- AMG preconditioners
+ HREF="node12.html">Multigrid Background
+
- Smoothed Aggregation + HREF="node13.html">AMG preconditioners
- Smoothers and coarsest-level solvers + HREF="node14.html">Smoothed Aggregation +
- Smoothers and coarsest-level solvers
- - Getting Started
-
- Examples
+ HREF="node16.html">Getting Started
+
- - User Interface
-
- Subroutine init
+ HREF="node18.html">User Interface
+
- Subroutine set + HREF="node19.html">Subroutine init
- Subroutine build + HREF="node20.html">Subroutine set
- Subroutine hierarchy_build + HREF="node21.html">Subroutine build
- Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build
- Subroutine apply + HREF="node23.html">Subroutine smoothers_build
- Subroutine free + HREF="node24.html">Subroutine apply
- Subroutine descr + HREF="node25.html">Subroutine free +
- Subroutine descr
- - Adding new smoother and solver objects to MLD2P4
- Error Handling + HREF="node27.html">Adding new smoother and solver objects to MLD2P4
- License + HREF="node28.html">Error Handling
- Bibliography + HREF="node29.html">License
- About this document ... + HREF="node30.html">Bibliography +
- About this document ...
diff --git a/docs/html/node1.html b/docs/html/node1.html index ce6ac925..537f9b98 100644 --- a/docs/html/node1.html +++ b/docs/html/node1.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds -
-
-
-
- Next: Next: Contents - Up: Up: userhtml - Previous: Previous: userhtml - Contents
@@ -92,26 +92,26 @@ the user interface of MLD2P4.
-
-
-
-
- Next: Next: Contents - Up: Up: userhtml - Previous: Previous: userhtml - Contents diff --git a/docs/html/node10.html b/docs/html/node10.html index 1eb7127a..0993cfcf 100644 --- a/docs/html/node10.html +++ b/docs/html/node10.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Example and test programs - +Bug reporting + @@ -18,65 +18,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds + - + -
-
+
-
- Next: Multigrid Background - Up: Configuring and Building MLD2P4 + Next: Example and test programs + Up: Configuring and Building MLD2P4 Previous: Bug reporting - Configuration options + Contents
--
-Example and test programs ++Bug reporting
-The package contains theexamplesandtestsdirectories; -both of them are further divided intofilereadand -pdegensubdirectories. Their purpose is as follows: --
-
- examples -
- contains a set of simple example programs with a - predefined choice of preconditioners, selectable via integer - values. These are intended to get an acquaintance with the - multi-level preconditioners available in MLD2P4. - -
- tests -
- contains a set of more sophisticated examples that
- will allow the user, via the input files in the
runs- subdirectories, to experiment with the full range of preconditioners - implemented in the package. -
-
filereaddirectories contain sample programs that read -sparse matrices from files, according to the Matrix Market or the -Harwell-Boeing storage format; thepdegenprograms generate -matrices in full parallel mode from the discretization of a sample partial -differential equation. +If you find any bugs in our codes, please send an email to
pasqua.dambra@cnr.it +
daniela.diserafino@unicampania.it +
salvatore.filippone@cranfield.ac.uk
+You should be aware that the amount of information needed to reproduce a problem +in a parallel program may vary quite a lot. ++
diff --git a/docs/html/node11.html b/docs/html/node11.html index 28beb567..6c27253f 100644 --- a/docs/html/node11.html +++ b/docs/html/node11.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Multigrid Background - +Example and test programs + @@ -18,9 +18,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + @@ -30,9 +29,9 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node12.html">
+ HREF="node6.html">
-
Next: AMG preconditioners + HREF="node12.html">Multigrid Background Up: userhtml - Previous: Example and test programs + HREF="node6.html">Configuring and Building MLD2P4 + Previous: Bug reporting Contents
-+
- - -
-Multigrid Background --Multigrid preconditioners, coupled with Krylov iterative -solvers, are widely used in the parallel solution of large and sparse linear systems, -because of their optimality in the solution of linear systems arising from the -discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids. -Optimality, also known as algorithmic scalability, is the property -of having a computational cost per iteration that depends linearly on -the problem size, and a convergence rate that is independent of the problem size. - -
-Multigrid preconditioners are based on a recursive application of a two-grid process -consisting of smoother iterations and a coarse-space (or coarse-level) correction. -The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones, -or more complex subspace-correction methods, such as the Schwarz ones. -The coarse-space correction consists of solving, in an appropriately chosen -coarse space, the residual equation associated with the approximate solution computed -by the smoother, and of using the solution of this equation to correct the -previous approximation. The transfer of information between the original -(fine) space and the coarse one is performed by using suitable restriction and -prolongation operators. The construction of the coarse space and the corresponding -transfer operators is carried out by applying a so-called coarsening algorithm to the system -matrix. Two main approaches can be used to perform coarsening: the geometric approach, -which exploits the knowledge of some physical grid associated with the matrix -and requires the user to define transfer operators from the fine -to the coarse level and vice versa, and the algebraic approach, which builds -the coarse-space correction and the associate transfer operators using only matrix -information. The first approach may be difficult when the system comes from -discretizations on complex geometries; -furthermore, ad hoc one-level smoothers may be required to get an efficient -interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients -are considered. The second approach performs a fully automatic coarsening and enforces the -interplay between fine and coarse level by suitably choosing the coarse space and -the coarse-to-fine interpolation (see, e.g., [3,23,21] for details.) +Example and test programs + +The package contains the
examplesandtestsdirectories; +both of them are further divided intofilereadand +pdegensubdirectories. Their purpose is as follows: +-
+
- examples +
- contains a set of simple example programs with a + predefined choice of preconditioners, selectable via integer + values. These are intended to get an acquaintance with the + multi-level preconditioners available in MLD2P4. + +
- tests +
- contains a set of more sophisticated examples that
+ will allow the user, via the input files in the
runs+ subdirectories, to experiment with the full range of preconditioners + implemented in the package. +
+
filereaddirectories contain sample programs that read +sparse matrices from files, according to the Matrix Market or the +Harwell-Boeing storage format; thepdegenprograms generate +matrices in full parallel mode from the discretization of a sample partial +differential equation. --MLD2P4 uses a pure algebraic approach, based on the smoothed -aggregation algorithm [2,25], -for building the sequence of coarse matrices and transfer operators, -starting from the original one. -A decoupled version of this algorithm is implemented, where the smoothed -aggregation is applied locally to each submatrix [24]. -A brief description of the AMG preconditioners implemented in MLD2P4 is given in -Sections 4.1-4.3. For further details the reader -is referred to [4,5,7,8]. - -
-We note that optimal multigrid preconditioners do not necessarily correspond -to minimum execution times in a parallel setting. Indeed, to obtain effective parallel -multigrid preconditioners, a tradeoff between the optimality and the cost of building and -applying the smoothers and the coarse-space corrections must be achieved. Effective -parallel preconditioners require algorithmic scalability to be coupled with implementation -scalability, i.e., a computational cost per iteration which remains (almost) constant as -the number of parallel processors increases. - -
-
- -Subsections - - - -
- - -
-
-
-
-
-
-
-
- Next: AMG preconditioners - Up: userhtml - Previous: Example and test programs - Contents - diff --git a/docs/html/node12.html b/docs/html/node12.html index 9de5d48c..bbb351f1 100644 --- a/docs/html/node12.html +++ b/docs/html/node12.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -AMG preconditioners - +Multigrid Background + @@ -18,293 +18,146 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + + -
-
+
-
-
- Next: Smoothed Aggregation - Up: Multigrid Background - Previous: Multigrid Background - Next: AMG preconditioners + Up: userhtml + Previous: Example and test programs + Contents
--
-
-AMG preconditioners --In order to describe the AMG preconditioners available in MLD2P4, we consider a -linear system - +
-where -
-
is a nonsingular sparse matrix;
-for ease of presentation we assume 
is real, but the
-results are valid for the complex case as well.
-Let us assume as finest index space the set of row (column) indices of
, i.e.,
-
-
.
-Any algebraic multilevel preconditioners implemented in MLD2P4 generates
-a hierarchy of index spaces and a corresponding hierarchy of matrices,
+Multigrid preconditioners, coupled with Krylov iterative
+solvers, are widely used in the parallel solution of large and sparse linear systems,
+because of their optimality in the solution of linear systems arising from the
+discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
+Optimality, also known as algorithmic scalability, is the property
+of having a computational cost per iteration that depends linearly on
+the problem size, and a convergence rate that is independent of the problem size.
-
-- - --
-
- -by using the information contained in, without assuming any
-knowledge of the geometry of the problem from which originates.
-A vector space
-
is associated with 
,
-where 
is the size of .
-For all 
, a restriction operator and a prolongation one are built,
-which connect two levels 
and 
:
++Multigrid preconditioners are based on a recursive application of a two-grid process +consisting of smoother iterations and a coarse-space (or coarse-level) correction. +The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones, +or more complex subspace-correction methods, such as the Schwarz ones. +The coarse-space correction consists of solving, in an appropriately chosen +coarse space, the residual equation associated with the approximate solution computed +by the smoother, and of using the solution of this equation to correct the +previous approximation. The transfer of information between the original +(fine) space and the coarse one is performed by using suitable restriction and +prolongation operators. The construction of the coarse space and the corresponding +transfer operators is carried out by applying a so-called coarsening algorithm to the system +matrix. Two main approaches can be used to perform coarsening: the geometric approach, +which exploits the knowledge of some physical grid associated with the matrix +and requires the user to define transfer operators from the fine +to the coarse level and vice versa, and the algebraic approach, which builds +the coarse-space correction and the associate transfer operators using only matrix +information. The first approach may be difficult when the system comes from +discretizations on complex geometries; +furthermore, ad hoc one-level smoothers may be required to get an efficient +interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients +are considered. The second approach performs a fully automatic coarsening and enforces the +interplay between fine and coarse level by suitably choosing the coarse space and +the coarse-to-fine interpolation (see, e.g., [3,23,21] for details.) -
-- - --
-
- -the matrix
is computed by using the previous operators according
-to the Galerkin approach, i.e.,
++MLD2P4 uses a pure algebraic approach, based on the smoothed +aggregation algorithm [2,25], +for building the sequence of coarse matrices and transfer operators, +starting from the original one. +A decoupled version of this algorithm is implemented, where the smoothed +aggregation is applied locally to each submatrix [24]. +A brief description of the AMG preconditioners implemented in MLD2P4 is given in +Sections 4.1-4.3. For further details the reader +is referred to [4,5,7,8]. -
-- - --
-
- -In the current implementation of MLD2P4 we have
-A smoother with iteration matrix 
is set up at each level , and a solver
-is set up at the coarsest level, so that they are ready for application
-(for example, setting up a solver based on the 
factorization means computing
-and storing the 
and 
factors). The construction of the hierarchy of AMG components
-described so far corresponds to the so-called build phase of the preconditioner.
++We note that optimal multigrid preconditioners do not necessarily correspond +to minimum execution times in a parallel setting. Indeed, to obtain effective parallel +multigrid preconditioners, a tradeoff between the optimality and the cost of building and +applying the smoothers and the coarse-space corrections must be achieved. Effective +parallel preconditioners require algorithmic scalability to be coupled with implementation +scalability, i.e., a computational cost per iteration which remains (almost) constant as +the number of parallel processors increases.
-
- --The components produced in the build phase may be combined in several ways -to obtain different multilevel preconditioners; -this is done in the application phase, i.e., in the computation of a vector -of type
, where 
denotes the preconditioner, usually within an iteration
-of a Krylov solver [20]. An example of such a combination, known as
-V-cycle, is given in Figure 1. In this case, a single iteration
-of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
-in the pre-smoothing and post-smoothing phases); however, different choices can be
-performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
-and W-cycle [3], and a version of the K-cycle described
-in [19].
-
--
+ + +
-
-
+
-
-
- Next: Smoothed Aggregation - Up: Multigrid Background - Previous: Multigrid Background - Next: AMG preconditioners + Up: userhtml + Previous: Example and test programs + Contents diff --git a/docs/html/node13.html b/docs/html/node13.html index 2e779069..ff0c3b63 100644 --- a/docs/html/node13.html +++ b/docs/html/node13.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Smoothed Aggregation - +AMG preconditioners + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node14.html">
+ HREF="node12.html">
@@ -40,336 +40,247 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Smoothers and coarsest-level solvers + HREF="node14.html">Smoothed Aggregation Up: Multigrid Background + HREF="node12.html">Multigrid Background Previous: AMG preconditioners + HREF="node12.html">Multigrid Background Contents
-+
-Smoothed Aggregation +AMG preconditioners-In order to define the prolongator
, used to compute
-the coarse-level matrix , MLD2P4 uses the smoothed aggregation
-algorithm described in [2,25].
-The basic idea of this algorithm is to build a coarse set of indices
-
by suitably grouping the indices of into disjoint
-subsets (aggregates), and to define the coarse-to-fine space transfer operator
- by applying a suitable smoother to a simple piecewise constant
-prolongation operator, with the aim of improving the quality of the coarse-space correction.
-
--Three main steps can be identified in the smoothed aggregation procedure: - -
-
-
- aggregation of the indices of to obtain ;
-
-
- construction of the prolongator ;
-
-
- application of and to build .
-
-
-In order to perform the coarsening step, the smoothed aggregation algorithm -described in [25] is used. In this algorithm, -each index -
corresponds to an aggregate 
of ,
-consisting of a suitably chosen index
-
and indices that are (usually) contained in a
-strongly-coupled neighborood of 
, i.e.,
+In order to describe the AMG preconditioners available in MLD2P4, we consider a
+linear system
-for a given threshold![$\theta \in [0,1]$](img32.png)
(see [25] for the details).
-Since this algorithm has a sequential nature, a decoupled
-version of it is applied, where each processor independently executes
-the algorithm on the set of indices 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 some nonuniform aggregates
-and is strongly dependent on the number of processors and on the initial partitioning
-of the matrix is a nonsingular sparse matrix;
+for ease of presentation we assume . Nevertheless, this parallel algorithm has been chosen for
-MLD2P4, since it has been shown to produce good results in practice
-[5,7,24].
+ ALT="$A$"> is real, but the
+results are valid for the complex case as well.
-The prolongator
is built starting from a tentative prolongator
+Let us assume as finest index space the set of row (column) indices of , i.e.,
, defined as
+ WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="img6.png"
+ ALT="$\Omega = \{1, 2, \ldots, n\}$">.
+Any algebraic multilevel preconditioners implemented in MLD2P4 generates
+a hierarchy of index spaces and a corresponding hierarchy of matrices,
-
-- +
+ -where is the aggregate of
-corresponding to the index
+
.
- is obtained by applying to 
a smoother
-
:
+ WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ SRC="img8.png"
+ ALT="$\mathbb{R}^{n_{k}}$"> is associated with ,
+where is the size of .
+For all , a restriction operator and a prolongation one are built,
+which connect two levels and :
+ WIDTH="254" HEIGHT="30" BORDER="0"
+ SRC="img14.png"
+ ALT="\begin{displaymath}
P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
R^k \in \mathbb{R}^{n_{k+1}\times n_k};
\end{displaymath}">
-in order to remove nonsmooth components from the range of the prolongator, -and hence to improve the convergence properties of the multi-level -method [2,23]. -A simple choice for
is the damped Jacobi smoother:
+the matrix is computed by using the previous operators according
+to the Galerkin approach, i.e.,
+ WIDTH="131" HEIGHT="27" BORDER="0"
+ SRC="img16.png"
+ ALT="\begin{displaymath}
A^{k+1}=R^kA^kP^k.
\end{displaymath}">
-where
is the diagonal matrix with the same diagonal entries as 
,
+In the current implementation of MLD2P4 we have
+A smoother with iteration matrix is set up at each level , and a solver
+is set up at the coarsest level, so that they are ready for application
+(for example, setting up a solver based on the factorization means computing
+and storing the and factors). The construction of the hierarchy of AMG components
+described so far corresponds to the so-called build phase of the preconditioner.
+
++ +
++
Figure 1: +Application phase of a V-cycle preconditioner. ++ + +
is the filtered matrix defined as
-
-
-+ WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0" + SRC="img22.png" + ALT="\framebox{ \begin{minipage}{.85\textwidth} \begin{tabbing} \quad \=\quad \=\quad... +...mm] \>endif [1mm] \>return $u^k$ [1mm] end \end{tabbing} \end{minipage} }"> - --and-
-
-(5) 
is an approximation of 
, where
-
is the spectral radius of
-
[2].
-In MLD2P4 this approximation is obtained by using
-
as an estimate
-of . Note that for systems coming from uniformly elliptic
-problems, filtering the matrix has little or no effect, and
- can be used instead of 
. The latter choice is the default in MLD2P4.
++The components produced in the build phase may be combined in several ways +to obtain different multilevel preconditioners; +this is done in the application phase, i.e., in the computation of a vector +of type
, where denotes the preconditioner, usually within an iteration
+of a Krylov solver [20]. An example of such a combination, known as
+V-cycle, is given in Figure 1. In this case, a single iteration
+of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
+in the pre-smoothing and post-smoothing phases); however, different choices can be
+performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
+and W-cycle [3], and a version of the K-cycle described
+in [19].
@@ -378,7 +289,7 @@ problems, filtering the matrix
+ HREF="node12.html">
@@ -388,11 +299,11 @@ problems, filtering the matrix
Next: Smoothers and coarsest-level solvers + HREF="node14.html">Smoothed Aggregation Up: Multigrid Background + HREF="node12.html">Multigrid Background Previous: AMG preconditioners + HREF="node12.html">Multigrid Background Contents diff --git a/docs/html/node14.html b/docs/html/node14.html index 25220388..0b19fb06 100644 --- a/docs/html/node14.html +++ b/docs/html/node14.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Smoothers and coarsest-level solvers - +Smoothed Aggregation + @@ -18,270 +18,382 @@ original version by: Nikos Drakos, CBLU, University of Leeds + - + -
-
+
-
- Next: Getting Started - Up: Multigrid Background + Next: Smoothers and coarsest-level solvers + Up: Multigrid Background Previous: Smoothed Aggregation - AMG preconditioners + Contents
-+
-Smoothers and coarsest-level solvers +Smoothed Aggregation-The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods, -a hybrid version of the forward and backward Gauss-Seidel methods, and the -additive Schwarz (AS) ones (see, e.g., [20,21]). +In order to define the prolongator
, used to compute
+the coarse-level matrix , MLD2P4 uses the smoothed aggregation
+algorithm described in [2,25].
+The basic idea of this algorithm is to build a coarse set of indices
+ by suitably grouping the indices of into disjoint
+subsets (aggregates), and to define the coarse-to-fine space transfer operator
+ by applying a suitable smoother to a simple piecewise constant
+prolongation operator, with the aim of improving the quality of the coarse-space correction.
-The hybrid Gauss-Seidel -version is considered because the original Gauss-Seidel method is inherently sequential. -At each iteration of the hybrid version, each parallel process uses the most recent values -of its own local variables and the values of the non-local variables computed at the -previous iteration, obtained by exchanging data with other processes before -the beginning of the current iteration. +Three main steps can be identified in the smoothed aggregation procedure: -
-In the AS methods, the index space
+
- aggregation of the indices of
+ is divided into 
-subsets to obtain
;
+- construction of the prolongator
+;
+- application of
+ + and to build .
