5%֔bQ0X.i0*(N%W|tgx^;}9uvV0s P$=vɺ] +`<է?!V%["(eZ9x=M##`[[Nx2 sR肆Or7bHH.2 oHw܊-(P1M4Ig-@H+3 w#.~~W@3J3,g W 1Ԛ"vJJ`g&2$Ta1ORѫ{2/82U- 1A*iBF2.>|)yyY +0L =;B[y{w5|PLMrF$ yy}WoJkrV ^#sx$0F쬫6C`-Wo1|>K0g. +3њ+4]<8T_YL:-`|R-וR,tF7rSEI`9z$xV"jť'pVwAӥ|3rT!}_[!CjS)\gwz +-WZLT +Y F9 QӜ!Rm:aDKݢFl0;vqz*X^uAAڸmoȽgJ^b;Gzp'{9ʨ9U E?1P囦rvjB#s掾QM]a*+dǧcv'uE _.r1~Sī1 ezoo'C/,X5)K."$W!ZLeJYu=RlF:-E%¦QOxӯ4R76='e=J)2v)[>@ ΡJ۠{u7G!Vć w#Ị@xCiQ^n ]VW!V<#&IlN<: x&2sZw +D0bא9OFoEXAYKY$=6n^aM:Snc%?3r؅E!zvxcXOWn&\s Z ԾD-/6BcCHg5%ۅ=}0=dvN ]QmASI8suE$K7;7=H28]Y l9H4\>;5Yr۰^8+`G#dT'9%߉)<D 6NCXƄ0(20BDF]mC]Zt1.e,G :d4O5'YRhWt0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 @@ -3926,41 +3443,41 @@ currentfile eexec cleartomark endstream endobj -102 0 obj << +93 0 obj << /Type /Font /Subtype /Type1 -/Encoding 234 0 R -/FirstChar 0 -/LastChar 107 -/Widths 235 0 R -/BaseFont /GDGSMG+CMSY10 -/FontDescriptor 100 0 R +/Encoding 297 0 R +/FirstChar 61 +/LastChar 121 +/Widths 298 0 R +/BaseFont /DPJNGB+CMMI10 +/FontDescriptor 91 0 R >> endobj -100 0 obj << -/Ascent 750 +91 0 obj << +/Ascent 694 /CapHeight 683 /Descent -194 -/FontName /GDGSMG+CMSY10 -/ItalicAngle -14.035 -/StemV 85 +/FontName /DPJNGB+CMMI10 +/ItalicAngle -14.04 +/StemV 72 /XHeight 431 -/FontBBox [-29 -960 1116 775] +/FontBBox [-32 -250 1048 750] /Flags 4 -/CharSet (/minus/bullet/greaterequal/bardbl) -/FontFile 101 0 R +/CharSet (/slash/A/D/M/b/k/o/p/t/x/y) +/FontFile 92 0 R >> endobj -235 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 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 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 0 500 ] +298 0 obj +[500 0 0 0 750 0 0 828 0 0 0 0 0 0 0 0 970 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 429 0 0 0 0 0 0 0 0 521 0 0 0 485 503 0 0 0 361 0 0 0 572 490 ] endobj -234 0 obj << +297 0 obj << /Type /Encoding -/Differences [ 0 /minus 1/.notdef 15/bullet 16/.notdef 21/greaterequal 22/.notdef 107/bardbl 108/.notdef] +/Differences [ 0 /.notdef 61/slash 62/.notdef 65/A 66/.notdef 68/D 69/.notdef 77/M 78/.notdef 98/b 99/.notdef 107/k 108/.notdef 111/o/p 113/.notdef 116/t 117/.notdef 120/x/y 122/.notdef] >> endobj -95 0 obj << -/Length1 1590 -/Length2 12214 +89 0 obj << +/Length1 1673 +/Length2 13400 /Length3 532 -/Length 14336 +/Length 15605 >> stream %!PS-AdobeFont-1.1: CMTI10 1.00B @@ -3976,14 +3493,16 @@ stream /ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /CDONCR+CMTI10 def +/FontName /TAPIAO+CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put +dup 12 /fi put dup 39 /quoteright put +dup 40 /parenleft put dup 44 /comma put dup 45 /hyphen put dup 46 /period put @@ -3992,6 +3511,7 @@ dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put +dup 52 /four put dup 53 /five put dup 57 /nine put dup 58 /colon put @@ -4003,6 +3523,7 @@ dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put +dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put @@ -4038,6 +3559,7 @@ dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put +dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end @@ -4053,53 +3575,52 @@ z8u .