++In order to perform the coarsening step, the smoothed aggregation algorithm +described in [25] is used. In this algorithm, +each index +
corresponds to an aggregate 
of size 
, possibly
-overlapping. For each of ,
+consisting of a suitably chosen index
+ and indices that are (usually) contained in a
+strongly-coupled neighborood of we consider the restriction
-operator
++
+
+(3)
+for a given threshold
-that maps a vector 
to the vector 
made of the components of
-with indices in , and the prolongation operator
+ WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ SRC="img32.png"
+ ALT="$\theta \in [0,1]$"> (see [25] for the details).
+Since this algorithm has a sequential nature, a decoupled
+version of it is applied, where each processor independently executes
+the algorithm on the set of indices 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 some nonuniform aggregates
+and is strongly dependent on the number of processors and on the initial partitioning
+of the matrix . Nevertheless, this parallel algorithm has been chosen for
+MLD2P4, since it has been shown to produce good results in practice
+[5,7,24].
+
++The prolongator
+where is built starting from a tentative prolongator
. These operators are then used to build
+ WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img33.png"
+ ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
+
+
+ is the aggregate of
+corresponding to the index
+.
, which is the restriction of to the index
-space .
-The classical AS preconditioner 
is defined as
+ SRC="img25.png"
+ ALT="$P^k$"> is obtained by applying to a smoother
+
+:
+ WIDTH="90" HEIGHT="30" BORDER="0"
+ SRC="img37.png"
+ ALT="\begin{displaymath}
P^k = S^k \bar{P}^k,
\end{displaymath}">
-where
is supposed to be nonsingular. We observe that an approximate
-inverse of is usually considered instead of 
.
-The setup of during the multilevel build phase
-involves
+in order to remove nonsmooth components from the range of the prolongator,
+and hence to improve the convergence properties of the multi-level
+method [2,23].
+A simple choice for is the damped Jacobi smoother:
--
-
- the definition of the index subspaces
and of the corresponding
- operators 
(and 
);
-
- - the computation of the submatrices ;
-
-
- the computation of their inverses (usually approximated - through some form of incomplete factorization). - -
, with
, during the
-multilevel application phase, requires
+ WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img42.png"
+ ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as
--
-
- the restriction of
to the subspaces
++
++(5)
+and is an approximation of , where
,
- i.e.
;
-
- - the computation of the vectors
;
-
- - the prolongation and the sum of the previous vectors,
- i.e.
+ WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ SRC="img48.png"
+ ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate
+of . Note that for systems coming from uniformly elliptic
+problems, filtering the matrix has little or no effect, and
.
-
-
-Direct solvers based on sparse LU factorizations, implemented in the -third-party libraries reported in Section 3.2, can be applied -as coarsest-level solvers by MLD2P4. Native inexact solvers based on -incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel, -and block Jacobi preconditioners are also available. Direct solvers usually -lead to more effective preconditioners in terms of algorithmic scalability; -however, this does not guarantee parallel efficiency. + WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" + SRC="img41.png" + ALT="$A^k$"> can be used instead of
. The latter choice is the default in MLD2P4.
-
+
-
-
+
-
- Next: Getting Started - Up: Multigrid Background + Next: Smoothers and coarsest-level solvers + Up: Multigrid Background Previous: Smoothed Aggregation - AMG preconditioners + Contents diff --git a/docs/html/node15.html b/docs/html/node15.html index c1007b17..6da0aee5 100644 --- a/docs/html/node15.html +++ b/docs/html/node15.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Getting Started - +Smoothers and coarsest-level solvers + @@ -18,9 +18,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + @@ -30,9 +29,9 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node16.html">
+ HREF="node12.html">
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Next: Examples + HREF="node16.html">Getting Started Up: userhtml - Previous: Smoothers and coarsest-level solvers + HREF="node12.html">Multigrid Background + Previous: Smoothed Aggregation Contents
-+
+Smoothers and coarsest-level solvers +
-Getting Started --We describe the basics for building and applying MLD2P4 one-level and multi-level -(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [13]. -The following steps are required: +The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods, +a hybrid version of the forward and backward Gauss-Seidel methods, and the +additive Schwarz (AS) ones (see, e.g., [20,21]). - -
-
-
- Declare the preconditioner data structure. It is a derived data type,
-
mld_xprec_type, where x may bes,d,c- orz, according to the basic data type of the sparse matrix - (s= real single precision;d= real double precision; -c= complex single precision;z= complex double precision). - This data structure is accessed by the user only through the MLD2P4 routines, - following an object-oriented approach. ++The hybrid Gauss-Seidel +version is considered because the original Gauss-Seidel method is inherently sequential. +At each iteration of the hybrid version, each parallel process uses the most recent values +of its own local variables and the values of the non-local variables computed at the +previous iteration, obtained by exchanging data with other processes before +the beginning of the current iteration. + +
+In the AS methods, the index space
+ is divided into
+subsets of size , possibly
+overlapping. For each we consider the restriction
+operator
+
+that maps a vector to the vector made of the components of
+with indices in , and the prolongation operator
+
+. These operators are then used to build
+
+, which is the restriction of to the index
+space .
+The classical AS preconditioner is defined as
+
++ + ++
+
+ +where is supposed to be nonsingular. We observe that an approximate
+inverse of is usually considered instead of .
+The setup of during the multilevel build phase
+involves
+
+-
+
- the definition of the index subspaces and of the corresponding
+ operators (and );
-
- Allocate and initialize the preconditioner data structure, according to
- a preconditioner type chosen by the user. This is performed by the routine
-
init, which also sets defaults for each preconditioner - type selected by the user. The preconditioner types and the defaults associated - with them are given in Table 1, where the strings used by -initto identify the preconditioner types are also given. - Note that these strings are valid also if uppercase letters are substituted by - corresponding lowercase ones. + - the computation of the submatrices ;
-
- Modify the selected preconditioner type, by properly setting
- preconditioner parameters. This is performed by the routine
set. - This routine must be called only if the user wants to modify the default values - of the parameters associated with the selected preconditioner type, to obtain a variant - of that preconditioner. Examples of use ofsetare given in - Section 5.1; a complete list of all the - preconditioner parameters and their allowed and default values is provided in - Section 6, Tables 2-8. + - the computation of their inverses (usually approximated + through some form of incomplete factorization). -
- Build the preconditioner for a given matrix. If the selected preconditioner
- is multi-level, then two steps must be performed, as specified next.
-
-
-
- 4.1 -
- Build the aggregation hierarchy for a given matrix. This is
-performed by the routine
hierarchy_build. -
- - 4.2 -
- Build the preconditioner for a given matrix. This is performed
-by the routine
smoothers_build. -
-
bld. +
, with
+, during the
+multilevel application phase, requires
+
+-
+
- the restriction of to the subspaces
+,
+ i.e.
+;
-
- Apply the preconditioner at each iteration of a Krylov solver.
- This is performed by the routine
aply. When using the PSBLAS Krylov solvers, - this step is completely transparent to the user, sinceaplyis called - by the PSBLAS routine implementing the Krylov solver (psb_krylov). + - the computation of the vectors
+;
-
- Free the preconditioner data structure. This is performed by
- the routine
free. This step is complementary to step 1 and should - be performed when the preconditioner is no more used. + - the prolongation and the sum of the previous vectors,
+ i.e.
+.
-
- the definition of the index subspaces
-All the previous routines are available as methods of the preconditioner object. -A detailed description of them is given in Section 6. -Examples showing the basic use of MLD2P4 are reported in Section 5.1. + +Variants of the classical AS method, which use modifications of the +restriction and prolongation operators, are also implemented in MLD2P4. +Among them, the Restricted AS (RAS) preconditioner usually +outperforms the classical AS preconditioner in terms of convergence +rate and of computation and communication time on parallel distributed-memory +computers, and is therefore the most widely used among the AS +preconditioners [6].
- -
----
-Table 1: -Preconditioner types, corresponding strings and default choices. - -- --
-TYPE -STRING -DEFAULT PRECONDITIONER -No preconditioner -'NOPREC'Considered only to use the PSBLAS - Krylov solvers with no preconditioner. -Diagonal -'DIAG'or'JACOBI'Diagonal preconditioner. - For any zero diagonal entry of the matrix to be preconditioned, - the corresponding entry of the preconditioner is set to 1. -Block Jacobi -'BJAC'Block-Jacobi with 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'V-cycle with one hybrid forward Gauss-Seidel - (GS) sweep as pre-smoother and one hybrid backward - GS sweep as post-smoother, basic smoothed aggregation - as coarsening algorithm, and LU (plus triangular solve) - as coarsest-level solver. See the default values in - Tables 2-8 - for further details of the preconditioner. -
--Note that the module
mld_prec_mod, containing the definition of the -preconditioner data type and the interfaces to the routines of MLD2P4, -must be used in any program calling such routines. -The modulespsb_base_mod, for the sparse matrix and communication descriptor -data types, andpsb_krylov_mod, for interfacing with the -Krylov solvers, must be also used (see Section 5.1). -
--Remark 1. Coarsest-level solvers based on the LU factorization, -such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist, -usually lead to smaller numbers of preconditioned Krylov -iterations than inexact solvers, when the linear system comes from -a standard discretization of basic scalar elliptic PDE problems. However, -this does not necessarily correspond to the smallest execution time -on parallel computers. +Direct solvers based on sparse LU factorizations, implemented in the +third-party libraries reported in Section 3.2, can be applied +as coarsest-level solvers by MLD2P4. Native inexact solvers based on +incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel, +and block Jacobi preconditioners are also available. Direct solvers usually +lead to more effective preconditioners in terms of algorithmic scalability; +however, this does not guarantee parallel efficiency. +
- -
- -Subsections - --
-
- Examples -
+
+ HREF="node12.html">
-
Next: Examples + HREF="node16.html">Getting Started Up: userhtml - Previous: Smoothers and coarsest-level solvers + HREF="node12.html">Multigrid Background + Previous: Smoothed Aggregation Contents diff --git a/docs/html/node16.html b/docs/html/node16.html index 9d368277..7a9da919 100644 --- a/docs/html/node16.html +++ b/docs/html/node16.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Examples - +Getting Started + @@ -18,297 +18,206 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - + + + -
-
+
-
-
- Next: User Interface - Up: Getting Started - Previous: Getting Started - Next: Examples + Up: userhtml + Previous: Smoothers and coarsest-level solvers + Contents
-+
-
-Examples --The code reported in Figure 2 shows how to set and apply the default -multi-level preconditioner available in the real double precision version -of MLD2P4 (see Table 1). This preconditioner is chosen -by simply specifying
'ML'as the second argument ofP%init-(a call toP%setis not needed) and is applied with the CG -solver provided by PSBLAS (the matrix of the system to be solved is -assumed to be positive definite). As previously observed, the modules -psb_base_mod,mld_prec_modandpsb_krylov_mod-must be used by the example program. - +Getting Started +-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; the statements concerning the deallocation of the PSBLAS -data structure are neglected too. -The complete code can be found in the example program file
mld_dexample_ml.f90, -in the directoryexamples/filereadof the MLD2P4 implementation (see -Section 3.5). A sample test problem along with the relevant -input data is available inexamples/fileread/runs. -For details on the use of the PSBLAS routines, see the PSBLAS User's -Guide [13]. +We describe the basics for building and applying MLD2P4 one-level and multi-level +(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [13]. +The following steps are required: + +-
+
- Declare the preconditioner data structure. It is a derived data type,
+
mld_xprec_type, where x may bes,d,c+ orz, according to the basic data type of the sparse matrix + (s= real single precision;d= real double precision; +c= complex single precision;z= complex double precision). + This data structure is accessed by the user only through the MLD2P4 routines, + following an object-oriented approach. +
+ - Allocate and initialize the preconditioner data structure, according to
+ a preconditioner type chosen by the user. This is performed by the routine
+
init, which also sets defaults for each preconditioner + type selected by the user. The preconditioner types and the defaults associated + with them are given in Table 1, where the strings used by +initto identify the preconditioner types are also given. + Note that these strings are valid also if uppercase letters are substituted by + corresponding lowercase ones. +
+ - Modify the selected preconditioner type, by properly setting
+ preconditioner parameters. This is performed by the routine
set. + This routine must be called only if the user wants to modify the default values + of the parameters associated with the selected preconditioner type, to obtain a variant + of that preconditioner. Examples of use ofsetare given in + Section 5.1; a complete list of all the + preconditioner parameters and their allowed and default values is provided in + Section 6, Tables 2-8. +
+ - Build the preconditioner for a given matrix. If the selected preconditioner
+ is multi-level, then two steps must be performed, as specified next.
+
-
+
- 4.1 +
- Build the aggregation hierarchy for a given matrix. This is
+performed by the routine
hierarchy_build. +
+ - 4.2 +
- Build the preconditioner for a given matrix. This is performed
+by the routine
smoothers_build. +
+
bld. +
+ - Apply the preconditioner at each iteration of a Krylov solver.
+ This is performed by the routine
aply. When using the PSBLAS Krylov solvers, + this step is completely transparent to the user, sinceaplyis called + by the PSBLAS routine implementing the Krylov solver (psb_krylov). +
+ - Free the preconditioner data structure. This is performed by
+ the routine
free. This step is complementary to step 1 and should + be performed when the preconditioner is no more used. +
+
-The setup and application of the default multi-level preconditioner -for the real single precision and the complex, single and double -precision, versions are obtained with straightforward modifications of the previous -example (see Section 6 for details). If these versions are installed, -the corresponding codes are available in
examples/fileread/. +All the previous routines are available as methods of the preconditioner object. +A detailed description of them is given in Section 6. +Examples showing the basic use of MLD2P4 are reported in Section 5.1.-
+
+-
Figure 2: -setup and application of the default multi-level preconditioner (example 1). + Table 1: +Preconditioner types, corresponding strings and default choices. --
- - -- - use psb_base_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_dprec_type) :: P -! right-hand side and solution vectors - type(psb_d_vect_type) :: b, x -... ... -! -! initialize the parallel environment - call psb_init(ictxt) - call psb_info(ictxt,iam,np) -... ... -! -! read and assemble the spd matrix A and the right-hand side b -! using PSBLAS routines for sparse matrix / vector management -... ... -! -! initialize the default multi-level preconditioner, i.e. V-cycle -! with basic smoothed aggregation, 1 hybrid forward/backward -! GS sweep as pre/post-smoother and UMFPACK as coarsest-level -! solver - call P%init('ML',info) -! -! build the preconditioner - call P%hierarchy_build(A,desc_A,info) - call P%smoothers_build(A,desc_A,info) - -! -! set the solver parameters and the initial guess - ... ... -! -! solve Ax=b with preconditioned CG - call psb_krylov('CG',A,P,b,x,tol,desc_A,info) - ... ... -! -! deallocate the preconditioner - call P%free(info) -! -! deallocate other data structures - ... ... -! -! exit the parallel environment - call psb_exit(ictxt) - stop --+
-TYPE +STRING +DEFAULT PRECONDITIONER +No preconditioner +'NOPREC'Considered only to use the PSBLAS + Krylov solvers with no preconditioner. +Diagonal +'DIAG'or'JACOBI'Diagonal preconditioner. + For any zero diagonal entry of the matrix to be preconditioned, + the corresponding entry of the preconditioner is set to 1. +Block Jacobi +'BJAC'Block-Jacobi with 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'V-cycle with one hybrid forward Gauss-Seidel + (GS) sweep as pre-smoother and one hybrid backward + GS sweep as post-smoother, basic smoothed aggregation + as coarsening algorithm, and LU (plus triangular solve) + as coarsest-level solver. See the default values in + Tables 2-8 + for further details of the preconditioner. +--Different versions of the multi-level preconditioner can be obtained by changing -the default values of the preconditioner parameters. The code reported in -Figure 3 shows how to set a V-cycle preconditioner -which applies 1 block-Jacobi sweep as pre- and post-smoother, -and solves the coarsest-level system with 8 block-Jacobi sweeps. -Note that the ILU(0) factorization (plus triangular solve) is used as -local solver for the block-Jacobi sweeps, since this is the default associated -with block-Jacobi and set by
P%init. -Furthermore, specifying block-Jacobi as coarsest-level -solver implies that the coarsest-level matrix is distributed -among the processes. -Figure 4 shows how to set a W-cycle preconditioner which -applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother, -and solves the coarsest-level system with the multifrontal LU factorization -implemented in MUMPS. It is specified that the coarsest-level -matrix is distributed, since MUMPS can be used on both -replicated and distributed matrices, and by default -it is used on replicated ones. The code fragments shown in Figures 3 and 4 are -included in the example program filemld_dexample_ml.f90too. - --Finally, Figure 5 shows the setup of a one-level -additive Schwarz preconditioner, i.e., RAS with overlap 2. -Note also that a Krylov method different from CG must be used to solve -the preconditioned system, since the preconditione in nonsymmetric. -The corresponding example program is available in the file -
mld_dexample_1lev.f90. - --For all the previous preconditioners, example programs where the sparse matrix and -the right-hand side are generated by discretizing a PDE with Dirichlet -boundary conditions are also available in the directory
examples/pdegen. - -- -
-- +Note that the module-
-Figure 3: -setup of a multi-level preconditioner -- --
-- -... ... -! build a V-cycle preconditioner with 1 block-Jacobi sweep (with -! ILU(0) on the blocks) as pre- and post-smoother, and 8 block-Jacobi -! sweeps (with ILU(0) on the blocks) as coarsest-level solver - call P%init('ML',info) - call_P%set('SMOOTHER_TYPE','BJAC',info) - call P%set('COARSE_SOLVE','BJAC',info) - call P%set('COARSE_SWEEPS',8,info) - call P%hierarchy_build(A,desc_A,info) - call P%smoothers_build(A,desc_A,info) -... ... ---+ +
-
-mld_prec_mod, containing the definition of the +preconditioner data type and the interfaces to the routines of MLD2P4, +must be used in any program calling such routines. +The modulespsb_base_mod, for the sparse matrix and communication descriptor +data types, andpsb_krylov_mod, for interfacing with the +Krylov solvers, must be also used (see Section 5.1). +
- -
-- +Remark 1. Coarsest-level solvers based on the LU factorization, +such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist, +usually lead to smaller numbers of preconditioned Krylov +iterations than inexact solvers, when the linear system comes from +a standard discretization of basic scalar elliptic PDE problems. However, +this does not necessarily correspond to the smallest execution time +on parallel computers.-
-Figure 4: -setup of a multi-level preconditioner -- --
-- -... ... -! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps -! as pre- and post-smoother, a distributed coarsest -! matrix, and MUMPS as coarsest-level solver - call P%init('ML',info) - call P%set('ML_CYCLE','WCYCLE',info) - call P%set('SMOOTHER_TYPE','FBGS',info) - call P%set('SMOOTHER_SWEEPS',2,info) - call P%set('COARSE_SOLVE','MUMPS',info) - call P%set('COARSE_MAT','DIST',info) - call P%hierarchy_build(A,desc_A,info) - call P%smoothers_build(A,desc_A,info) -... ... --- --
-- --
-Figure 5: -setup of a one-level Schwarz preconditioner. -- --
-- -... ... -! set RAS with overlap 2 and ILU(0) on the local blocks - call P%init('AS',info) - call P%set('SUB_OVR',2,info) - call P%bld(A,desc_A,info) -... ... -! solve Ax=b with preconditioned BiCGSTAB - call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info) ---
+-
+
- Examples +
-
-
+
-
-
- Next: User Interface - Up: Getting Started - Previous: Getting Started - Next: Examples + Up: userhtml + Previous: Smoothers and coarsest-level solvers + Contents diff --git a/docs/html/node17.html b/docs/html/node17.html index c5dc1094..3d749561 100644 --- a/docs/html/node17.html +++ b/docs/html/node17.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -User Interface - +Examples + @@ -18,9 +18,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + @@ -30,9 +29,9 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node18.html">
+ HREF="node16.html">
-
Next: Subroutine init + HREF="node18.html">User Interface Up: userhtml - Previous: Examples + HREF="node16.html">Getting Started + Previous: Getting Started Contents
-+
+Examples + +
-User Interface -+The code reported in Figure 2 shows how to set and apply the default +multi-level preconditioner available in the real double precision version +of MLD2P4 (see Table 1). This preconditioner is chosen +by simply specifying
'ML'as the second argument ofP%init+(a call toP%setis not needed) and is applied with the CG +solver provided by PSBLAS (the matrix of the system to be solved is +assumed to be positive definite). As previously observed, the modules +psb_base_mod,mld_prec_modandpsb_krylov_mod+must be used by the example program. + ++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; the statements concerning the deallocation of the PSBLAS +data structure are neglected too. +The complete code can be found in the example program file
mld_dexample_ml.f90, +in the directoryexamples/filereadof the MLD2P4 implementation (see +Section 3.5). A sample test problem along with the relevant +input data is available inexamples/fileread/runs. +For details on the use of the PSBLAS routines, see the PSBLAS User's +Guide [13]. + ++The setup and application of the default multi-level preconditioner +for the real single precision and the complex, single and double +precision, versions are obtained with straightforward modifications of the previous +example (see Section 6 for details). If these versions are installed, +the corresponding codes are available in
examples/fileread/. + ++ +
+++
+Figure 2: +setup and application of the default multi-level preconditioner (example 1). + ++ ++
++ + use psb_base_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_dprec_type) :: P +! right-hand side and solution vectors + type(psb_d_vect_type) :: b, x +... ... +! +! initialize the parallel environment + call psb_init(ictxt) + call psb_info(ictxt,iam,np) +... ... +! +! read and assemble the spd matrix A and the right-hand side b +! using PSBLAS routines for sparse matrix / vector management +... ... +! +! initialize the default multi-level preconditioner, i.e. V-cycle +! with basic smoothed aggregation, 1 hybrid forward/backward +! GS sweep as pre/post-smoother and UMFPACK as coarsest-level +! solver + call P%init('ML',info) +! +! build the preconditioner + call P%hierarchy_build(A,desc_A,info) + call P%smoothers_build(A,desc_A,info) + +! +! set the solver parameters and the initial guess + ... ... +! +! solve Ax=b with preconditioned CG + call psb_krylov('CG',A,P,b,x,tol,desc_A,info) + ... ... +! +! deallocate the preconditioner + call P%free(info) +! +! deallocate other data structures + ... ... +! +! exit the parallel environment + call psb_exit(ictxt) + stop +++ +-The basic user interface of MLD2P4 consists of eight routines. The six -routines
init,set, -hierarchy_build,smoothers_build, -bld, andapplyencapsulate all the -functionalities for the setup and the application of any multi-level and one-level -preconditioner implemented in the package. -The routinefreedeallocates the preconditioner data structure, while -descrprints a description of the preconditioner setup by the user. +Different versions of the multi-level preconditioner can be obtained by changing +the default values of the preconditioner parameters. The code reported in +Figure 3 shows how to set a V-cycle preconditioner +which applies 1 block-Jacobi sweep as pre- and post-smoother, +and solves the coarsest-level system with 8 block-Jacobi sweeps. +Note that the ILU(0) factorization (plus triangular solve) is used as +local solver for the block-Jacobi sweeps, since this is the default associated +with block-Jacobi and set byP%init. +Furthermore, specifying block-Jacobi as coarsest-level +solver implies that the coarsest-level matrix is distributed +among the processes. +Figure 4 shows how to set a W-cycle preconditioner which +applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother, +and solves the coarsest-level system with the multifrontal LU factorization +implemented in MUMPS. It is specified that the coarsest-level +matrix is distributed, since MUMPS can be used on both +replicated and distributed matrices, and by default +it is used on replicated ones. The code fragments shown in Figures 3 and 4 are +included in the example program filemld_dexample_ml.f90too.-All the routines are available as methods of the preconditioner object. -For each routine, the same user interface is overloaded with -respect to the real/ complex case and the single/double precision; -arguments with appropriate data types must be passed to the routine, -i.e., +Finally, Figure 5 shows the setup of a one-level +additive Schwarz preconditioner, i.e., RAS with overlap 2. +Note also that a Krylov method different from CG must be used to solve +the preconditioned system, since the preconditione in nonsymmetric. +The corresponding example program is available in the file +
mld_dexample_1lev.f90. --
-
- the sparse matrix data structure, containing the matrix to be
- preconditioned, must be of type
psb_xspmat_type- with x =sfor real single precision, x =d- for real double precision, x =cfor complex single precision, - x =zfor complex double precision; -
- - the preconditioner data structure must be of type
-
mld_xprec_type, with x = -s,d,c,z, according to the sparse - matrix data structure; -
- - the arrays containing the vectors
and 
involved in
- the preconditioner application must be of type
- psb_xvect_typewith x = -s,d,c,z, in a manner completely - analogous to the sparse matrix type; -
- - real parameters defining the preconditioner must be declared - according to the precision of the sparse matrix and preconditioner - data structures (see Section 6.2). - -
+For all the previous preconditioners, example programs where the sparse matrix and +the right-hand side are generated by discretizing a PDE with Dirichlet +boundary conditions are also available in the directory
examples/pdegen.+
+++
+Figure 3: +setup of a multi-level preconditioner ++ ++
++ +... ... +! build a V-cycle preconditioner with 1 block-Jacobi sweep (with +! ILU(0) on the blocks) as pre- and post-smoother, and 8 block-Jacobi +! sweeps (with ILU(0) on the blocks) as coarsest-level solver + call P%init('ML',info) + call_P%set('SMOOTHER_TYPE','BJAC',info) + call P%set('COARSE_SOLVE','BJAC',info) + call P%set('COARSE_SWEEPS',8,info) + call P%hierarchy_build(A,desc_A,info) + call P%smoothers_build(A,desc_A,info) +... ... +++++
+-
- -Subsections +++ ++
+Figure 4: +setup of a multi-level preconditioner ++ ++
++ +... ... +! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps +! as pre- and post-smoother, a distributed coarsest +! matrix, and MUMPS as coarsest-level solver + call P%init('ML',info) + call P%set('ML_CYCLE','WCYCLE',info) + call P%set('SMOOTHER_TYPE','FBGS',info) + call P%set('SMOOTHER_SWEEPS',2,info) + call P%set('COARSE_SOLVE','MUMPS',info) + call P%set('COARSE_MAT','DIST',info) + call P%hierarchy_build(A,desc_A,info) + call P%smoothers_build(A,desc_A,info) +... ... +++ +
++ ++
+Figure 5: +setup of a one-level Schwarz preconditioner. ++ ++
++ +... ... +! set RAS with overlap 2 and ILU(0) on the local blocks + call P%init('AS',info) + call P%set('SUB_OVR',2,info) + call P%bld(A,desc_A,info) +... ... +! solve Ax=b with preconditioned BiCGSTAB + call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info) +++ ++
+ HREF="node16.html">
-
Next: Subroutine init + HREF="node18.html">User Interface Up: userhtml - Previous: Examples + HREF="node16.html">Getting Started + Previous: Getting Started Contents diff --git a/docs/html/node18.html b/docs/html/node18.html index 4479d15b..8a65faa0 100644 --- a/docs/html/node18.html +++ b/docs/html/node18.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine init - +User Interface + @@ -18,112 +18,143 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + + -
-
+
-
-
- Next: Subroutine set - Up: User Interface - Previous: User Interface - Next: Subroutine init + Up: userhtml + Previous: Examples + Contents
-+
+User Interface +
-Subroutine init -- -
-call p%init(ptype,info)-- -This routine allocates and initializes the preconditioner -
p, according to the preconditioner type chosen by the user. +The basic user interface of MLD2P4 consists of eight routines. The six +routinesinit,set, +hierarchy_build,smoothers_build, +bld, andapplyencapsulate all the +functionalities for the setup and the application of any multi-level and one-level +preconditioner implemented in the package. +The routinefreedeallocates the preconditioner data structure, while +descrprints a description of the preconditioner setup by the user.-Arguments -
- --
-
-ptypecharacter(len=*), intent(in).- -The type of preconditioner. Its values are specified - in Table 1. -- -Note that the strings are case insensitive. -- -infointeger, intent(out).- -Error code. If no error, 0 is returned. See Section 8 for details. --For compatibility with the previous versions of MLD2P4, this routine can be also invoked -as follows: +All the routines are available as methods of the preconditioner object. +For each routine, the same user interface is overloaded with +respect to the real/ complex case and the single/double precision; +arguments with appropriate data types must be passed to the routine, +i.e., + +
-
+
- the sparse matrix data structure, containing the matrix to be
+ preconditioned, must be of type
psb_xspmat_type+ with x =sfor real single precision, x =d+ for real double precision, x =cfor complex single precision, + x =zfor complex double precision; +
+ - the preconditioner data structure must be of type
+
mld_xprec_type, with x = +s,d,c,z, according to the sparse + matrix data structure; +
+ - the arrays containing the vectors and involved in
+ the preconditioner application must be of type
+
psb_xvect_typewith x = +s,d,c,z, in a manner completely + analogous to the sparse matrix type; +
+ - real parameters defining the preconditioner must be declared + according to the precision of the sparse matrix and preconditioner + data structures (see Section 6.2). + +
-
call mld_precinit(p,ptype,info)--
-
+
+ +Subsections + +-
+
- Subroutine init +
- Subroutine set +
- Subroutine build +
- Subroutine hierarchy_build +
- Subroutine smoothers_build +
- Subroutine apply +
- Subroutine free +
- Subroutine descr +
-
-
+
-
-
- Next: Subroutine set - Up: User Interface - Previous: User Interface - Next: Subroutine init + Up: userhtml + Previous: Examples + Contents diff --git a/docs/html/node19.html b/docs/html/node19.html index 9f0ff2a6..2211e4c2 100644 --- a/docs/html/node19.html +++ b/docs/html/node19.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine set - +Subroutine init + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node20.html">
+ HREF="node18.html">
@@ -40,30 +40,29 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Subroutine build + HREF="node20.html">Subroutine set Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine init + HREF="node18.html">User Interface Contents
-+
-Subroutine set +Subroutine init-
call p%set(what,val,info [,ilev, ilmax, pos])+call p%init(ptype,info)-This routine sets the parameters defining the preconditioner
p. More -precisely, the parameter identified bywhatis assigned the value -contained inval. +This routine allocates and initializes the preconditioner +p, according to the preconditioner type chosen by the user.Arguments @@ -71,28 +70,17 @@ contained in
val.-
whatcharacter(len=*).ptypecharacter(len=*), intent(in).The parameter to be set. It can be specified through its name; - the string is case-insensitive. See - Tables 2-8. -- -valintegerorcharacter(len=*)or -real(psb_spk_)orreal(psb_dpk_), -intent(in).The type of preconditioner. Its values are specified + in Table 1. The value of the parameter to be set. The list of allowed - values and the corresponding data types is given in - Tables 2-8. - When the value is of type +character(len=*), - it is also treated as case insensitive.Note that the strings are case insensitive. infoval.Error code. If no error, 0 is returned. See Section 8 - for details. -- -ilevinteger, optional, intent(in).- -For the multi-level preconditioner, the level at which the - preconditioner parameter has to be set. - The levels are numbered in increasing - order starting from the finest one, i.e., level 1 is the finest level. - If -ilevis not present, the parameter identified bywhat- is set at all the appropriate levels (see - Tables 2-8).- -ilmaxinteger, optional, intent(in).- -For the multi-level preconditioner, when both - -ilevandilmaxare present, the settings - are applied at all levelsilev:ilmax. When -ilevis present butilmaxis not, then - the default isilmax=ilev. - The levels are numbered in increasing - order starting from the finest one, i.e., level 1 is the finest level.- -poscharater(len=*), optional, intent(in).- -Whether the other arguments apply only to the pre-smoother ( +'PRE') - or to the post-smoother ('POST'). Ifposis not present, - the other arguments are applied to both smoothers. - If the preconditioner is one-level or the parameter identified bywhat- does not concern the smoothers,posis ignored. -Error code. If no error, 0 is returned. See Section 8 for details. @@ -151,731 +97,9 @@ as follows:
-
call mld_precset(p,what,val,info)+call mld_precinit(p,ptype,info)- -However, in this case the optional arguments
ilev,ilmax, andpos-cannot be used. -
--A variety of preconditioners can be obtained -by a suitable setting of the preconditioner parameters. These parameters -can be logically divided into four groups, i.e., parameters defining - -
-
-
- the type of multi-level cycle and how many cycles must be applied; - -
- the aggregation algorithm; - -
- the coarse-space correction at the coarsest level (for multi-level - preconditioners only); - -
- the smoother of the multi-level preconditioners, or the one-level
- preconditioner.
-
-
-
-
--Remark 2. A smoother is usually obtained by combining two objects: -a smoother (
SMOOTHER_TYPE) and a local solver (SUB_SOLVE), -as specified in Tables 7-8. -For example, the block-Jacobi smoother using -ILU(0) on the blocks is obtained by combining the block-Jacobi smoother -object with the ILU(0) solver object. Similarly, -the hybrid Gauss-Seidel smoother (see Note in Table 7) -is obtained by combining the block-Jacobi smoother object with a single sweep -of the Gauss-Seidel solver object, while the point-Jacobi smoother is the -result of combining the block-Jacobi smoother object with a single sweep -of the pointwise-Jacobi solver object. However, for simplicity, shortcuts are -provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and -hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined -by setting onlySMOOTHER_TYPEto appropriate values (see -Tables 7), i.e., without setting -SUB_SOLVEtoo. - --The smoother and solver objects are arranged in a -hierarchical manner. When specifying a smoother object, its parameters, -including the local solver, are set to their default values, and when a solver -object is specified, its defaults are also set, overriding in both -cases any previous settings even if explicitly specified. Therefore if -the user sets a smoother, and wishes to use a solver -different from the default one, the call to set the solver must come -after the call to set the smoother. - -
-Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi -coarsest-level solvers, and shortcuts are available -in this case too (see Table 5). -
--Remark 3. In general, a coarsest-level solver cannot be used with -both the replicated and distributed coarsest-matrix layout; -therefore, setting the solver after the layout may change the layout. -Similarly, setting the layout after the solver may change the solver. - -
-More precisely, UMFPACK and SuperLU require the coarsest-level -matrix to be replicated, while SuperLU_Dist requires it to be distributed. -In these cases, setting the coarsest-level solver implies that -the layout is redefined according to the solver, ovverriding any -previous settings. MUMPS, point-Jacobi, -hybrid Gauss-Seidel and block-Jacobi can be applied to -replicated and distributed matrices, thus their choice -does not modify any previously specified layout. -It is worth noting that, when the matrix is replicated, -the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers -reduce to the corresponding local solver objects (see Remark 2). -For the point-Jacobi and Gauss-Seidel solvers, these objects -correspond to a single point-Jacobi sweep and a single -Gauss-Seidel sweep, respectively, which are very poor solvers. - -
-On the other hand, the distributed layout can be used with any solver -but UMFPACK and SuperLU; therefore, if any of these two solvers has already -been selected, the coarsest-level solver is changed to block-Jacobi, -with the previously chosen solver applied to the local blocks. -Likewise, the replicated layout can be used with any solver but SuperLu_Dist; -therefore, if SuperLu_Dist has been previously set, the coarsest-level -solver is changed to the default sequential solver. - -
- -
----
-Table 2: -Parameters defining the multi-level cycle and the number of cycles to -be applied. - -- ---
-whatDATA TYPE -valDEFAULT -COMMENTS -'ML_CYCLE'character(len=*)'VCYCLE' - --'WCYCLE' -
-'KCYCLE' -
-'MULT' -
-'ADD'
'VCYCLE' -Multi-level cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz, - and Additive Schwarz. - --Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and - is included for compatibility with previous versions of MLD2P4.
'OUTER_SWEEPS'integer -Any integer - --number
1 -Number of multi-level cycles. -
-- -
----
-Table 3: -Parameters defining the aggregation algorithm. - -- ---
-whatDATA TYPE -valDEFAULT -COMMENTS -'MIN_COARSE_SIZE'integerAny number - --
- -![$\lfloor 40 \sqrt[3]{n} \rfloor$](img76.png)
, where 
is the dimension
- of the matrix at the finest levelCoarse size threshold. The aggregation stops - if the global number of variables of the - computed coarsest matrix - is lower than or equal to this threshold - (see Note). -'MIN_CR_RATIO'realAny number - --
1.5 -Minimum coarsening ratio. The aggregation stops - if the ratio between the matrix dimensions - at two consecutive levels is lower than or equal to this - threshold (see Note). -'MAX_LEVS'integerAny integer - --number
20 -Maximum number of levels. The aggregation stops - if the number of levels reaches this value (see Note). -'PAR_AGGR'character(len=*)'DEC', 'SYMDEC' -'DEC' -Parallel aggregation algorithm. - --Currently, only the - decoupled aggregation (
DEC) is available; the -SYMDECoption applies decoupled - aggregation to the sparsity pattern - of
.'AGGR_TYPE'character(len=*)'VMB' -'VMB' -Type of aggregation algorithm: currently, the scalar aggregation - algorithm by Vanek, Mandel and Brezina is implemented - [25]. -'AGGR_PROL'character(len=*)'SMOOTHED', 'UNSMOOTHED' -'SMOOTHED' -Prolongator used by the aggregation algorithm: smoothed or unsmoothed - (i.e., tentative prolongator). -Note. The aggregation algorithm stops when -at least one of the following criteria is met: -the coarse size threshold, the -maximum coarsening ratio, or the maximum number -of levels is reached. Therefore, the actual number of levels may be -smaller than the specified maximum number -of levels. -
-- -
----
-Table 4: -Parameters defining the aggregation algorithm (continued). - -- ---
-whatDATA TYPE -valDEFAULT -COMMENTS -'AGGR_ORD'character(len=*)'NATURAL' - --'DEGREE'
'NATURAL' -Initial ordering of indices for the aggregation - algorithm: either natural ordering or sorted by - descending degrees of the nodes in the - matrix graph. -'AGGR_THRESH'real(kind_parameter)Any real - --number
![$\in [0, 1]$](img80.png)
0.05 -The threshold -
in the aggregation algorithm,
- see (3) in Section 4.2.