{IQ7^N-+C #̲]\M؅1 pk2 \t,o)2TTڮw}Ҏvq `;lSgs3c-h}W9;ڤ9Ö`2 ${};;m򉅘^l1'C*~O`XUn0x|cɾ'iD%WNCەU.9R*R?nLWˆs/[趡ۆ8>H@#ٻx wf} ayBR C1|P|'r,߰INJU`DkUQe ]Am\"GmFO )2rIuW\SW,'vdy]eqXH0#~Z9|pŞPOHau`Cb鬅<=?t8!-~C0VY-aht`rxK[3,f @=]wG2C#܎dVe"τ~fFE"ʨ IlB#{ FV<,L0 AXF,M19r@3Ipl,S~bF![=ҩ]co%5 x\Ґ|wfdG@ -_N(C*8h\k -ڮEKLK?eF[N#7)if(?e`o {K+\/ Bc T#< kTv\&Fȍez]3 k>!0[l~TE& 2j7Hmu.CE] e8 ?)X#'JvkRkJ7#Yֵfa9i#8!VOP~z30{chU!aDL24H\qtc5_U3oؕ8##[t4D6e> 4l[1:f~1G%M_fH~u1pbc1C PQ$j8#pbJN:L mqڣ BMĂd2\&?1J݊K$4m$[&>/EXPe&+ ` mLDg%/]uUٻa6߆`}o1p@` s(26Z8To `$Q%޺gnKa חXө ={tUNԏ(Lˊjjff0*r}s/ڈK63.``~_MLOt A;ed0b:d>\6j䂼 - P(!"JyAksDQƃ@qMMN8: F}ˀ8\$Dmn$Ph -'%p錬C<t헴ՊTG6{}G!~-6UB=`NFfvoEbzaG -<&fhqLiS8zF"TE@z 1Jŕ: ~럪nX!`aV=SQ7W, t=̪nL?oXב1 \GXcyBp; 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=<;4VmuV9Z+<?d6IJU)P=f0A[8"= S#{oŚE';'o=}lBUUO>%'ڧw$fol5)z +qw$Y+Y$C [ +@A)櫭yX 4S177gU*tyAy(ej3^?@wʬATvy6:¹N>Cv98~Ճ Z˨Li4 Y2`Įc HwPOL*q .\17yb){1 ov>c-KѬӋ39-|lD܅(`(jL(7>]KMBnpeOC̏_z7n-wbmWT3ЀB.>iplf,|@ +NTh:8#8lݪ~inU` k弳`71=D J +U(K1-,%o( B|ek +VI+O#2{c|C6\ vYڌiꬌW0>iu4?Aѥ c^E!ȏ8ZM4)4jn\'N8௛WOI Z}q7-;?=Tن]T_SY)8Z| +ͣbf%z)-tOkJ&2- nN +alC֯v t9@W. ?Tmh86gCdPUceJ!JJ} +LYϯw:rDKB n;+zeQ^ѥWm_ oGE ~^fGHiTZ6(g0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 @@ -4110,41 +3631,41 @@ p cleartomark endstream endobj -96 0 obj << +90 0 obj << /Type /Font /Subtype /Type1 -/Encoding 236 0 R +/Encoding 299 0 R /FirstChar 11 -/LastChar 121 -/Widths 237 0 R -/BaseFont /CDONCR+CMTI10 -/FontDescriptor 94 0 R +/LastChar 122 +/Widths 300 0 R +/BaseFont /TAPIAO+CMTI10 +/FontDescriptor 88 0 R >> endobj -94 0 obj << +88 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /CDONCR+CMTI10 +/FontName /TAPIAO+CMTI10 /ItalicAngle -14.04 /StemV 68 /XHeight 431 /FontBBox [-163 -250 1146 969] /Flags 4 -/CharSet (/ff/quoteright/comma/hyphen/period/slash/zero/one/two/three/five/nine/colon/A/B/C/D/E/F/G/I/L/M/N/O/P/R/S/T/U/V/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) -/FontFile 95 0 R +/CharSet (/ff/fi/quoteright/parenleft/comma/hyphen/period/slash/zero/one/two/three/four/five/nine/colon/A/B/C/D/E/F/G/I/K/L/M/N/O/P/R/S/T/U/V/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) +/FontFile 89 0 R >> endobj -237 0 obj -[613 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 0 307 0 0 0 0 307 358 307 511 511 511 511 511 0 511 0 0 0 511 307 0 0 0 0 0 0 743 704 716 755 678 653 774 0 386 0 0 627 897 743 767 678 0 729 562 716 743 743 0 0 0 0 0 0 0 0 0 0 511 460 460 511 460 307 460 511 307 307 460 256 818 562 511 511 460 422 409 332 537 460 664 464 486 ] +300 0 obj +[613 562 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 0 307 409 0 0 0 307 358 307 511 511 511 511 511 511 511 0 0 0 511 307 0 0 0 0 0 0 743 704 716 755 678 653 774 0 386 0 769 627 897 743 767 678 0 729 562 716 743 743 0 0 0 0 0 0 0 0 0 0 511 460 460 511 460 307 460 511 307 307 460 256 818 562 511 511 460 422 409 332 537 460 664 464 486 409 ] endobj -236 0 obj << +299 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 11/ff 12/.notdef 39/quoteright 40/.notdef 44/comma/hyphen/period/slash/zero/one/two/three 52/.notdef 53/five 54/.notdef 57/nine/colon 59/.notdef 65/A/B/C/D/E/F/G 72/.notdef 73/I 74/.notdef 76/L/M/N/O/P 81/.notdef 82/R/S/T/U/V 87/.notdef 97/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 122/.notdef] +/Differences [ 0 /.notdef 11/ff/fi 13/.notdef 39/quoteright/parenleft 41/.notdef 44/comma/hyphen/period/slash/zero/one/two/three/four/five 54/.notdef 57/nine/colon 59/.notdef 65/A/B/C/D/E/F/G 72/.notdef 73/I 74/.notdef 75/K/L/M/N/O/P 81/.notdef 82/R/S/T/U/V 87/.notdef 97/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] >> endobj -65 0 obj << -/Length1 1979 -/Length2 14326 +77 0 obj << +/Length1 2084 +/Length2 15132 /Length3 532 -/Length 16837 +/Length 17748 >> stream %!PS-AdobeFont-1.1: CMR10 1.00B @@ -4160,7 +3681,7 @@ stream /ItalicAngle 0 def /isFixedPitch false def end readonly def -/FontName /PQEPRA+CMR10 def +/FontName /CNXWOW+CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -4169,6 +3690,7 @@ end readonly def dup 11 /ff put dup 12 /fi put dup 14 /ffi put +dup 20 /caron put dup 34 /quotedblright put dup 38 /ampersand put dup 39 /quoteright put @@ -4178,6 +3700,7 @@ dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put +dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put @@ -4191,6 +3714,7 @@ dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put +dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put @@ -4213,6 +3737,8 @@ dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put +dup 88 /X put +dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put @@ -4243,6 +3769,7 @@ dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put +dup 124 /emdash put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end @@ -4254,58 +3781,55 @@ i Hs(}CE c;[e/SmV& i2ǡmc\0kC'sp)X0@$2jOGG9%fLJD@m2b5\cdy!iz+lxa+2S@uIY5F>qgROEowK*;/sDGǡx=>J ujLDfdžk-j_~hNa0GF{Bad&p+ZlN>#\S ;rCԿk Yce2->fBi߬,uOLu #s6-;s Jt%K|pX;p+,ArwV\jU<6^ YS|?Ɯu`x*,ܺvdz`;t/nQ ,՟=)uv1Q|Afg] 0)``[ߺAv&eӈPmayak3,OD(,de5lN,n@Uy;ye <OK3`r -h I&@e/JHx1iZiPt&9Be8=^gl%Q/2T^G[ojDeQ#o?B9$3% 'ݨmL |/U 8BQ3;:& %#La}Z.b ohm\ʅ9&ˈ}¢MpUndE: z$wIS 8FQ^lJ@QA\/ "4#;}/ƔFExh `Oq?1D߀œf1ޡɊ`Ћ%>5h?