- See also the note at the bottom of this table.'AGGR_FILTER'character(len=*)'FILTER' - --'NOFILTER'
'NOFILTER' -Matrix used in computing the smoothed - prolongator: filtered or unfiltered (see (5) in Section 4.2). -Note. Different thresholds at different levels, such as -those used in [25, Section 5.1], can be easily set by -invoking the rou- -tine set with -the parameter ilev. -
-- -
----
-Table 5: -Parameters defining the coarse-space correction at the coarsest -level. -- ---
-whatDATA TYPE -valDEFAULT -COMMENTS -'COARSE_MAT'character(len=*)'DIST' - --'REPL'
'REPL' -Coarsest matrix layout: distributed among the processes or - replicated on each of them. -'COARSE_SOLVE'character(len=*)'MUMPS' - --'UMF' -
-'SLU' -
-'SLUDIST' -
-'JACOBI' -
-'GS' -
-'BJAC'
See Note. -Solver used at the coarsest level: sequential - LU from MUMPS, UMFPACK, or SuperLU - (plus triangular solve); - distributed LU from MUMPS or SuperLU_Dist - (plus triangular solve); - point-Jacobi, hybrid Gauss-Seidel or block-Jacobi. - --Note that UMF and SLU require the coarsest - matrix to be replicated, SLUDIST, JACOBI, - GS and BJAC require it to be - distributed, MUMPS can be used with either - a replicated or a distributed matrix. When any of the previous - solvers is specified, the matrix layout is set to a default - value - which allows the use - value UMFPACK and SuperLU_Dist - are available only in double precision.
'COARSE_SUBSOLVE'character(len=*)'ILU' - --'ILUT' -
-'MILU' -
-'MUMPS' -
-'SLU' -
-'UMF'
See Note. -Solver for the diagonal blocks of the coarse matrix, - in case the block Jacobi solver - is chosen as coarsest-level solver: ILU( -
), ILU(
),
- MILU(), LU from MUMPS, SuperLU or UMFPACK
- (plus triangular solve).
- Note that UMFPACK and SuperLU_Dist
- are available only in double precision.Note. Defaults for COARSE_SOLVE and -COARSE_SUBSOLVE are chosen in the following order: -single precision version - MUMPS if installed, - then SLU if installed, - ILU otherwise; -double precision version - UMF if installed, - then MUMPS if installed, then SLU if - installed, ILU otherwise. -
-- -
----
-Table 6: -Parameters defining the coarse-space correction at the coarsest -level (continued). -- ---
-whatDATA TYPE -valDEFAULT -COMMENTS -'COARSE_SWEEPS'integerAny integer - --number
10 -Number of sweeps when -JACOBI,GSorBJAC- is chosen as coarsest-level solver.'COARSE_FILLIN'integerAny integer - --number
0 -Fill-in level - of the ILU factorizations.'COARSE_ILUTHRS'real(kind_parameter)Any real - --number
0 -Drop tolerance -
in the ILU() factorization.
-- -
----
-Table 7: -Parameters defining the smoother or the details of the one-level preconditioner. - -- ---
- -whatDATA TYPE -valDEFAULT --COMMENTS -'SMOOTHER_TYPE'character(len=*)-'JACOBI'--
'GS'--
'BGS'--
'BJAC'- --
'AS'-'FBGS'-Type of smoother used in the multi-level preconditioner: - point-Jacobi, hybrid (forward) Gauss-Seidel, - hybrid backward Gauss-Seidel, block-Jacobi, and - Additive Schwarz. - --It is ignored by one-level preconditioners.
'SUB_SOLVE'character(len=*)-'JACOBI' - --'GS' -
-'BGS' -
-'ILU' -
-'ILUT' -
-'MILU' -
-'MUMPS' -
-'SLU' -
-'UMF' -
GS and BGS for pre- and post-smoothers - of multi-level preconditioners, respectively - --ILU for block-Jacobi and Additive Schwarz - one-level preconditioners -
The local solver to be used with the smoother or one-level - preconditioner (see Remark 2, page 24): point-Jacobi, - hybrid (forward) Gauss-Seidel, hybrid backward - Gauss-Seidel, ILU( -), ILU(), MILU(),
- LU from MUMPS, SuperLU or UMFPACK
- (plus triangular solve). See Note for details on hybrid
- Gauss-Seidel. 'SMOOTHER_SWEEPS'integer-Any integer - --number
- 1 - -Number of sweeps of the smoother or one-level preconditioner. - In the multi-level case, no pre-smother or - post-smoother is used if this parameter is set to 0 - together with -pos='PRE'orpos='POST, - respectively.'SUB_OVR'integer-Any integer - --number
- 1 - -Number of overlap layers, for Additive Schwarz only. -
-- -
----
-Table 8: -Parameters defining the smoother or the details of the one-level preconditioner -(continued). -- ---
- -whatDATA TYPE -valDEFAULT --COMMENTS -'SUB_RESTR'character(len=*)-'HALO' - --'NONE' -
'HALO' - -Type of restriction operator, for Additive Schwarz only: - HALO for taking into account the overlap, NONE - for neglecting it. - --Note that HALO must be chosen for - the classical Addditive Schwarz smoother and its RAS variant.
'SUB_PROL'character(len=*)-'SUM' - --'NONE' -
'NONE' - -Type of prolongation operator, for Additive Schwarz only: - SUM for adding the contributions from the overlap, NONE - for neglecting them. - --Note that SUM must be chosen for the classical Additive - Schwarz smoother, and NONE for its RAS variant.
'SUB_FILLIN'integer-Any integer - --number
- 0 - -Fill-in level - of the incomplete LU factorizations. 'SUB_ILUTHRS'real(kind_parameter)-Any real number -
- 0 - -Drop tolerance - in the ILU() factorization. - - - -
-
@@ -884,7 +108,7 @@ Parameters defining the smoother or the details of the one-level preconditioner HREF="node20.html">
+ HREF="node18.html">
@@ -894,11 +118,11 @@ Parameters defining the smoother or the details of the one-level preconditioner
Next: Subroutine build + HREF="node20.html">Subroutine set Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine init + HREF="node18.html">User Interface Contents diff --git a/docs/html/node2.html b/docs/html/node2.html index a3236629..d1f05927 100644 --- a/docs/html/node2.html +++ b/docs/html/node2.html @@ -26,21 +26,21 @@ original version by: Nikos Drakos, CBLU, University of Leeds -
-
-
- Next: Next: General Overview - Up: Up: userhtml - Previous: Previous: Abstract
@@ -53,72 +53,72 @@ Contents-
-
- General Overview
- Code Distribution + HREF="node3.html">General Overview
- Configuring and Building MLD2P4
-
-
+ HREF="node4.html">Code Distribution
- Prerequisites
+ HREF="node6.html">Configuring and Building MLD2P4
+
- Optional third party libraries + HREF="node7.html">Prerequisites
- Configuration options + HREF="node8.html">Optional third party libraries
- Bug reporting + HREF="node9.html">Configuration options
- Example and test programs + HREF="node10.html">Bug reporting +
- Example and test programs
- - Multigrid Background
-
- AMG preconditioners
+ HREF="node12.html">Multigrid Background
+
- Smoothed Aggregation + HREF="node13.html">AMG preconditioners
- Smoothers and coarsest-level solvers + HREF="node14.html">Smoothed Aggregation +
- Smoothers and coarsest-level solvers
- - Getting Started
-
- Examples
+ HREF="node16.html">Getting Started
+
- - User Interface
-
- Subroutine init
+ HREF="node18.html">User Interface
+
- Subroutine set + HREF="node19.html">Subroutine init
- Subroutine build + HREF="node20.html">Subroutine set
- Subroutine hierarchy_build + HREF="node21.html">Subroutine build
- Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build
- Subroutine apply + HREF="node23.html">Subroutine smoothers_build
- Subroutine free + HREF="node24.html">Subroutine apply
- Subroutine descr + HREF="node25.html">Subroutine free +
- Subroutine descr
- - Adding new smoother and solver objects to MLD2P4
- Error Handling + HREF="node27.html">Adding new smoother and solver objects to MLD2P4
- License + HREF="node28.html">Error Handling
- Bibliography + HREF="node29.html">License +
- Bibliography
- Subroutine init
+ HREF="node18.html">User Interface
+
- Examples
+ HREF="node16.html">Getting Started
+
- AMG preconditioners
+ HREF="node12.html">Multigrid Background
+
- Prerequisites
+ HREF="node6.html">Configuring and Building MLD2P4
+
Subroutine build - +Subroutine set + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node21.html">
+ HREF="node18.html">
@@ -40,30 +40,30 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Subroutine hierarchy_build + HREF="node21.html">Subroutine build Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine set + HREF="node19.html">Subroutine init Contents
-+
-Subroutine build +Subroutine set-
+call p%build(a,desc_a,info)-call p%set(what,val,info [,ilev, ilmax, pos])+-This routine builds the one-level preconditioner
paccording to the requirements -made by the user through the routinesinitandset-(see Sections 6.4 and 6.5 for multi-level preconditioners). +This routine sets the parameters defining the preconditionerp. More +precisely, the parameter identified bywhatis assigned the value +contained inval.Arguments @@ -71,34 +71,78 @@ made by the user through the routines
initandset-
- -atype(psb_xspmat_type), intent(in).whatcharacter(len=*).The sparse matrix structure containing the local part of the - matrix to be preconditioned. Note that x must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use. - See the PSBLAS User's Guide for details [13]. +The parameter to be set. It can be specified through its name; + the string is case-insensitive. See + Tables 2-8. - -desc_atype(psb_desc_type), intent(in).+ +valintegerorcharacter(len=*)or +real(psb_spk_)orreal(psb_dpk_), +intent(in).The communication descriptor of +a. See the PSBLAS User's Guide for - details [13].The value of the parameter to be set. The list of allowed + values and the corresponding data types is given in + Tables 2-8. + When the value is of type character(len=*), + it is also treated as case insensitive.info+ infointeger, intent(out).Error code. If no error, 0 is returned. See Section 8 for details. +Error code. If no error, 0 is returned. See Section 8 + for details. ++ +ilevinteger, optional, intent(in).+ +For the multi-level preconditioner, the level at which the + preconditioner parameter has to be set. + The levels are numbered in increasing + order starting from the finest one, i.e., level 1 is the finest level. + If +ilevis not present, the parameter identified bywhat+ is set at all the appropriate levels (see + Tables 2-8).+ +ilmaxinteger, optional, intent(in).+ +For the multi-level preconditioner, when both + +ilevandilmaxare present, the settings + are applied at all levelsilev:ilmax. When +ilevis present butilmaxis not, then + the default isilmax=ilev. + The levels are numbered in increasing + order starting from the finest one, i.e., level 1 is the finest level.+ +poscharater(len=*), optional, intent(in).+ +Whether the other arguments apply only to the pre-smoother ( 'PRE') + or to the post-smoother ('POST'). Ifposis not present, + the other arguments are applied to both smoothers. + If the preconditioner is one-level or the parameter identified bywhat+ does not concern the smoothers,posis ignored. +@@ -107,13 +151,731 @@ as follows:
-
call mld_precbld(p,what,val,info)+call mld_precset(p,what,val,info)-In this case, the routine can be used to build multi-level preconditioners too. +However, in this case the optional arguments
ilev,ilmax, andpos+cannot be used. +
++A variety of preconditioners can be obtained +by a suitable setting of the preconditioner parameters. These parameters +can be logically divided into four groups, i.e., parameters defining + +
-
+
- the type of multi-level cycle and how many cycles must be applied; + +
- the aggregation algorithm; + +
- the coarse-space correction at the coarsest level (for multi-level + preconditioners only); + +
- the smoother of the multi-level preconditioners, or the one-level
+ preconditioner.
+
+
+
+
++Remark 2. A smoother is usually obtained by combining two objects: +a smoother (
SMOOTHER_TYPE) and a local solver (SUB_SOLVE), +as specified in Tables 7-8. +For example, the block-Jacobi smoother using +ILU(0) on the blocks is obtained by combining the block-Jacobi smoother +object with the ILU(0) solver object. Similarly, +the hybrid Gauss-Seidel smoother (see Note in Table 7) +is obtained by combining the block-Jacobi smoother object with a single sweep +of the Gauss-Seidel solver object, while the point-Jacobi smoother is the +result of combining the block-Jacobi smoother object with a single sweep +of the pointwise-Jacobi solver object. However, for simplicity, shortcuts are +provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and +hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined +by setting onlySMOOTHER_TYPEto appropriate values (see +Tables 7), i.e., without setting +SUB_SOLVEtoo.+The smoother and solver objects are arranged in a +hierarchical manner. When specifying a smoother object, its parameters, +including the local solver, are set to their default values, and when a solver +object is specified, its defaults are also set, overriding in both +cases any previous settings even if explicitly specified. Therefore if +the user sets a smoother, and wishes to use a solver +different from the default one, the call to set the solver must come +after the call to set the smoother. + +
+Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi +coarsest-level solvers, and shortcuts are available +in this case too (see Table 5). +
++Remark 3. In general, a coarsest-level solver cannot be used with +both the replicated and distributed coarsest-matrix layout; +therefore, setting the solver after the layout may change the layout. +Similarly, setting the layout after the solver may change the solver. + +
+More precisely, UMFPACK and SuperLU require the coarsest-level +matrix to be replicated, while SuperLU_Dist requires it to be distributed. +In these cases, setting the coarsest-level solver implies that +the layout is redefined according to the solver, ovverriding any +previous settings. MUMPS, point-Jacobi, +hybrid Gauss-Seidel and block-Jacobi can be applied to +replicated and distributed matrices, thus their choice +does not modify any previously specified layout. +It is worth noting that, when the matrix is replicated, +the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers +reduce to the corresponding local solver objects (see Remark 2). +For the point-Jacobi and Gauss-Seidel solvers, these objects +correspond to a single point-Jacobi sweep and a single +Gauss-Seidel sweep, respectively, which are very poor solvers. + +
+On the other hand, the distributed layout can be used with any solver +but UMFPACK and SuperLU; therefore, if any of these two solvers has already +been selected, the coarsest-level solver is changed to block-Jacobi, +with the previously chosen solver applied to the local blocks. +Likewise, the replicated layout can be used with any solver but SuperLu_Dist; +therefore, if SuperLu_Dist has been previously set, the coarsest-level +solver is changed to the default sequential solver. + +
+ +
++++
+Table 2: +Parameters defining the multi-level cycle and the number of cycles to +be applied. + ++ +++
+whatDATA TYPE +valDEFAULT +COMMENTS +'ML_CYCLE'character(len=*)'VCYCLE' + ++'WCYCLE' +
+'KCYCLE' +
+'MULT' +
+'ADD'
'VCYCLE' +Multi-level cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz, + and Additive Schwarz. + ++Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and + is included for compatibility with previous versions of MLD2P4.
'OUTER_SWEEPS'integer +Any integer + ++number
1 +Number of multi-level cycles. +
++ +
++++
+Table 3: +Parameters defining the aggregation algorithm. + ++ +++
+whatDATA TYPE +valDEFAULT +COMMENTS +'MIN_COARSE_SIZE'integerAny number + ++
+ +, where is the dimension
+ of the matrix at the finest levelCoarse size threshold. The aggregation stops + if the global number of variables of the + computed coarsest matrix + is lower than or equal to this threshold + (see Note). +'MIN_CR_RATIO'realAny number + ++
1.5 +Minimum coarsening ratio. The aggregation stops + if the ratio between the matrix dimensions + at two consecutive levels is lower than or equal to this + threshold (see Note). +'MAX_LEVS'integerAny integer + ++number
20 +Maximum number of levels. The aggregation stops + if the number of levels reaches this value (see Note). +'PAR_AGGR'character(len=*)'DEC', 'SYMDEC' +'DEC' +Parallel aggregation algorithm. + ++Currently, only the + decoupled aggregation (
DEC) is available; the +SYMDECoption applies decoupled + aggregation to the sparsity pattern + of.'AGGR_TYPE'character(len=*)'VMB' +'VMB' +Type of aggregation algorithm: currently, the scalar aggregation + algorithm by Vanek, Mandel and Brezina is implemented + [25]. +'AGGR_PROL'character(len=*)'SMOOTHED', 'UNSMOOTHED' +'SMOOTHED' +Prolongator used by the aggregation algorithm: smoothed or unsmoothed + (i.e., tentative prolongator). +Note. The aggregation algorithm stops when +at least one of the following criteria is met: +the coarse size threshold, the +maximum coarsening ratio, or the maximum number +of levels is reached. Therefore, the actual number of levels may be +smaller than the specified maximum number +of levels. +
++ +
++++
+Table 4: +Parameters defining the aggregation algorithm (continued). + ++ +++
+whatDATA TYPE +valDEFAULT +COMMENTS +'AGGR_ORD'character(len=*)'NATURAL' + ++'DEGREE'
'NATURAL' +Initial ordering of indices for the aggregation + algorithm: either natural ordering or sorted by + descending degrees of the nodes in the + matrix graph. +'AGGR_THRESH'real(kind_parameter)Any real + ++number
0.05 +The threshold + in the aggregation algorithm,
+ see (3) in Section 4.2.
+ See also the note at the bottom of this table.'AGGR_FILTER'character(len=*)'FILTER' + ++'NOFILTER'
'NOFILTER' +Matrix used in computing the smoothed + prolongator: filtered or unfiltered (see (5) in Section 4.2). +Note. Different thresholds at different levels, such as +those used in [25, Section 5.1], can be easily set by +invoking the rou- +tine set with +the parameter ilev. +
++ +
++++
+Table 5: +Parameters defining the coarse-space correction at the coarsest +level. ++ +++
+whatDATA TYPE +valDEFAULT +COMMENTS +'COARSE_MAT'character(len=*)'DIST' + ++'REPL'
'REPL' +Coarsest matrix layout: distributed among the processes or + replicated on each of them. +'COARSE_SOLVE'character(len=*)'MUMPS' + ++'UMF' +
+'SLU' +
+'SLUDIST' +
+'JACOBI' +
+'GS' +
+'BJAC'
See Note. +Solver used at the coarsest level: sequential + LU from MUMPS, UMFPACK, or SuperLU + (plus triangular solve); + distributed LU from MUMPS or SuperLU_Dist + (plus triangular solve); + point-Jacobi, hybrid Gauss-Seidel or block-Jacobi. + ++Note that UMF and SLU require the coarsest + matrix to be replicated, SLUDIST, JACOBI, + GS and BJAC require it to be + distributed, MUMPS can be used with either + a replicated or a distributed matrix. When any of the previous + solvers is specified, the matrix layout is set to a default + value + which allows the use + value UMFPACK and SuperLU_Dist + are available only in double precision.
'COARSE_SUBSOLVE'character(len=*)'ILU' + ++'ILUT' +
+'MILU' +
+'MUMPS' +
+'SLU' +
+'UMF'
See Note. +Solver for the diagonal blocks of the coarse matrix, + in case the block Jacobi solver + is chosen as coarsest-level solver: ILU( +), ILU(),
+ MILU(), LU from MUMPS, SuperLU or UMFPACK
+ (plus triangular solve).