|zKW%r?R!ZйuD2 uxO!&7%Fw'kn%5r?Mr>J2ט̥._uy=4#ج)2sP~YB.jj0^ -Mh8{jHjL4QU=seE֏^ȶ vĺ[D9*Pp SU;'PPSNf $_ b[Z,رξjë}06%XPb?aٷ=!Y3TxC Hp yw̘ͽyH2Y }S0qǧ<;sIG|Oγ 'BegpNVtԙ Rp -܇Kg97"״\pZ_IjseAA7s:vӅ7dHNՂ OՂ_,+ Nx\2` Xkp85opݠkJ݉A&|O(Y UR11 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Tܐ%`I}՜'oaWAb'\B@+`pC[Qづg}p@+/:tz,؀HL4$+ifQ2ys]~{8bSsup(ՐQN Wn<4^ݨ+b|5k}ˣȌIAS;%whrt{{gWP9(|N[[ z >NUSu*!'u0 y\BksܴPjuD$jYk+Zf"gj)oVU^2Z&A` vzɎ8xBYCa )mѹx Eo Vy2ѝ-=2ҋ$>(”P˝ϊNu͙ &f" ?Ci#p}(#'a96kC۠9f`GC"# +q2c`;<1;J#^]nrٝ3/aL,QеdɨIRE;~2xAyfaxy`'me]x/~}u; Z l/'|wU+̤S#E4 XW'^nU<7Pxa UgSHu3XHy?r=A5 Ui#Kkf:-h55H1^rMO a<+:NPrDs.ݪע[t%'l$[e<0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 @@ -4316,41 +3840,41 @@ WJ cleartomark endstream endobj -66 0 obj << +78 0 obj << /Type /Font /Subtype /Type1 -/Encoding 238 0 R +/Encoding 301 0 R /FirstChar 11 -/LastChar 123 -/Widths 239 0 R -/BaseFont /PQEPRA+CMR10 -/FontDescriptor 64 0 R +/LastChar 124 +/Widths 302 0 R +/BaseFont /CNXWOW+CMR10 +/FontDescriptor 76 0 R >> endobj -64 0 obj << +76 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /PQEPRA+CMR10 +/FontName /CNXWOW+CMR10 /ItalicAngle 0 /StemV 69 /XHeight 431 /FontBBox [-251 -250 1009 969] /Flags 4 -/CharSet (/ff/fi/ffi/quotedblright/ampersand/quoteright/parenleft/parenright/plus/comma/hyphen/period/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/equal/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/R/S/T/U/V/W/bracketleft/quotedblleft/bracketright/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/endash) -/FontFile 65 0 R +/CharSet (/ff/fi/ffi/caron/quotedblright/ampersand/quoteright/parenleft/parenright/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/equal/question/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/R/S/T/U/V/W/X/Y/bracketleft/quotedblleft/bracketright/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/endash/emdash) +/FontFile 77 0 R >> endobj -239 0 obj -[583 556 0 833 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 0 0 778 278 389 389 0 778 278 333 278 0 500 500 500 500 500 500 500 500 500 500 278 278 0 778 0 0 0 750 708 722 764 681 653 785 750 361 514 778 625 917 750 778 681 0 736 556 722 750 750 1028 0 0 0 278 500 278 0 0 0 500 556 444 556 444 306 500 556 278 306 528 278 833 556 500 556 528 392 394 389 556 528 722 528 528 444 500 ] +302 0 obj +[583 556 0 833 0 0 0 0 0 500 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 0 0 778 278 389 389 0 778 278 333 278 500 500 500 500 500 500 500 500 500 500 500 278 278 0 778 0 472 0 750 708 722 764 681 653 785 750 361 514 778 625 917 750 778 681 0 736 556 722 750 750 1028 750 750 0 278 500 278 0 0 0 500 556 444 556 444 306 500 556 278 306 528 278 833 556 500 556 528 392 394 389 556 528 722 528 528 444 500 1000 ] endobj -238 0 obj << +301 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 11/ff/fi 13/.notdef 14/ffi 15/.notdef 34/quotedblright 35/.notdef 38/ampersand/quoteright/parenleft/parenright 42/.notdef 43/plus/comma/hyphen/period 47/.notdef 48/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon 60/.notdef 61/equal 62/.notdef 65/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P 81/.notdef 82/R/S/T/U/V/W 88/.notdef 91/bracketleft/quotedblleft/bracketright 94/.notdef 97/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/endash 124/.notdef] +/Differences [ 0 /.notdef 11/ff/fi 13/.notdef 14/ffi 15/.notdef 20/caron 21/.notdef 34/quotedblright 35/.notdef 38/ampersand/quoteright/parenleft/parenright 42/.notdef 43/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon 60/.notdef 61/equal 62/.notdef 63/question 64/.notdef 65/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P 81/.notdef 82/R/S/T/U/V/W/X/Y 90/.notdef 91/bracketleft/quotedblleft/bracketright 94/.notdef 97/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/endash/emdash 125/.notdef] >> endobj -62 0 obj << -/Length1 1466 -/Length2 9551 +74 0 obj << +/Length1 1735 +/Length2 11890 /Length3 532 -/Length 11549 +/Length 14157 >> stream %!PS-AdobeFont-1.1: CMBX10 1.00B @@ -4366,7 +3890,7 @@ stream /ItalicAngle 0 def /isFixedPitch false def end readonly def -/FontName /SLFAMK+CMBX10 def +/FontName /CHBINR+CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -4374,11 +3898,24 @@ end readonly def 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 39 /quoteright put +dup 40 /parenleft put +dup 41 /parenright put +dup 44 /comma put +dup 45 /hyphen put +dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put +dup 52 /four put +dup 53 /five put +dup 54 /six put +dup 55 /seven put +dup 56 /eight put +dup 57 /nine put dup 58 /colon put +dup 59 /semicolon put +dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put @@ -4386,8 +3923,8 @@ dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put +dup 72 /H put dup 73 /I put -dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put @@ -4396,6 +3933,9 @@ dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put +dup 85 /U put +dup 86 /V put +dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put @@ -4418,9 +3958,8 @@ dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put -dup 120 /x put dup 121 /y put -dup 123 /endash put +dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end @@ -4430,43 +3969,45 @@ currentfile eexec N(-C :ߒw .B bw0$["PU~NY*64C>N^?0^~o&fZ9-(*ϟVV9 |J7@s.$DFYMi+M즒DSW['1qy4qZ;]6y@F< V&N3n]Db a[sK kԃ9$[F=e,&Ȩ)uv1DVIjaIç# -5k~QaMjFJL3.0lVem; YnⱪyRyh$&'gYx Qi- o 7kacg92X3]*#5sJ^fXגbA>!đkLa\wi\p d³JKWpt+2~-OC=<ߨ7H'4FDS٤d4isCD3M`=eO7mfz~6H36&Uԍ&23`}VBr![/9tp'iq$$+ʼ1> ҐRQE+9d-$CkdH5 -ASHHmmK9>SU$7=ڍu4 DKLW_<|ha 6b-D^N"cMJHԶ>W Z 6(;XhǺW^1m#a19F!