+ Note that UMFPACK and SuperLU_Dist
+ are available only in double precision.Note. Defaults for COARSE_SOLVE and +COARSE_SUBSOLVE are chosen in the following order: +single precision version - MUMPS if installed, + then SLU if installed, + ILU otherwise; +double precision version - UMF if installed, + then MUMPS if installed, then SLU if + installed, ILU otherwise. +
++ +
++++
+Table 6: +Parameters defining the coarse-space correction at the coarsest +level (continued). ++ +++
+whatDATA TYPE +valDEFAULT +COMMENTS +'COARSE_SWEEPS'integerAny integer + ++number
10 +Number of sweeps when +JACOBI,GSorBJAC+ is chosen as coarsest-level solver.'COARSE_FILLIN'integerAny integer + ++number
0 +Fill-in level + of the ILU factorizations.'COARSE_ILUTHRS'real(kind_parameter)Any real + ++number
0 +Drop tolerance + in the ILU() factorization.
++ +
++++
+Table 7: +Parameters defining the smoother or the details of the one-level preconditioner. + ++ +++
+ +whatDATA TYPE +valDEFAULT ++COMMENTS +'SMOOTHER_TYPE'character(len=*)+'JACOBI'++
'GS'++
'BGS'++
'BJAC'+ ++
'AS'+'FBGS'+Type of smoother used in the multi-level preconditioner: + point-Jacobi, hybrid (forward) Gauss-Seidel, + hybrid backward Gauss-Seidel, block-Jacobi, and + Additive Schwarz. + ++It is ignored by one-level preconditioners.
'SUB_SOLVE'character(len=*)+'JACOBI' + ++'GS' +
+'BGS' +
+'ILU' +
+'ILUT' +
+'MILU' +
+'MUMPS' +
+'SLU' +
+'UMF' +
GS and BGS for pre- and post-smoothers + of multi-level preconditioners, respectively + ++ILU for block-Jacobi and Additive Schwarz + one-level preconditioners +
The local solver to be used with the smoother or one-level + preconditioner (see Remark 2, page 24): point-Jacobi, + hybrid (forward) Gauss-Seidel, hybrid backward + Gauss-Seidel, ILU( +), ILU(), MILU(),
+ LU from MUMPS, SuperLU or UMFPACK
+ (plus triangular solve). See Note for details on hybrid
+ Gauss-Seidel. 'SMOOTHER_SWEEPS'integer+Any integer + ++number
+ 1 + +Number of sweeps of the smoother or one-level preconditioner. + In the multi-level case, no pre-smother or + post-smoother is used if this parameter is set to 0 + together with +pos='PRE'orpos='POST, + respectively.'SUB_OVR'integer+Any integer + ++number
+ 1 + +Number of overlap layers, for Additive Schwarz only. +
++ +
++++
+Table 8: +Parameters defining the smoother or the details of the one-level preconditioner +(continued). ++ +++
+ +whatDATA TYPE +valDEFAULT ++COMMENTS +'SUB_RESTR'character(len=*)+'HALO' + ++'NONE' +
'HALO' + +Type of restriction operator, for Additive Schwarz only: + HALO for taking into account the overlap, NONE + for neglecting it. + ++Note that HALO must be chosen for + the classical Addditive Schwarz smoother and its RAS variant.
'SUB_PROL'character(len=*)+'SUM' + ++'NONE' +
'NONE' + +Type of prolongation operator, for Additive Schwarz only: + SUM for adding the contributions from the overlap, NONE + for neglecting them. + ++Note that SUM must be chosen for the classical Additive + Schwarz smoother, and NONE for its RAS variant.
'SUB_FILLIN'integer+Any integer + ++number
+ 0 + +Fill-in level + of the incomplete LU factorizations. 'SUB_ILUTHRS'real(kind_parameter)+Any real number +
+ 0 + +Drop tolerance + in the ILU() factorization. + + + +
+
@@ -122,7 +884,7 @@ In this case, the routine can be used to build multi-level preconditioners too. HREF="node21.html">
+ HREF="node18.html">
@@ -132,11 +894,11 @@ In this case, the routine can be used to build multi-level preconditioners too.
Next: Subroutine hierarchy_build + HREF="node21.html">Subroutine build Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine set + HREF="node19.html">Subroutine init Contents diff --git a/docs/html/node21.html b/docs/html/node21.html index f1e5c65a..b998987f 100644 --- a/docs/html/node21.html +++ b/docs/html/node21.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine hierarchy_build - +Subroutine build + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node22.html">
+ HREF="node18.html">
@@ -40,30 +40,30 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine build + HREF="node20.html">Subroutine set Contents
-+
-Subroutine hierarchy_build +Subroutine build-
call p%hierarchy_build(a,desc_a,info)+call p%build(a,desc_a,info)-This routine builds the hierarchy of matrices and restriction/prolongation -operators for the multi-level preconditioner
p, according to the requirements -made by the user through the routinesinitandset. +This routine builds the one-level preconditionerpaccording to the requirements +made by the user through the routinesinitandset+(see Sections 6.4 and 6.5 for multi-level preconditioners).Arguments @@ -72,17 +72,16 @@ made by the user through the routines
initandset.- +aatype(psb_xspmat_type), intent(in).The sparse matrix structure containing the local part of the matrix to be preconditioned. Note that x must be chosen according - to the real/complex, -single/double precision version of MLD2P4 under use. + to the real/complex, single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [13]. + HREF="node30.html#PSBLASGUIDE">13].desc_aThe communication descriptor of + HREF="node30.html#PSBLASGUIDE">13].a. See the PSBLAS User's Guide for details [13].infointeger, intent(out).Error code. If no error, 0 is returned. See Section 8 for details. +Error code. If no error, 0 is returned. See Section 8 for details. +For compatibility with the previous versions of MLD2P4, this routine can be also invoked +as follows: + +
+ +
+call mld_precbld(p,what,val,info)++ +In this case, the routine can be used to build multi-level preconditioners too. + +
@@ -111,7 +122,7 @@ single/double precision version of MLD2P4 under use. HREF="node22.html">
+ HREF="node18.html">
@@ -121,11 +132,11 @@ single/double precision version of MLD2P4 under use.
Next: Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine build + HREF="node20.html">Subroutine set Contents diff --git a/docs/html/node22.html b/docs/html/node22.html index 4647922a..344ae846 100644 --- a/docs/html/node22.html +++ b/docs/html/node22.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine smoothers_build - +Subroutine hierarchy_build + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node23.html">
+ HREF="node18.html">
@@ -40,32 +40,30 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Subroutine apply + HREF="node23.html">Subroutine smoothers_build Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine hierarchy_build + HREF="node21.html">Subroutine build Contents
-+
-Subroutine smoothers_build +Subroutine hierarchy_build-
call p%smoothers_build(a,desc_a,p,info)+call p%hierarchy_build(a,desc_a,info)-This routine builds the smoothers and the coarsest-level solvers for the -multi-level preconditioner
p, according to the requirements made by -the user through the routinesinitandset, and based on the aggregation -hierarchy produced by a previous call tohierarchy_build-(see Section 6.4). +This routine builds the hierarchy of matrices and restriction/prolongation +operators for the multi-level preconditionerp, according to the requirements +made by the user through the routinesinitandset.Arguments @@ -81,9 +79,10 @@ hierarchy produced by a previous call to
hierarchy_buildThe sparse matrix structure containing the local part of the matrix to be preconditioned. Note that x must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use. + to the real/complex, +single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [13]. + HREF="node30.html#PSBLASGUIDE">13].desc_ahierarchy_buildThe communication descriptor of + HREF="node30.html#PSBLASGUIDE">13].a. See the PSBLAS User's Guide for details [13].infointeger, intent(out).Error code. If no error, 0 is returned. See Section 8 for details. +Error code. If no error, 0 is returned. See Section 8 for details. + +
+ HREF="node18.html">
@@ -120,11 +121,11 @@ hierarchy produced by a previous call to hierarchy_build
Next: Subroutine apply + HREF="node23.html">Subroutine smoothers_build Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine hierarchy_build + HREF="node21.html">Subroutine build Contents diff --git a/docs/html/node23.html b/docs/html/node23.html index fb1abe9a..44b9e1d5 100644 --- a/docs/html/node23.html +++ b/docs/html/node23.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine apply - +Subroutine smoothers_build + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node24.html">
+ HREF="node18.html">
@@ -40,46 +40,32 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Subroutine free + HREF="node24.html">Subroutine apply Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build Contents
-+
-Subroutine apply +Subroutine smoothers_build-
call p%apply(x,y,desc_a,info [,trans,work])+call p%smoothers_build(a,desc_a,p,info)-This routine computes -
, where is a previously built
-preconditioner, stored into p, and
-denotes the preconditioner itself or its transpose, according to
-the value of trans. -Note that, when MLD2P4 is used with a Krylov solver from PSBLAS, -p%applyis called within the PSBLAS routinepsb_krylov-and hence it is completely transparent to the user. +This routine builds the smoothers and the coarsest-level solvers for the +multi-level preconditionerp, according to the requirements made by +the user through the routinesinitandset, and based on the aggregation +hierarchy produced by a previous call tohierarchy_build+(see Section 6.4).Arguments @@ -87,30 +73,17 @@ and hence it is completely transparent to the user.
-
xtype +(kind_parameter), dimension(:), intent(in).+ +atype(psb_xspmat_type), intent(in).The local part of the vector -
. Note that type and
- kind_parameter must be chosen according
- to the real/complex, single/double precision version of MLD2P4 under use.- -ytype -(kind_parameter), dimension(:), intent(out).- -The local part of the vector +
. Note that type and
- kind_parameter must be chosen according
- to the real/complex, single/double precision version of MLD2P4 under use.The sparse matrix structure containing the local part of the + matrix to be preconditioned. Note that x must be chosen according + to the real/complex, single/double precision version of MLD2P4 under use. + See the PSBLAS User's Guide for details [13]. desc_aThe communication descriptor associated to the matrix to be - preconditioned. +The communication descriptor of a. See the PSBLAS User's Guide for + details [13].- +infoinfointeger, intent(out).Error code. If no error, 0 is returned. See Section 8 for details. -- -transcharacter(len=1), optional, intent(in).- -If -trans='N','n'then -
;
- if trans='T','t'then -
- (transpose of 
; if trans='C','c'then -
- (conjugate transpose of .- -worktype -(kind_parameter), dimension(:), optional, target.- -Workspace. Its size should be at - least +4 * psb_cd_get_local_cols(desc_a)(see the PSBLAS User's Guide). - Note that type and kind_parameter must be chosen according - to the real/complex, single/double precision version of MLD2P4 under use.Error code. If no error, 0 is returned. See Section 8 for details. -For compatibility with the previous versions of MLD2P4, this routine can be also invoked -as follows: - -
- -
-call mld_precaply(p,what,val,info)-- -
+ HREF="node18.html">
@@ -203,11 +120,11 @@ as follows:
Next: Subroutine free + HREF="node24.html">Subroutine apply Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build Contents diff --git a/docs/html/node24.html b/docs/html/node24.html index 53a2959e..8364aec6 100644 --- a/docs/html/node24.html +++ b/docs/html/node24.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine free - +Subroutine apply + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node25.html">
+ HREF="node18.html">
@@ -40,28 +40,46 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Subroutine descr + HREF="node25.html">Subroutine free Up: User Interface + HREF="node18.html">User Interface Previous: Subroutine apply + HREF="node23.html">Subroutine smoothers_build Contents
-+
-Subroutine free +Subroutine apply-
call p%free(p,info)+call p%apply(x,y,desc_a,info [,trans,work])-This routine deallocates the preconditioner data structure
p. +This routine computes +, where is a previously built
+preconditioner, stored into p, and
+denotes the preconditioner itself or its transpose, according to
+the value of trans. +Note that, when MLD2P4 is used with a Krylov solver from PSBLAS, +p%applyis called within the PSBLAS routinepsb_krylov+and hence it is completely transparent to the user.Arguments @@ -69,12 +87,93 @@ This routine deallocates the preconditioner data structure
p.-
infointeger, intent(out).xtype (kind_parameter), dimension(:), intent(in).Error code. If no error, 0 is returned. See Section 8 for details. +The local part of the vector +. Note that type and
+ kind_parameter must be chosen according
+ to the real/complex, single/double precision version of MLD2P4 under use.+ +ytype +(kind_parameter), dimension(:), intent(out).+ +The local part of the vector +. Note that type and
+ kind_parameter must be chosen according
+ to the real/complex, single/double precision version of MLD2P4 under use.+ +desc_atype(psb_desc_type), intent(in).+ +The communication descriptor associated to the matrix to be + preconditioned. ++ +infointeger, intent(out).+ +Error code. If no error, 0 is returned. See Section 8 for details. ++ +transcharacter(len=1), optional, intent(in).+ +If +trans='N','n'then +;
+ if trans='T','t'then +
+ (transpose of ; if trans='C','c'then +
+ (conjugate transpose of .+ +worktype +(kind_parameter), dimension(:), optional, target.+ +Workspace. Its size should be at + least 4 * psb_cd_get_local_cols(desc_a)(see the PSBLAS User's Guide). + Note that type and kind_parameter must be chosen according + to the real/complex, single/double precision version of MLD2P4 under use.@@ -83,13 +182,35 @@ as follows:
-
call mld_precfree(p,info)+call mld_precaply(p,what,val,info)- -
+
+ + +
+
+
+
+
+
+
+
+ Next: Subroutine free + Up: User Interface + Previous: Subroutine smoothers_build + Contents + diff --git a/docs/html/node25.html b/docs/html/node25.html index e2357c3c..7dc4fa28 100644 --- a/docs/html/node25.html +++ b/docs/html/node25.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Subroutine descr - +Subroutine free + @@ -18,51 +18,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds + - + -
-
+
-
- Next: Adding new smoother and - Up: User Interface + Next: Subroutine descr + Up: User Interface Previous: Subroutine free - Subroutine apply + Contents
-+
-Subroutine descr +Subroutine free-
call p%descr(info, [iout])+call p%free(p,info)-This routine prints a description of the preconditioner
pto the standard output or -to a file. It must be called afterhierachy_buildandsmoothers_build, -orbuild, have been called. +This routine deallocates the preconditioner data structurep.Arguments @@ -71,20 +70,11 @@ or
build, have been called.infointeger, intent(out).- -Error code. If no error, 0 is returned. See Section 8 for details. -- -ioutinteger, intent(in), optional.integer, intent(out).The id of the file where the preconditioner description - will be printed; the default is the standard output. +Error code. If no error, 0 is returned. See Section 8 for details. @@ -93,37 +83,13 @@ as follows:
-
call mld_precdescr(p,info [,iout])+call mld_precfree(p,info)- -
-
- - -
-
-
-
-
-
-
-
- Next: Adding new smoother and - Up: User Interface - Previous: Subroutine free - Contents - + +
diff --git a/docs/html/node26.html b/docs/html/node26.html index c0265b99..9bbcd020 100644 --- a/docs/html/node26.html +++ b/docs/html/node26.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Adding new smoother and solver objects to MLD2P4 - +Subroutine descr + @@ -18,9 +18,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + @@ -30,9 +29,9 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node27.html">
+ HREF="node18.html">
-
Next: Error Handling + HREF="node27.html">Adding new smoother and Up: userhtml - Previous: Subroutine descr + HREF="node18.html">User Interface + Previous: Subroutine free Contents
-+
+Subroutine descr +
-Adding new smoother and solver objects to MLD2P4 --Developers can add completely new smoother and/or solver classes -derived from the base objects in the library (see Remark 2 in Section 6.2), -without recompiling the library itself. - + +
call p%descr(info, [iout])+-To do so, it is necessary first to select the base type to be extended. -In our experience, it is quite likely that the new application needs -only the definition of a ``solver'' object, which is almost -always acting only on the local part of the distributed matrix. -The parallel actions required to connect the various solver objects -are most often already provided by the block-Jacobi or the additive -Schwarz smoothers. To define a new solver, the developer will then -have to define its components and methods, perhaps taking one of the -predefined solvers as a starting point, if possible. + +This routine prints a description of the preconditioner
pto the standard output or +to a file. It must be called afterhierachy_buildandsmoothers_build, +orbuild, have been called.-Once the new smoother/solver class has been developed, to use it in -the context of the multilevel preconditioners it is necessary to: - -
-
-
- declare in the application program a variable of the new type; - -
- pass that variable as the argument to the
setroutine as in the -following: ---call p%set(smoother,info [,ilev,ilmax,pos])-call p%set(solver,info [,ilev,ilmax,pos])- -
- - link the code implementing the various methods into the application executable. - -
-It is possible to define new values for the keyword
WHATin the -setroutine; if the library code does not recognize a keyword, -it passes it down the composition hierarchy (levels containing -smoothers containing in turn solvers), so that it can be eventually caught by -the new solver. - --An example is provided in the source code distribution under the -folder
+tests/newslv. In this example we are implementing a new -incomplete factorization variant (which is simply the ILU(0) -factorization under a new name). Because of the specifics of this case, it is -possible to reuse the basic structure of the ILU solver, with its -L/D/U components and the methods needed to apply the solver; only a -few methods, such as the description and most importantly the build, -need to be ovverridden (rewritten). - +Arguments +-The interfaces for the calls shown above are defined using - -
+-
+- -smootherclass(mld_x_base_smoother_type)+
-The other arguments are defined in the way described in -Sec. 6.2. As an example, in theinfointeger, intent(out).+ The user-defined new smoother to be employed in the - preconditioner. +Error code. If no error, 0 is returned. See Section 8 for details. - -solverclass(mld_x_base_solver_type)+ +ioutinteger, intent(in), optional.+ The user-defined new solver to be employed in the - preconditioner. - +The id of the file where the preconditioner description + will be printed; the default is the standard output. tests/newslv-code we define a new object of typemld_d_tlu_solver_type, and -we pass it as follows: -- ! sparse matrix and preconditioner - type(psb_dspmat_type) :: a - type(mld_dprec_type) :: prec - type(mld_d_tlu_solver_type) :: tlusv - -...... - ! - ! prepare the preconditioner: an ML with defaults, but with TLU solver at - ! intermediate levels. All other parameters are at default values. - ! - call prec%init('ML', info) - call prec%hierarchy_build(a,desc_a,info) - nlv = prec%get_nlevs() - call prec%set(tlusv, info,ilev=1,ilmax=max(1,nlv-1)) - call prec%smoothers_build(a,desc_a,info) --
++For compatibility with the previous versions of MLD2P4, this routine can be also invoked +as follows: + +
+ +
+call mld_precdescr(p,info [,iout])++ +
+ +
+
+ HREF="node18.html">
-
Next: Error Handling + HREF="node27.html">Adding new smoother and Up: userhtml - Previous: Subroutine descr + HREF="node18.html">User Interface + Previous: Subroutine free Contents diff --git a/docs/html/node27.html b/docs/html/node27.html index 1be7a8f9..9c785a13 100644 --- a/docs/html/node27.html +++ b/docs/html/node27.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Error Handling - +Adding new smoother and solver objects to MLD2P4 + @@ -19,7 +19,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -40,39 +40,146 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: License + HREF="node28.html">Error Handling Up: userhtml Previous: Adding new smoother and + HREF="node26.html">Subroutine descr Contents
-+
-Error Handling +Adding new smoother and solver objects to MLD2P4-The error handling in MLD2P4 is based on the PSBLAS (version 2) error -handling. Error conditions are signaled via an integer argument -
info; whenever an error condition is detected, an error trace -stack is built by the library up to the top-level, user-callable -routine. This routine will then decide, according to the user -preferences, whether the error should be handled by terminating the -program or by returning the error condition to the user code, which -will then take action, and whether -an error message should be printed. These options may be set by using -the PSBLAS error handling routines; for further details see the PSBLAS -User's Guide [13]. +Developers can add completely new smoother and/or solver classes +derived from the base objects in the library (see Remark 2 in Section 6.2), +without recompiling the library itself.- +To do so, it is necessary first to select the base type to be extended. +In our experience, it is quite likely that the new application needs +only the definition of a ``solver'' object, which is almost +always acting only on the local part of the distributed matrix. +The parallel actions required to connect the various solver objects +are most often already provided by the block-Jacobi or the additive +Schwarz smoothers. To define a new solver, the developer will then +have to define its components and methods, perhaps taking one of the +predefined solvers as a starting point, if possible. -
++Once the new smoother/solver class has been developed, to use it in +the context of the multilevel preconditioners it is necessary to: + +
-
+
- declare in the application program a variable of the new type; + +
- pass that variable as the argument to the
setroutine as in the +following: +++call p%set(smoother,info [,ilev,ilmax,pos])+call p%set(solver,info [,ilev,ilmax,pos])+ +
+ - link the code implementing the various methods into the application executable. + +
+It is possible to define new values for the keyword
WHATin the +setroutine; if the library code does not recognize a keyword, +it passes it down the composition hierarchy (levels containing +smoothers containing in turn solvers), so that it can be eventually caught by +the new solver. + ++An example is provided in the source code distribution under the +folder
tests/newslv. In this example we are implementing a new +incomplete factorization variant (which is simply the ILU(0) +factorization under a new name). Because of the specifics of this case, it is +possible to reuse the basic structure of the ILU solver, with its +L/D/U components and the methods needed to apply the solver; only a +few methods, such as the description and most importantly the build, +need to be ovverridden (rewritten). + ++The interfaces for the calls shown above are defined using + +
+The other arguments are defined in the way described in +Sec. 6.2. As an example, in the+
+ +smootherclass(mld_x_base_smoother_type)+ +The user-defined new smoother to be employed in the + preconditioner. ++ +solverclass(mld_x_base_solver_type)+ +The user-defined new solver to be employed in the + preconditioner. + +tests/newslv+code we define a new object of typemld_d_tlu_solver_type, and +we pass it as follows: ++ ! sparse matrix and preconditioner + type(psb_dspmat_type) :: a + type(mld_dprec_type) :: prec + type(mld_d_tlu_solver_type) :: tlusv + +...... + ! + ! prepare the preconditioner: an ML with defaults, but with TLU solver at + ! intermediate levels. All other parameters are at default values. + ! + call prec%init('ML', info) + call prec%hierarchy_build(a,desc_a,info) + nlv = prec%get_nlevs() + call prec%set(tlusv, info,ilev=1,ilmax=max(1,nlv-1)) + call prec%smoothers_build(a,desc_a,info) ++
+ + +
+
+
+
+
+
+
+
+ Next: Error Handling + Up: userhtml + Previous: Subroutine descr + Contents + diff --git a/docs/html/node28.html b/docs/html/node28.html index 58e4f069..b7194cdb 100644 --- a/docs/html/node28.html +++ b/docs/html/node28.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -License - +Error Handling + @@ -40,63 +40,37 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Bibliography + HREF="node29.html">License Up: userhtml Previous: Error Handling + HREF="node27.html">Adding new smoother and Contents
-+
-License +Error Handling-The MLD2P4 is freely distributable under the following copyright -terms:
- - - MLD2P4 version 2.1 - MultiLevel Domain Decomposition Parallel Preconditioners Package - based on PSBLAS (Parallel Sparse BLAS version 3.4) - - (C) Copyright 2008, 2010, 2012, 2017 - - Salvatore Filippone Cranfield University, Cranfield, UK - Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT - Alfredo Buttari CNRS-IRIT, Toulouse, FR - Pasqua D'Ambra IAC-CNR, Naples, IT - Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT - - 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. -
- +The error handling in MLD2P4 is based on the PSBLAS (version 2) error +handling. Error conditions are signaled via an integer argument +info; whenever an error condition is detected, an error trace +stack is built by the library up to the top-level, user-callable +routine. This routine will then decide, according to the user +preferences, whether the error should be handled by terminating the +program or by returning the error condition to the user code, which +will then take action, and whether +an error message should be printed. These options may be set by using +the PSBLAS error handling routines; for further details see the PSBLAS +User's Guide [13]. + ++
diff --git a/docs/html/node29.html b/docs/html/node29.html index ab187489..0af35633 100644 --- a/docs/html/node29.html +++ b/docs/html/node29.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Bibliography - +License + @@ -40,140 +40,63 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: About this document ... + HREF="node30.html">Bibliography Up: userhtml Previous: License + HREF="node28.html">Error Handling Contents -
+
+
- --Bibliography -
-
+
+
+License +-
- 1 -
- -P. R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker, -Improving multifrontal methods by means of block low-rank representations, -SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474. -See also http://mumps.enseeiht.fr.