|(nou[B A2tϚnLo׺i ML 2+> -^MBܷGIPCQ94+"?8 -( -$o-7Dz2WU)GЄRj~qp ŃpBR,bF#q'KI3C0P\lj?%o sYʦX3Onؑ{`~b$9/U -Wa#TA,ǣz!wȆZeӏD}HƇƺd"a03ݔK\<3? 4PW[S>&ΡGpu`V]"d04}\bt0'Mtuô˶eL淵lj?yK4$wNlzPZF(5KQYfCjCA).A\TWݩ=;\|O_rLI ΚM\vLf擾qNO^gp𠑢>OmvDc5Jff[EG䤒Oϯ/s#UQ7/>΁# ]AibF.O9:Kb=,w;@ #0PB1UMDy -#0^BJ֪Hl؁ZE &t⺶2e't7 ٕ`HѨ(۹fo e9eQ?aN ѵF8AhR;-[2M oX&*J) .Ϩ,yoJf19Jezu:J?<Ǚ9*r/Wq4)4~k2*׆g作S \[V ڀ4U 8O>)fK{?DP8(Hǀ%p1$=!]”ٹ.n0)zu{@QEe* 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A]\(_aʆfywDbg{KlK꾞̼EO26zp4clAZs]"OqQ:7vXzՑ," kđ:/ ˲QGO +Jx "rKTFi{s 4""46 9 +D@yTܭ56"=OTsi pғA5?BOu3"j +G6qNDre +VK3<{ X#uRTȬ +%\Hr j.t˂ +iucd;[ޏs͟7,v6_y D3ڣ6b-}if +~z3u lyP\50_Px>'Ynv)Ȧ55{ӓd DA_/N3 +Uq ZκR 5 +X %X. vKtF׎ :n@S c m~hfP@TV4d1b"Aơ~ n,6r$i[ՓUmF +ٝ3.rY;7z *U6u&a9!xzjkGtŜhLqUtFՆҢJ ;^\Jd(eD7=<_՞粭3Atc(~le8 !GD"bxmf:f5PzH b18!F7DY<Ť+]?"AfnO5& "0 R x [Lof80(y,)CJȏwk t3pV*5llKvnAdg.G< X!C&7rҎ6Fj` Y_FDσ2=;A2/S!gWZ"*cAIjۑ "ؿ똢a6,z]+*7*bw=)ur DS_E, +ۊh{T%vg8ɚ.*]i;f="9C[JLY@Ѹϝiu$Vl.8\^,fPp:ܡǸyG AJo W>w߼6@CS~?7J>b4u==EHh7.ez|3[ kDM2I\]ca{::@U뙣6D-/( SU0Q)vaV8-ǓVR }{Bk<*KhhD<'ok2%g&UӪe@]/3HFW@(M%}-LD2U8D$՞18,l!@G CUSttdTyrx3.ހ ;ÖM%l:[C1s ] Y@6:+V;8ڀm!-0?K^.U'*+P_Hf;{sUr՗KmC3y$.\dTIJA9{Q7蔤y?Vs@5lIehѨ]ٖEGLI~h`D uX)T!tLlڧlUUhx}XW_gK‚ԍXT Vdz@Ɲ٪j:%f#. *Kx^IRIZp4-͗gpUjtCefDY" h.«2m-+[;:®6=wCAH@L43Ѕy_t[G3(u+?U8ɠy_ݪʜ1 Pt F{H(n|4_b羚.ɱşGYJNZ)ըVC7B.#!:fA{B y!V6ܪEfH6B7?,KZ0'=qWSnWڝUw&޳Mi?5"Twl0?W!1Ccfa׆OO=_ q0ĄXPYa@s0S3(vT|u l#EerI( Tq\'> endobj -61 0 obj << +73 0 obj << /Ascent 694 /CapHeight 686 /Descent -194 -/FontName /SLFAMK+CMBX10 +/FontName /CHBINR+CMBX10 /ItalicAngle 0 /StemV 114 /XHeight 444 /FontBBox [-301 -250 1164 946] /Flags 4 -/CharSet (/fi/quoteright/zero/one/two/three/colon/A/B/C/D/E/F/G/I/J/L/M/N/O/P/R/S/T/a/b/c/d/e/f/g/h/i/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/endash) -/FontFile 62 0 R +/CharSet (/fi/quoteright/parenleft/parenright/comma/hyphen/period/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/question/A/B/C/D/E/F/G/H/I/L/M/N/O/P/R/S/T/U/V/X/a/b/c/d/e/f/g/h/i/k/l/m/n/o/p/q/r/s/t/u/v/w/y/z) +/FontFile 74 0 R >> endobj -241 0 obj -[639 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 0 319 0 0 0 0 0 0 0 0 575 575 575 575 0 0 0 0 0 0 319 0 0 0 0 0 0 869 818 831 882 756 724 904 0 436 594 0 692 1092 900 864 786 0 862 639 800 0 0 0 0 0 0 0 0 0 0 0 0 559 639 511 639 527 351 575 639 319 0 607 319 958 639 575 639 607 474 454 447 639 607 831 607 607 0 575 ] +304 0 obj +[639 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 0 319 447 447 0 0 319 383 319 0 575 575 575 575 575 575 575 575 575 575 319 319 0 0 0 543 0 869 818 831 882 756 724 904 900 436 0 0 692 1092 900 864 786 0 862 639 800 885 869 0 869 0 0 0 0 0 0 0 0 559 639 511 