- 2 -
- -M. Brezina, P. Vanek, -A Black-Box Iterative Solver Based on a Two-Level Schwarz Method, -Computing, 63, 1999, 233-263.
- 3 -
- -W. L. Briggs, V. E. Henson, S. F. McCormick, -A Multigrid Tutorial, Second Edition, -SIAM, 2000.
- 4 -
- -A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, -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, Lecture Notes in Computer Science, -Springer, 2005, 593-602.
- 5 -
- -A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, -2LEV-D2P4: a package of high-performance preconditioners -for scientific and engineering applications, -Applicable Algebra in Engineering, Communications and Computing, -18 (3) 2007, 223-239.
- 6 -
- -X. C. Cai, M. Sarkis, -A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems, -SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.
- 7 -
- -P. D'Ambra, S. Filippone, D. di Serafino, -On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners, -Applied Numerical Mathematics, Elsevier Science, -57 (11-12), 2007, 1181-1196.
- 8 -
- -P. D'Ambra, D. di Serafino, S. Filippone, -MLD2P4: a Package of Parallel Multilevel -Algebraic Domain Decomposition Preconditioners -in Fortran 95, ACM Trans. Math. Softw., 37(3), 2010, art. 30.
- 9 -
- -T. A. Davis, -Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal -Method with a Column Pre-ordering Strategy, -ACM Transactions on Mathematical Software, 30, 2004, 196-199. -(See also http://www.cise.ufl.edu/ davis/)
- 10 -
- -J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu, -A supernodal approach to sparse partial pivoting, -SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.
- 11 -
- -J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling, -A set of Level 3 Basic Linear Algebra Subprograms, -ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.
- 12 -
- -J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson, -An extended set of FORTRAN Basic Linear Algebra Subprograms, -ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.
- 13 -
- -S. Filippone, A. Buttari, -PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library, 2012, -available from http://www.ce.uniroma2.it/psblas/.
- 14 -
- -S. Filippone, A. Buttari, -Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003. -ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.
- 15 -
- -S. Filippone, M. Colajanni, -PSBLAS: A Library for Parallel Linear Algebra -Computation on Sparse Matrices, -ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.
- 16 -
- -W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir, -MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions, -MIT Press, 1998.
- 17 -
- -C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh, -Basic Linear Algebra Subprograms for FORTRAN usage, -ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.
- 18 -
- -X. S. Li, J. W. Demmel, -SuperLU_DIST: A Scalable Distributed-memory -Sparse Direct Solver for Unsymmetric Linear Systems, -ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.
- 19 -
- -Y. Notay, P. S. Vassilevski, -Recursive Krylov-based multigrid cycles, -Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487.
- 20 -
- -Y. Saad, -Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.
- 21 -
- -B. Smith, P. Bjorstad, W. Gropp, -Domain Decomposition: Parallel Multilevel Methods for Elliptic -Partial Differential Equations, -Cambridge University Press, 1996.
- 22 -
- -M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra, -MPI: The Complete Reference. Volume 1 - The MPI Core, second edition, -MIT Press, 1998.
- 23 -
- -K. Stüben, -An Introduction to Algebraic Multigrid, -in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, -Academic Press, 2001.
- 24 -
- -R. S. Tuminaro, C. Tong, -Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.
- 25 -
-
-P. Vanek, J. Mandel, M. Brezina,
-Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems,
-Computing, 56 (3) 1996, 179-196.
+The MLD2P4 is freely distributable under the following copyright
+terms:
+ + + MLD2P4 version 2.1 + MultiLevel Domain Decomposition Parallel Preconditioners Package + based on PSBLAS (Parallel Sparse BLAS version 3.4) + + (C) Copyright 2008, 2010, 2012, 2017 + + Salvatore Filippone Cranfield University, Cranfield, UK + Pasqua D'Ambra IAC-CNR, Naples, IT + Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT + + 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. +
--
-
+
diff --git a/docs/html/node3.html b/docs/html/node3.html index 07fff983..b8c2c7a7 100644 --- a/docs/html/node3.html +++ b/docs/html/node3.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds -
-
-
-
- Next: Next: Code Distribution - Up: Up: userhtml - Previous: Previous: Contents - Contents
@@ -59,9 +59,9 @@ General Overview The MULTI-LEVEL DOMAIN DECOMPOSITION PARALLEL PRECONDITIONERS PACKAGE BASED ON PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain Decomposition preconditioners (see, e.g., [3,23,21]), + HREF="node30.html#Briggs2000">3,23,21]), to be used in the iterative solution of linear systems,
@@ -95,8 +95,8 @@ multi-level cycles and smoothers widely used in multigrid methods. The multi-level preconditioners implemented in MLD2P4 are obtained by combining AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and K-cycles [3,19] are available, which allow to define + HREF="node30.html#Briggs2000">3,19] are available, which allow to define almost all the preconditioners in the package, including the multi-level hybrid Schwarz ones; a specific cycle is implemented to obtain multi-level additive Schwarz preconditioners. The Jacobi, hybridforward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods @@ -104,8 +104,8 @@ are available as smoothers. An algebraic approach is used to generate a hierarch coarse-level matrices and operators, without explicitly using any information on the geometry of the original problem, e.g., the discretization of a PDE. To this end, the smoothed aggregation technique [2,25] + HREF="node30.html#BREZINA_VANEK">2,25] is applied. Either exact or approximate solvers can be used on the coarsest-level system. Specifically, different sparse LU factorizations from external packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel, @@ -126,8 +126,8 @@ interface. MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) computational framework [15,14]. PSBLAS provides basic linear algebra + HREF="node30.html#psblas_00">15,14]. PSBLAS provides basic linear algebra operators and data management facilities for distributed sparse matrices, as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners. The choice of PSBLAS has been mainly motivated by the need of having @@ -150,55 +150,55 @@ few black-box routines at the upper layer allow all users to easily build and apply any preconditioner available in MLD2P4; facilities are also available allowing expert users to extend the set of smoothers and solvers for building new versions of the preconditioners (see -Section 7). +Section 7).We note that the user interface of MLD2P4 2.1 has been extended with respect to the previous versions in order to separate the construction of the multi-level hierarchy from the construction of the smoothers and solvers, and to allow for more flexibility at each level. The software architecture described in [8] has significantly + HREF="node30.html#MLD2P4_TOMS">8] has significantly evolved too, in order to fully exploit the Fortran 2003 features implemented in PSBLAS 3. However, compatibility with previous versions has been preserved.
This guide is organized as follows. General information on the distribution of the source code is reported in Section 2, while details on the configuration -and installation of the package are given in Section 3. A short description -of the preconditioners implemented in MLD2P4 is provided in Section 4, +and installation of the package are given in Section 3. A short description +of the preconditioners implemented in MLD2P4 is provided in Section 4, to help the users in choosing among them. The basics for building and applying the preconditioners with the Krylov solvers implemented in PSBLAS are reported -in Section 5, where the Fortran codes of a few sample programs +in Section 5, where the Fortran codes of a few sample programs are also shown. A reference guide for the user interface routines is provided -in Section 6. Information on the extension of the package -through the addition of new smoothers and solvers is reported in Section 7. +in Section 6. Information on the extension of the package +through the addition of new smoothers and solvers is reported in Section 7. The error handling mechanism used by the package -is briefly described in Section 8. The copyright terms concerning the -distribution and modification of MLD2P4 are reported in Appendix A. +is briefly described in Section 8. The copyright terms concerning the +distribution and modification of MLD2P4 are reported in Appendix A.
-
-
-
-
- Next: Next: Code Distribution - Up: Up: userhtml - Previous: Previous: Contents - Contents diff --git a/docs/html/node30.html b/docs/html/node30.html index f348d8d3..f599d623 100644 --- a/docs/html/node30.html +++ b/docs/html/node30.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -About this document ... - +Bibliography + @@ -18,53 +18,162 @@ original version by: Nikos Drakos, CBLU, University of Leeds + + -
-
+
+
-
- Up: Next: About this document ... + Up: userhtml Previous: Bibliography - License + Contents -
-
+
+ ++Bibliography +
-
-
-About this document ... -
--This document was generated using the -LaTeX2HTML translator Version 2012 (1.2)
-Copyright © 1993, 1994, 1995, 1996, -Nikos Drakos, -Computer Based Learning Unit, University of Leeds. -
-Copyright © 1997, 1998, 1999, -Ross Moore, -Mathematics Department, Macquarie University, Sydney. + - 1 +
- +P. R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker, +Improving multifrontal methods by means of block low-rank representations, +SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474. +See also http://mumps.enseeiht.fr.
- 2 +
- +M. Brezina, P. Vanek, +A Black-Box Iterative Solver Based on a Two-Level Schwarz Method, +Computing, 63, 1999, 233-263.
- 3 +
- +W. L. Briggs, V. E. Henson, S. F. McCormick, +A Multigrid Tutorial, Second Edition, +SIAM, 2000.
- 4 +
- +A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, +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, Lecture Notes in Computer Science, +Springer, 2005, 593-602.
- 5 +
- +A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, +2LEV-D2P4: a package of high-performance preconditioners +for scientific and engineering applications, +Applicable Algebra in Engineering, Communications and Computing, +18 (3) 2007, 223-239.
- 6 +
- +X. C. Cai, M. Sarkis, +A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems, +SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.
- 7 +
- +P. D'Ambra, S. Filippone, D. di Serafino, +On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners, +Applied Numerical Mathematics, Elsevier Science, +57 (11-12), 2007, 1181-1196.
- 8 +
- +P. D'Ambra, D. di Serafino, S. Filippone, +MLD2P4: a Package of Parallel Multilevel +Algebraic Domain Decomposition Preconditioners +in Fortran 95, ACM Trans. Math. Softw., 37(3), 2010, art. 30.
- 9 +
- +T. A. Davis, +Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal +Method with a Column Pre-ordering Strategy, +ACM Transactions on Mathematical Software, 30, 2004, 196-199. +(See also http://www.cise.ufl.edu/ davis/)
- 10 +
- +J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu, +A supernodal approach to sparse partial pivoting, +SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.
- 11 +
- +J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling, +A set of Level 3 Basic Linear Algebra Subprograms, +ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.
- 12 +
- +J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson, +An extended set of FORTRAN Basic Linear Algebra Subprograms, +ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.
- 13 +
- +S. Filippone, A. Buttari, +PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library, 2012, +available from http://www.ce.uniroma2.it/psblas/.
- 14 +
- +S. Filippone, A. Buttari, +Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003. +ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.
- 15 +
- +S. Filippone, M. Colajanni, +PSBLAS: A Library for Parallel Linear Algebra +Computation on Sparse Matrices, +ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.
- 16 +
- +W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir, +MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions, +MIT Press, 1998.
- 17 +
- +C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh, +Basic Linear Algebra Subprograms for FORTRAN usage, +ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.
- 18 +
- +X. S. Li, J. W. Demmel, +SuperLU_DIST: A Scalable Distributed-memory +Sparse Direct Solver for Unsymmetric Linear Systems, +ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.
- 19 +
- +Y. Notay, P. S. Vassilevski, +Recursive Krylov-based multigrid cycles, +Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487.
- 20 +
- +Y. Saad, +Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.
- 21 +
- +B. Smith, P. Bjorstad, W. Gropp, +Domain Decomposition: Parallel Multilevel Methods for Elliptic +Partial Differential Equations, +Cambridge University Press, 1996.
- 22 +
- +M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra, +MPI: The Complete Reference. Volume 1 - The MPI Core, second edition, +MIT Press, 1998.
- 23 +
- +K. Stüben, +An Introduction to Algebraic Multigrid, +in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, +Academic Press, 2001.
- 24 +
- +R. S. Tuminaro, C. Tong, +Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.
- 25 +
-
+P. Vanek, J. Mandel, M. Brezina,
+Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems,
+Computing, 56 (3) 1996, 179-196.
+
-The command line arguments were:
- latex2html -local_icons -noaddress -dir ../../html userhtml.tex +
-The translation was initiated by Salvatore Filippone on 2017-07-25 -
+
diff --git a/docs/html/node31.html b/docs/html/node31.html index 2f3088d1..13a02ef9 100644 --- a/docs/html/node31.html +++ b/docs/html/node31.html @@ -1,6 +1,6 @@ - + - -
-
+
+
-
-
+
-
-
+
-
+
- Up: Up: userhtml - Previous: Previous: Bibliography - Contents
-
-+
+About this document ... -
-+
This document was generated using the -LaTeX2HTML translator Version 2002-2-1 (1.71) +LaTeX2HTML translator Version 2012 (1.2)
Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, @@ -66,13 +61,10 @@ Copyright © 1997, 1998, 1999, Mathematics Department, Macquarie University, Sydney.
The command line arguments were:
- latex2html -html_version 4.0 -dir ../../html userhtml.tex + latex2html -local_icons -noaddress -dir ../../html userhtml.tex-The translation was initiated by Salvatore Filippone on 2008-07-23 +The translation was initiated by Salvatore Filippone on 2017-08-09
- -Salvatore Filippone -2008-07-23 - + diff --git a/docs/html/node4.html b/docs/html/node4.html index cb8f1623..c97dade1 100644 --- a/docs/html/node4.html +++ b/docs/html/node4.html @@ -18,7 +18,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds -
-
-
-
- Next: Configuring and Building MLD2P4 - Up: Next: Contributors + Up: userhtml - Previous: Previous: General Overview - Contents
@@ -65,7 +65,7 @@ where contact points for further information can be also found.The software is available under a modified BSD license, as specified -in Appendix A; please note that some of the optional +in Appendix A; please note that some of the optional third party libraries may be licensed under a different and more stringent license, most notably the GPL, and this should be taken into account when treating derived works. @@ -89,8 +89,19 @@ constant
-whose current value is2.1.0+whose current value is2.1.0. ++ +
+ +Subsections + +-
+
- Contributors +
diff --git a/docs/html/node5.html b/docs/html/node5.html index 7a4528fb..ffcb232a 100644 --- a/docs/html/node5.html +++ b/docs/html/node5.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Configuring and Building MLD2P4 - +Contributors + @@ -18,9 +18,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - + @@ -30,9 +29,9 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node6.html">
+ HREF="node4.html">
-
Next: Prerequisites + HREF="node6.html">Configuring and Building MLD2P4 Up: userhtml - Previous: Code Distribution + Previous: Code Distribution Contents @@ -51,80 +50,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
--
- -In order to build MLD2P4 it is necessary to set up a Makefile with appropriate -system-dependent variables; this is done by means of the
-Configuring and Building MLD2P4 -configure-script. The distribution also includes the autoconf and automake -sources employed to generate the script, but usually this is not needed -to build the software. ++Contributors +
+Contributors to version 2: --MLD2P4 is implemented almost entirely in Fortran 2003, with some -interfaces to external libraries in C; the Fortran compiler -must support the Fortran 2003 standard plus the extension
MOLD=-feature, which enhances the usability ofALLOCATE. -Many compilers do this; in particular, this is -supported by the GNU Fortran compiler, for which we -recommend to use at least version 4.8. -The software defines data types and interfaces for -real and complex data, in both single and double precision. +-
+
- Salvatore Filippone, Cranfield University, UK; + +
- Pasqua D'Ambra, IAC-CNR, Naples, IT; + +
- Daniela di Serafino, University of Campania ``L. Vanvitelli'', Caserta, IT; + +
- Ambra Abdullahi Hassan, University of Rome ``Tor Vergata'', IT. + +
-Building MLD2P4 requires some base libraries (see Section 3.1); -interfaces to optional third-party libraries, which extend the functionalities of MLD2P4 -(see Section 3.2), are also available. Many Linux distributions -(e.g., Ubuntu, Fedora, CentOS) provide precompiled packages for the prerequisite and -optional software. In many cases these packages are split between a runtime part and a -``developer'' part; in order to build MLD2P4 you need both. A description of the base and -optional software used by MLD2P4 is given in the next sections. +
-
+
- Salvatore Filippone; + +
- Pasqua D'Ambra; + +
- Daniela di Serafino; + +
- Alfredo Buttari, CNRS-IRIT, Toulouse, F. + +
-
- -Subsections - --
-
- Prerequisites -
- Optional third party libraries -
- Configuration options -
- Bug reporting -
- Example and test programs -
- - -
-
-
-
-
-
-
-
- Next: Prerequisites - Up: userhtml - Previous: Code Distribution - Contents - diff --git a/docs/html/node6.html b/docs/html/node6.html index a5398c73..5e8fed72 100644 --- a/docs/html/node6.html +++ b/docs/html/node6.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Prerequisites - +Configuring and Building MLD2P4 + @@ -18,116 +18,111 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - + + + -
-
+
-
-
- Next: Optional third party libraries - Up: Configuring and Building MLD2P4 - Previous: Configuring and Building MLD2P4 - Next: Prerequisites + Up: userhtml + Previous: Contributors + Contents
-+
+Configuring and Building MLD2P4 + + +In order to build MLD2P4 it is necessary to set up a Makefile with appropriate +system-dependent variables; this is done by means of the
-Prerequisites -configure+script. The distribution also includes the autoconf and automake +sources employed to generate the script, but usually this is not needed +to build the software. +-The following base libraries are needed: -
-
-
- BLAS -
- [11,12,17] Many vendors provide optimized versions - of BLAS; if no vendor version is - available for a given platform, the ATLAS software - (math-atlas.sourceforge.net) - may be employed. The reference BLAS from Netlib - (www.netlib.org/blas) are meant to define the standard - behaviour of the BLAS interface, so they are not optimized for any - particular plaftorm, and should only be used as a last - resort. Note that BLAS computations form a relatively small part of - the MLD2P4/PSBLAS computations; they are however critical when using - preconditioners based on MUMPS, UMFPACK or SuperLU third party - libraries. Note that UMFPACK requires a full LAPACK library; our -experience is that configuring ATLAS for building full LAPACK does not -work in the correct way. Our advice is first to download the LAPACK tarfile from -www.netlib.org/lapack and install it independently of ATLAS. In this case, -you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option -in the make.inc file of the LAPACK library. - -
- MPI -
- [16,22] A version of MPI is available on most - high-performance computing systems. - - -
- PSBLAS -
- [13,15] Parallel Sparse BLAS (PSBLAS) is - available from www.ce.uniroma2.it/psblas; version - 3.5.0 (or later) is required. Indeed, all the prerequisites - listed so far are also prerequisites of PSBLAS. - -
MOLD=+feature, which enhances the usability ofALLOCATE. +Many compilers do this; in particular, this is +supported by the GNU Fortran compiler, for which we +recommend to use at least version 4.8. +The software defines data types and interfaces for +real and complex data, in both single and double precision.-
+Building MLD2P4 requires some base libraries (see Section 3.1); +interfaces to optional third-party libraries, which extend the functionalities of MLD2P4 +(see Section 3.2), are also available. Many Linux distributions +(e.g., Ubuntu, Fedora, CentOS) provide precompiled packages for the prerequisite and +optional software. In many cases these packages are split between a runtime part and a +``developer'' part; in order to build MLD2P4 you need both. A description of the base and +optional software used by MLD2P4 is given in the next sections. + ++ +
+ +Subsections + +-
+
- Prerequisites +
- Optional third party libraries +
- Configuration options +
- Bug reporting +
- Example and test programs +
-
-
+
-
-
- Next: Optional third party libraries - Up: Configuring and Building MLD2P4 - Previous: Configuring and Building MLD2P4 - Next: Prerequisites + Up: userhtml + Previous: Contributors + Contents diff --git a/docs/html/node7.html b/docs/html/node7.html index f87e992d..cd53709f 100644 --- a/docs/html/node7.html +++ b/docs/html/node7.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Optional third party libraries - +Prerequisites + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node8.html">
+ HREF="node6.html">
@@ -40,74 +40,71 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Configuration options + HREF="node8.html">Optional third party libraries Up: Configuring and Building MLD2P4 + HREF="node6.html">Configuring and Building MLD2P4 Previous: Prerequisites + HREF="node6.html">Configuring and Building MLD2P4 Contents
-+
-Optional third party libraries +Prerequisites-We provide interfaces to the following third-party software libraries; -note that these are optional, but if you enable them some defaults -for multi-level preconditioners may change to reflect their presence. - -
-
-
-
- UMFPACK +The following base libraries are needed: +
- BLAS
- [9]
- A sparse LU factorization package included in the SuiteSparse library, available from
- faculty.cse.tamu.edu/davis/suitesparse.html;
- it provides sequential factorization and triangular system solution for double
- precision real and complex data. We tested version 4.5.4 of SuiteSparse.
- Note that for configuring SuiteSparse you should provide the right path to the BLAS
- and LAPACK libraries in the
SuiteSparse_config/SuiteSparse_config.mkfile. + HREF="node30.html#blas3">11,12,17] Many vendors provide optimized versions + of BLAS; if no vendor version is + available for a given platform, the ATLAS software + (math-atlas.sourceforge.net) + may be employed. The reference BLAS from Netlib + (www.netlib.org/blas) are meant to define the standard + behaviour of the BLAS interface, so they are not optimized for any + particular plaftorm, and should only be used as a last + resort. Note that BLAS computations form a relatively small part of + the MLD2P4/PSBLAS computations; they are however critical when using + preconditioners based on MUMPS, UMFPACK or SuperLU third party + libraries. Note that UMFPACK requires a full LAPACK library; our +experience is that configuring ATLAS for building full LAPACK does not +work in the correct way. Our advice is first to download the LAPACK tarfile from +www.netlib.org/lapack and install it independently of ATLAS. In this case, +you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option +in the make.inc file of the LAPACK library.
- - MUMPS +
- MPI
- [1] - A sparse LU factorization package available from mumps.enseeiht.fr; - it provides sequential and parallel factorizations and triangular system solution - for single and double precision, real and complex data. - We tested versions 4.10.0 and 5.0.1. - -
- SuperLU -
- [10] - A sparse LU factorization package available from - crd.lbl.gov/~xiaoye/SuperLU/; it provides sequential - factorization and triangular system solution for single and double precision, - real and complex data. We tested versions 4.3 and 5.0. If you installed BLAS from - ATLAS, remember to define the BLASLIB variable in the make.inc file. + HREF="node30.html#MPI2">16,22] A version of MPI is available on most + high-performance computing systems. -
- SuperLU_Dist +
- PSBLAS
- [18]
- A sparse LU factorization package available
- from the same site as SuperLU; it provides parallel factorization and
- triangular system solution for double precision real and complex data.
- We tested versions 3.3 and 4.2. If you installed BLAS from
- ATLAS, remember to define the BLASLIB variable in the make.inc file and
- to add the
-std=c99option to the C compiler options. - Note that this library requires the ParMETIS - library for parallel graph partitioning and fill-reducing matrix ordering, available from - glaros.dtc.umn.edu/gkhome/metis/parmetis/overview. + HREF="node30.html#PSBLASGUIDE">13,15] Parallel Sparse BLAS (PSBLAS) is + available from www.ce.uniroma2.it/psblas; version + 3.5.0 (or later) is required. Indeed, all the prerequisites + listed so far are also prerequisites of PSBLAS.
-
-
+
@@ -115,7 +112,7 @@ for multi-level preconditioners may change to reflect their presence. HREF="node8.html">
+ HREF="node6.html">
@@ -125,11 +122,11 @@ for multi-level preconditioners may change to reflect their presence.
Next: Configuration options + HREF="node8.html">Optional third party libraries Up: Configuring and Building MLD2P4 + HREF="node6.html">Configuring and Building MLD2P4 Previous: Prerequisites + HREF="node6.html">Configuring and Building MLD2P4 Contents diff --git a/docs/html/node8.html b/docs/html/node8.html index db0eaab4..fc7bb0fd 100644 --- a/docs/html/node8.html +++ b/docs/html/node8.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Configuration options - +Optional third party libraries + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node9.html">
+ HREF="node6.html">
@@ -40,231 +40,82 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Bug reporting + HREF="node9.html">Configuration options Up: Configuring and Building MLD2P4 + HREF="node6.html">Configuring and Building MLD2P4 Previous: Optional third party libraries + HREF="node7.html">Prerequisites Contents
--Configuration options +
+
+Optional third party libraries-In order to build MLD2P4, the first step is to use the
configurescript -in the main directory to generate the necessary makefile. +We provide interfaces to the following third-party software libraries; +note that these are optional, but if you enable them some defaults +for multi-level preconditioners may change to reflect their presence.-As a minimal example consider the following: -
-./configure --with-psblas=PSB-INSTALL-DIR -
-which assumes that the various MPI compilers and support libraries are -available in the standard directories on the system, and specifies -only the PSBLAS install directory (note that the latter directory must -be specified with an absolute path). -The full set of options may be looked at by issuing the command -./configure --help, which produces: --`configure' configures MLD2P4 2.1 to adapt to many kinds of systems. - -Usage: ./configure [OPTION]... [VAR=VALUE]... - -To assign environment variables (e.g., CC, CFLAGS...), specify them as -VAR=VALUE. See below for descriptions of some of the useful variables. - -Defaults for the options are specified in brackets. - -Configuration: - -h, --help display this help and exit - --help=short display options specific to this package - --help=recursive display the short help of all the included packages - -V, --version display version information and exit - -q, --quiet, --silent do not print `checking...' messages - --cache-file=FILE cache test results in FILE [disabled] - -C, --config-cache alias for `--cache-file=config.cache' - -n, --no-create do not create output files - --srcdir=DIR find the sources in DIR [configure dir or `..'] - -Installation directories: - --prefix=PREFIX install architecture-independent files in PREFIX - [/usr/local] - --exec-prefix=EPREFIX install architecture-dependent files in EPREFIX - [PREFIX] - -By default, `make install' will install all the files in -`/usr/local/bin', `/usr/local/lib' etc. You can specify -an installation prefix other than `/usr/local' using `--prefix', -for instance `--prefix=$HOME'. - -For better control, use the options below. - -Fine tuning of the installation directories: - --bindir=DIR user executables [EPREFIX/bin] - --sbindir=DIR system admin executables [EPREFIX/sbin] - --libexecdir=DIR program executables [EPREFIX/libexec] - --sysconfdir=DIR read-only single-machine data [PREFIX/etc] - --sharedstatedir=DIR modifiable architecture-independent data [PREFIX/com] - --localstatedir=DIR modifiable single-machine data [PREFIX/var] - --libdir=DIR object code libraries [EPREFIX/lib] - --includedir=DIR C header files [PREFIX/include] - --oldincludedir=DIR C header files for non-gcc [/usr/include] - --datarootdir=DIR read-only arch.-independent data root [PREFIX/share] - --datadir=DIR read-only architecture-independent data [DATAROOTDIR] - --infodir=DIR info documentation [DATAROOTDIR/info] - --localedir=DIR locale-dependent data [DATAROOTDIR/locale] - --mandir=DIR man documentation [DATAROOTDIR/man] - --docdir=DIR documentation root [DATAROOTDIR/doc/mld2p4] - --htmldir=DIR html documentation [DOCDIR] - --dvidir=DIR dvi documentation [DOCDIR] - --pdfdir=DIR pdf documentation [DOCDIR] - --psdir=DIR ps documentation [DOCDIR] - -Program names: - --program-prefix=PREFIX prepend PREFIX to installed program names - --program-suffix=SUFFIX append SUFFIX to installed program names - --program-transform-name=PROGRAM run sed PROGRAM on installed program names - -Optional Features: - --disable-option-checking ignore unrecognized --enable/--with options - --disable-FEATURE do not include FEATURE (same as --enable-FEATURE=no) - --enable-FEATURE[=ARG] include FEATURE [ARG=yes] - --disable-dependency-tracking speeds up one-time build - --enable-dependency-tracking do not reject slow dependency extractors - --enable-serial Specify whether to enable a fake mpi library to run - in serial mode. - --enable-long-integers Specify usage of 64 bits integers. - -Optional Packages: - --with-PACKAGE[=ARG] use PACKAGE [ARG=yes] - --without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no) - --with-psblas=DIR The install directory for PSBLAS, for example, - --with-psblas=/opt/packages/psblas-3.5 - --with-psblas-incdir=DIR - Specify the directory for PSBLAS includes. - --with-psblas-libdir=DIR - Specify the directory for PSBLAS library. - --with-ccopt additional CCOPT flags to be added: will prepend - to CCOPT - --with-fcopt additional FCOPT flags to be added: will prepend - to FCOPT - --with-libs List additional link flags here. For example, - --with-libs=-lspecial_system_lib or - --with-libs=-L/path/to/libs - --with-clibs additional CLIBS flags to be added: will prepend - to CLIBS - --with-flibs additional FLIBS flags to be added: will prepend - to FLIBS - --with-library-path additional LIBRARYPATH flags to be added: will - prepend to LIBRARYPATH - --with-include-path additional INCLUDEPATH flags to be added: will - prepend to INCLUDEPATH - --with-module-path additional MODULE_PATH flags to be added: will - prepend to MODULE_PATH - --with-extra-libs List additional link flags here. For example, - --with-extra-libs=-lspecial_system_lib or - --with-extra-libs=-L/path/to/libs - --with-blas=<lib> use BLAS library <lib> - --with-blasdir=<dir> search for BLAS library in <dir> - --with-lapack=<lib> use LAPACK library <lib> - --with-mumps=LIBNAME Specify the libname for MUMPS. Default: autodetect - with minimum "-lmumps_common -lpord" - --with-mumpsdir=DIR Specify the directory for MUMPS library and - includes. Note: you will need to add auxiliary - libraries with --extra-libs; this depends on how - MUMPS was configured and installed, at a minimum you - will need SCALAPACK and BLAS - --with-mumpsincdir=DIR Specify the directory for MUMPS includes. - --with-mumpslibdir=DIR Specify the directory for MUMPS library. - --with-umfpack=LIBNAME Specify the library name for UMFPACK and its support - libraries. Default: "-lumfpack -lamd" - --with-umfpackdir=DIR Specify the directory for UMFPACK library and - includes. - --with-umfpackincdir=DIR - Specify the directory for UMFPACK includes. - --with-umfpacklibdir=DIR - Specify the directory for UMFPACK library. - --with-superlu=LIBNAME Specify the library name for SUPERLU library. - Default: "-lsuperlu" - --with-superludir=DIR Specify the directory for SUPERLU library and - includes. - --with-superluincdir=DIR - Specify the directory for SUPERLU includes. - --with-superlulibdir=DIR - Specify the directory for SUPERLU library. - --with-superludist=LIBNAME - Specify the libname for SUPERLUDIST library. - Requires you also specify SuperLU. Default: - "-lsuperlu_dist" - --with-superludistdir=DIR - Specify the directory for SUPERLUDIST library and - includes. - --with-superludistincdir=DIR - Specify the directory for SUPERLUDIST includes. - --with-superludistlibdir=DIR - Specify the directory for SUPERLUDIST library. - -Some influential environment variables: - FC Fortran compiler command - FCFLAGS Fortran compiler flags - LDFLAGS linker flags, e.g. -L<lib dir> if you have libraries in a - nonstandard directory <lib dir> - LIBS libraries to pass to the linker, e.g. -l<library> - CC C compiler command - CFLAGS C compiler flags - CPPFLAGS C/C++/Objective C preprocessor flags, e.g. -I<include dir> if - you have headers in a nonstandard directory <include dir> - MPICC MPI C compiler command - MPIFC MPI Fortran compiler command - CPP C preprocessor - -Use these variables to override the choices made by `configure' or to help -it to find libraries and programs with nonstandard names/locations. - -Report bugs to <bugreport@mld2p4.it>. -
--For instance, if a user has built and installed PSBLAS 3.5 under the -
/optdirectory and is -using the SuiteSparse package (which includes UMFPACK), then MLD2P4 -might be configured with: -- ./configure --with-psblas=/opt/psblas-3.5/ \ - --with-umfpackincdir=/usr/include/suitesparse/ -
-Once the configure script has completed execution, it will have -generated the fileMake.incwhich will then be used by all -Makefiles in the directory tree; this file will be copied in the -install directory under the nameMake.inc.MLD2P4. - --To use the MUMPS solver package, -the user has to add the appropriate options to the configure script; -by default we are looking for the libraries -
-ldmumps -lsmumps-lzmumps -lcmumps -mumps_common -lpord. -MUMPS often uses additional packages such as ScaLAPACK, ParMETIS, -SCOTCH, as well as enabling OpenMP; in such cases it is necessary to -add linker options with the--with-extra-libsconfigure option. - +-
+
- UMFPACK +
- [9]
+ A sparse LU factorization package included in the SuiteSparse library, available from
+ faculty.cse.tamu.edu/davis/suitesparse.html;
+ it provides sequential factorization and triangular system solution for double
+ precision real and complex data. We tested version 4.5.4 of SuiteSparse.