639 527 351 575 639 319 0 607 319 958 639 575 639 607 474 454 447 639 607 831 0 607 511 ] endobj -240 0 obj << +303 0 obj << /Type /Encoding -/Differences [ 0 /.notdef 12/fi 13/.notdef 39/quoteright 40/.notdef 48/zero/one/two/three 52/.notdef 58/colon 59/.notdef 65/A/B/C/D/E/F/G 72/.notdef 73/I/J 75/.notdef 76/L/M/N/O/P 81/.notdef 82/R/S/T 85/.notdef 97/a/b/c/d/e/f/g/h/i 106/.notdef 107/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y 122/.notdef 123/endash 124/.notdef] +/Differences [ 0 /.notdef 12/fi 13/.notdef 39/quoteright/parenleft/parenright 42/.notdef 44/comma/hyphen/period 47/.notdef 48/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon 60/.notdef 63/question 64/.notdef 65/A/B/C/D/E/F/G/H/I 74/.notdef 76/L/M/N/O/P 81/.notdef 82/R/S/T/U/V 87/.notdef 88/X 89/.notdef 97/a/b/c/d/e/f/g/h/i 106/.notdef 107/k/l/m/n/o/p/q/r/s/t/u/v/w 120/.notdef 121/y/z 123/.notdef] >> endobj -59 0 obj << +71 0 obj << /Length1 1125 /Length2 6015 /Length3 532 @@ -4527,7 +4068,7 @@ stream /ItalicAngle -14.04 def /isFixedPitch false def end readonly def -/FontName /TSJTZG+CMTI12 def +/FontName /VWJGWQ+CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def @@ -4600,41 +4141,41 @@ X0F-( cleartomark endstream endobj -60 0 obj << +72 0 obj << /Type /Font /Subtype /Type1 -/Encoding 242 0 R +/Encoding 305 0 R /FirstChar 65 /LastChar 118 -/Widths 243 0 R -/BaseFont /TSJTZG+CMTI12 -/FontDescriptor 58 0 R +/Widths 306 0 R +/BaseFont /VWJGWQ+CMTI12 +/FontDescriptor 70 0 R >> endobj -58 0 obj << +70 0 obj << /Ascent 694 /CapHeight 683 /Descent -194 -/FontName /TSJTZG+CMTI12 +/FontName /VWJGWQ+CMTI12 /ItalicAngle -14.04 /StemV 63 /XHeight 431 /FontBBox [-36 -251 1103 750] /Flags 4 /CharSet (/A/B/D/L/M/P/S/a/b/c/d/e/f/g/h/i/k/l/m/n/o/p/r/s/t/u/v) -/FontFile 59 0 R +/FontFile 71 0 R >> endobj -243 0 obj +306 0 obj [727 688 0 738 0 0 0 0 0 0 0 613 877 0 0 663 0 0 550 0 0 0 0 0 0 0 0 0 0 0 0 0 500 450 450 500 450 300 450 500 300 0 450 250 800 550 500 500 0 413 400 325 525 450 ] endobj -242 0 obj << +305 0 obj << /Type /Encoding /Differences [ 0 /.notdef 65/A/B 67/.notdef 68/D 69/.notdef 76/L/M 78/.notdef 80/P 81/.notdef 83/S 84/.notdef 97/a/b/c/d/e/f/g/h/i 106/.notdef 107/k/l/m/n/o/p 113/.notdef 114/r/s/t/u/v 119/.notdef] >> endobj -56 0 obj << -/Length1 1398 -/Length2 7256 +68 0 obj << +/Length1 1505 +/Length2 8591 /Length3 532 -/Length 9186 +/Length 10628 >> stream %!PS-AdobeFont-1.1: CMBX12 1.0 @@ -4650,12 +4191,13 @@ stream /ItalicAngle 0 def /isFixedPitch false def end readonly def -/FontName /ZRPMDE+CMBX12 def +/FontName /JOKBMQ+CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for +dup 12 /fi put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put @@ 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