+ Note that for configuring SuiteSparse you should provide the right path to the BLAS
+ and LAPACK libraries in the
SuiteSparse_config/SuiteSparse_config.mkfile. +
+ - MUMPS +
- [1] + A sparse LU factorization package available from mumps.enseeiht.fr; + it provides sequential and parallel factorizations and triangular system solution + for single and double precision, real and complex data. + We tested versions 4.10.0 and 5.0.1. + +
- SuperLU +
- [10] + A sparse LU factorization package available from + crd.lbl.gov/~xiaoye/SuperLU/; it provides sequential + factorization and triangular system solution for single and double precision, + real and complex data. We tested versions 4.3 and 5.0. If you installed BLAS from + ATLAS, remember to define the BLASLIB variable in the make.inc file. + + +
- SuperLU_Dist +
- [18]
+ A sparse LU factorization package available
+ from the same site as SuperLU; it provides parallel factorization and
+ triangular system solution for double precision real and complex data.
+ We tested versions 3.3 and 4.2. If you installed BLAS from
+ ATLAS, remember to define the BLASLIB variable in the make.inc file and
+ to add the
-std=c99option to the C compiler options. + Note that this library requires the ParMETIS + library for parallel graph partitioning and fill-reducing matrix ordering, available from + glaros.dtc.umn.edu/gkhome/metis/parmetis/overview. +
+
-To build the library the user will now enter -
-make -
-followed (optionally) by --make install -
+
+ HREF="node6.html">
@@ -274,11 +125,11 @@ make install
Next: Bug reporting + HREF="node9.html">Configuration options Up: Configuring and Building MLD2P4 + HREF="node6.html">Configuring and Building MLD2P4 Previous: Optional third party libraries + HREF="node7.html">Prerequisites Contents diff --git a/docs/html/node9.html b/docs/html/node9.html index 6a26d105..0bec52db 100644 --- a/docs/html/node9.html +++ b/docs/html/node9.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Bug reporting - +Configuration options + @@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - + @@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds HREF="node10.html">
+ HREF="node6.html">
@@ -40,29 +40,248 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Next: Example and test programs + HREF="node10.html">Bug reporting Up: Configuring and Building MLD2P4 + HREF="node6.html">Configuring and Building MLD2P4 Previous: Configuration options + HREF="node8.html">Optional third party libraries Contents
--Bug reporting -
-If you find any bugs in our codes, please send an email to
pasqua.dambra@cnr.it -
daniela.diserafino@unicampania.it -
salvatore.filippone@cranfield.ac.uk
-You should be aware that the amount of information needed to reproduce a problem -in a parallel program may vary quite a lot. ++Configuration options +
++In order to build MLD2P4, the first step is to use the
configurescript +in the main directory to generate the necessary makefile. + ++As a minimal example consider the following: +
+./configure --with-psblas=PSB-INSTALL-DIR +
+which assumes that the various MPI compilers and support libraries are +available in the standard directories on the system, and specifies +only the PSBLAS install directory (note that the latter directory must +be specified with an absolute path). +The full set of options may be looked at by issuing the command +./configure --help, which produces: ++`configure' configures MLD2P4 2.1 to adapt to many kinds of systems. + +Usage: ./configure [OPTION]... [VAR=VALUE]... + +To assign environment variables (e.g., CC, CFLAGS...), specify them as +VAR=VALUE. See below for descriptions of some of the useful variables. + +Defaults for the options are specified in brackets. + +Configuration: + -h, --help display this help and exit + --help=short display options specific to this package + --help=recursive display the short help of all the included packages + -V, --version display version information and exit + -q, --quiet, --silent do not print `checking...' messages + --cache-file=FILE cache test results in FILE [disabled] + -C, --config-cache alias for `--cache-file=config.cache' + -n, --no-create do not create output files + --srcdir=DIR find the sources in DIR [configure dir or `..'] + +Installation directories: + --prefix=PREFIX install architecture-independent files in PREFIX + [/usr/local] + --exec-prefix=EPREFIX install architecture-dependent files in EPREFIX + [PREFIX] + +By default, `make install' will install all the files in +`/usr/local/bin', `/usr/local/lib' etc. You can specify +an installation prefix other than `/usr/local' using `--prefix', +for instance `--prefix=$HOME'. + +For better control, use the options below. + +Fine tuning of the installation directories: + --bindir=DIR user executables [EPREFIX/bin] + --sbindir=DIR system admin executables [EPREFIX/sbin] + --libexecdir=DIR program executables [EPREFIX/libexec] + --sysconfdir=DIR read-only single-machine data [PREFIX/etc] + --sharedstatedir=DIR modifiable architecture-independent data [PREFIX/com] + --localstatedir=DIR modifiable single-machine data [PREFIX/var] + --libdir=DIR object code libraries [EPREFIX/lib] + --includedir=DIR C header files [PREFIX/include] + --oldincludedir=DIR C header files for non-gcc [/usr/include] + --datarootdir=DIR read-only arch.-independent data root [PREFIX/share] + --datadir=DIR read-only architecture-independent data [DATAROOTDIR] + --infodir=DIR info documentation [DATAROOTDIR/info] + --localedir=DIR locale-dependent data [DATAROOTDIR/locale] + --mandir=DIR man documentation [DATAROOTDIR/man] + --docdir=DIR documentation root [DATAROOTDIR/doc/mld2p4] + --htmldir=DIR html documentation [DOCDIR] + --dvidir=DIR dvi documentation [DOCDIR] + --pdfdir=DIR pdf documentation [DOCDIR] + --psdir=DIR ps documentation [DOCDIR] + +Program names: + --program-prefix=PREFIX prepend PREFIX to installed program names + --program-suffix=SUFFIX append SUFFIX to installed program names + --program-transform-name=PROGRAM run sed PROGRAM on installed program names + +Optional Features: + --disable-option-checking ignore unrecognized --enable/--with options + --disable-FEATURE do not include FEATURE (same as --enable-FEATURE=no) + --enable-FEATURE[=ARG] include FEATURE [ARG=yes] + --disable-dependency-tracking speeds up one-time build + --enable-dependency-tracking do not reject slow dependency extractors + --enable-serial Specify whether to enable a fake mpi library to run + in serial mode. + --enable-long-integers Specify usage of 64 bits integers. + +Optional Packages: + --with-PACKAGE[=ARG] use PACKAGE [ARG=yes] + --without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no) + --with-psblas=DIR The install directory for PSBLAS, for example, + --with-psblas=/opt/packages/psblas-3.5 + --with-psblas-incdir=DIR + Specify the directory for PSBLAS includes. + --with-psblas-libdir=DIR + Specify the directory for PSBLAS library. + --with-ccopt additional CCOPT flags to be added: will prepend + to CCOPT + --with-fcopt additional FCOPT flags to be added: will prepend + to FCOPT + --with-libs List additional link flags here. For example, + --with-libs=-lspecial_system_lib or + --with-libs=-L/path/to/libs + --with-clibs additional CLIBS flags to be added: will prepend + to CLIBS + --with-flibs additional FLIBS flags to be added: will prepend + to FLIBS + --with-library-path additional LIBRARYPATH flags to be added: will + prepend to LIBRARYPATH + --with-include-path additional INCLUDEPATH flags to be added: will + prepend to INCLUDEPATH + --with-module-path additional MODULE_PATH flags to be added: will + prepend to MODULE_PATH + --with-extra-libs List additional link flags here. For example, + --with-extra-libs=-lspecial_system_lib or + --with-extra-libs=-L/path/to/libs + --with-blas=<lib> use BLAS library <lib> + --with-blasdir=<dir> search for BLAS library in <dir> + --with-lapack=<lib> use LAPACK library <lib> + --with-mumps=LIBNAME Specify the libname for MUMPS. Default: autodetect + with minimum "-lmumps_common -lpord" + --with-mumpsdir=DIR Specify the directory for MUMPS library and + includes. Note: you will need to add auxiliary + libraries with --extra-libs; this depends on how + MUMPS was configured and installed, at a minimum you + will need SCALAPACK and BLAS + --with-mumpsincdir=DIR Specify the directory for MUMPS includes. + --with-mumpslibdir=DIR Specify the directory for MUMPS library. + --with-umfpack=LIBNAME Specify the library name for UMFPACK and its support + libraries. Default: "-lumfpack -lamd" + --with-umfpackdir=DIR Specify the directory for UMFPACK library and + includes. + --with-umfpackincdir=DIR + Specify the directory for UMFPACK includes. + --with-umfpacklibdir=DIR + Specify the directory for UMFPACK library. + --with-superlu=LIBNAME Specify the library name for SUPERLU library. + Default: "-lsuperlu" + --with-superludir=DIR Specify the directory for SUPERLU library and + includes. + --with-superluincdir=DIR + Specify the directory for SUPERLU includes. + --with-superlulibdir=DIR + Specify the directory for SUPERLU library. + --with-superludist=LIBNAME + Specify the libname for SUPERLUDIST library. + Requires you also specify SuperLU. Default: + "-lsuperlu_dist" + --with-superludistdir=DIR + Specify the directory for SUPERLUDIST library and + includes. + --with-superludistincdir=DIR + Specify the directory for SUPERLUDIST includes. + --with-superludistlibdir=DIR + Specify the directory for SUPERLUDIST library. + +Some influential environment variables: + FC Fortran compiler command + FCFLAGS Fortran compiler flags + LDFLAGS linker flags, e.g. -L<lib dir> if you have libraries in a + nonstandard directory <lib dir> + LIBS libraries to pass to the linker, e.g. -l<library> + CC C compiler command + CFLAGS C compiler flags + CPPFLAGS C/C++/Objective C preprocessor flags, e.g. -I<include dir> if + you have headers in a nonstandard directory <include dir> + MPICC MPI C compiler command + MPIFC MPI Fortran compiler command + CPP C preprocessor + +Use these variables to override the choices made by `configure' or to help +it to find libraries and programs with nonstandard names/locations. + +Report bugs to <bugreport@mld2p4.it>. +
++For instance, if a user has built and installed PSBLAS 3.5 under the +
/optdirectory and is +using the SuiteSparse package (which includes UMFPACK), then MLD2P4 +might be configured with: ++ ./configure --with-psblas=/opt/psblas-3.5/ \ + --with-umfpackincdir=/usr/include/suitesparse/ +
+Once the configure script has completed execution, it will have +generated the fileMake.incwhich will then be used by all +Makefiles in the directory tree; this file will be copied in the +install directory under the nameMake.inc.MLD2P4. + ++To use the MUMPS solver package, +the user has to add the appropriate options to the configure script; +by default we are looking for the libraries +
-ldmumps -lsmumps-lzmumps -lcmumps -mumps_common -lpord. +MUMPS often uses additional packages such as ScaLAPACK, ParMETIS, +SCOTCH, as well as enabling OpenMP; in such cases it is necessary to +add linker options with the--with-extra-libsconfigure option.- -
+To build the library the user will now enter ++make +
+followed (optionally) by ++make install +
+ + +
+
+
+
+
+
+
+
+ Next: Bug reporting + Up: Configuring and Building MLD2P4 + Previous: Optional third party libraries + Contents + diff --git a/docs/html/userhtml.html b/docs/html/userhtml.html index 82c86e2e..dfbb0955 100644 --- a/docs/html/userhtml.html +++ b/docs/html/userhtml.html @@ -98,70 +98,75 @@ July 31, 2017 HREF="node3.html">General Overview- Code Distribution -
- Configuring and Building MLD2P4
-
+
- Contributors +
- Prerequisites + HREF="node6.html">Configuring and Building MLD2P4 +
- Optional third party libraries + HREF="node7.html">Prerequisites
- Configuration options + HREF="node8.html">Optional third party libraries
- Bug reporting + HREF="node9.html">Configuration options
- Example and test programs + HREF="node10.html">Bug reporting +
- Example and test programs
-- Multigrid Background -
- AMG preconditioners
+ HREF="node12.html">Multigrid Background
+
- Smoothed Aggregation + HREF="node13.html">AMG preconditioners
- Smoothers and coarsest-level solvers + HREF="node14.html">Smoothed Aggregation +
- Smoothers and coarsest-level solvers
- - Getting Started
-
- Examples
+ HREF="node16.html">Getting Started
+
- - User Interface
-
- Subroutine init
+ HREF="node18.html">User Interface
+
- Subroutine set + HREF="node19.html">Subroutine init
- Subroutine build + HREF="node20.html">Subroutine set
- Subroutine hierarchy_build + HREF="node21.html">Subroutine build
- Subroutine smoothers_build + HREF="node22.html">Subroutine hierarchy_build
- Subroutine apply + HREF="node23.html">Subroutine smoothers_build
- Subroutine free + HREF="node24.html">Subroutine apply
- Subroutine descr + HREF="node25.html">Subroutine free +
- Subroutine descr
- - Adding new smoother and solver objects to MLD2P4
- Error Handling + HREF="node27.html">Adding new smoother and solver objects to MLD2P4
- License + HREF="node28.html">Error Handling
- Bibliography + HREF="node29.html">License
- About this document ... + HREF="node30.html">Bibliography +
- About this document ...
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+6386,7 @@ ET endstream endobj -591 0 obj +592 0 obj << /Length 7114 >> @@ -6436,187 +6464,33 @@ ET endstream endobj -610 0 obj +537 0 obj << -/Length1 2121 -/Length2 17358 -/Length3 0 -/Length 19479 +/Type /ObjStm +/N 100 +/First 893 +/Length 16048 >> stream -%!PS-AdobeFont-1.0: CMBX10 003.002 -%%Title: CMBX10 -%Version: 003.002 -%%CreationDate: Mon Jul 13 16:17:00 2009 -%%Creator: David M. Jones -%Copyright: Copyright (c) 1997, 2009 American Mathematical Society -%Copyright: (݈*zjA3?U*}b.y=-^ Kc5fJŞGQ"zZ -6>ur+lW>ntq/˖. hں̲ G C -@?TW\3'1xZNK;Ɨn!A\ 2r/L۪*Q$s2ӊ;hJy}E-M=>A9J s3BK=h-[c<ݟ H^t|ݺf?Dѯ2V{D# '26:G GRğ\*A_SFE8JټTص y_] [wC|$+[۱`q{/(WBzgVɞ - H{ϗ GFϔ$rţ Tx-h)'RTTFѫ*H_5$ ]5e腈 6= Fʄ3 l.gZxH1 BػsM- W߀ ;Y׀m %ȅc]{)D`?GbYV(9^{zۆ -쯿CW!۴ hX0>bN벗ГԹ#('ڹݟH !]2Dx ڱf/Xbgy/xA6\ c4ޯ, 4+E %@˅~G<|tJ3QftI W#DXL͠ M m!0QѬN r= ( H5_t~C%_܍D=,N:}Ew #!ѧ ,x@@ֿ-IG;
i95%L_@=ՃLjc7.dtX8w In憧bHؒtޛ;19 x攓=& }s[]lĹmpOvMh|M0 hX&M{ŖS bj'%VV`M[wo^@j/xڞ3 61X?F2z! 0PlmSC'wNB& <|l*IszLp~@@%Zg! < {MLs5'.? ?*P`،zh!W~7;ЍNkbP~VUԳ.H foU ~y ߆w#$ 9 pVjYxɵw=^_TV>GoOI-oǁNlkwY_ Wc$ 2?'f#" ucn똛 0Bh:sP' * tӑɼ .t\Kl}e-[ v /͍ ]UƊϽɝfB֝e U*Q1SG3.kHDTfx-L [W͉uB Ԥ/mzo`r%DQ[' 9J}/m 3QJE|LLA sdGҷy9-ۑ b w'͏ilu~m .U"~N-m\ݩuUL+^ |5.2|"].Mcs%'5͓Ucu 2" - ӗp^Ub{VQ]rV [ =g9=_!kr" _u&hꢵNZ˗cX]ϟnYiVT -_ xX1X8{ S=@U ɂ'qPqU]7a[xnPז LˇKdP؍ -vR| WDbi磣"ytR@q(ʚjbT߭5VL-GpF>zr:KhMA| |!w>D>6R -aW] -Д\Wn e5'}2%)Q,4? v,қ;AeE: ybI/B{>1r1c& kQ5^$'XgZ o%#s;ƍ$=AL-GuJ`| (iOi ncnGȞApC@k<In9V6cн5tBfD&GWPE&<<74P([pyR /-n _tGg}(z) #d7:W! a%;LA7 #Lc#]L0 \վcAɈQ$3cc2gɃ+? ^Sۦ ^(}5PLo^a -!, KUQTc'g\ -U?dcvZBWb: --٫DFOń)u4!\7H/lZ'51kb/2G 4 UBl} eDM!# dnզpt] vV%4Mtj6#]W$ 3]u/+([U 9GmAv2!}SO] 3]e 3`̛ jײǻ풕KnN Xʔq { -MonDˎq<#)ꋜU#{(ûDcM$h++.MwM=uHb9yQ\;Qi Js6Bw+/75;_M@ -OAR b37Pr ^RQ.vU! DhAxecy6O7:LLuqBn!xh1ow4S^"R t/e+!2`5}#NM2*.#!ѺQ 굝e*.&>B =8P. a`/آ@D]
DS0zM4)JZcb}`kxSu 浺c$Vk|f+Sۋ\4SY)L<`⻢? \xu* -OofJ% [7̕[qio+GvQdþ հ窲AT@LWC>^M90s ;Y%d҃r<[4s$̂o] B 38< yi_=@X Y]CiRcj> 3x<~b&y%hĥp?=Nz9V j UկBNEF4/#7{@? phV2Śs*z v|Wq/({ Apːz۔LEXJg @D1cL:*|o&M snr6NŁ0T8a}__EW+tWpgi b3%A9)Ỷt8}+>nqõE;(gjܺ~\#<&%-p&"ֿwD xhO ΨV= tu gjU} ?ѩ`< ݈*zjA3?U*}b.y=-^ Kc5fJŞGQ"zZ +6>ur+lW>ntq/˖. hں̲ G C +@?TW\3'1xZNK;Ɨn!A\ 2r/L۪*Q$s2ӊ;hJy}E-M=>A9J s3BK=h-[c<ݟ H^t|ݺf?Dѯ2V{D# '26:G GRğ\*A_SFE8JټTص y_] [wC|$+[۱`q{/(WBzgVɞ + H{ϗ GFϔ$rţ Tx-h)'RTTFѫ*H_5$ ]5e腈 6= Fʄ3 l.gZxH1 BػsM- W߀ ;Y׀m %ȅc]{)D`?GbYV(9^{zۆ +쯿CW!۴ hX0>bN벗ГԹ#('ڹݟH !]2Dx ڱf/Xbgy/xA6\ c4ޯ, 4+E %@˅~G<|tJ3QftI W#DXL͠ M m!0QѬN r= ( H5_t~C%_܍D=,N:}Ew #!ѧ ,x@@ֿ-IG;
i95%L_@=ՃLjc7.dtX8w In憧bHؒtޛ;19 x攓=& }s[]lĹmpOvMh|M0 hX&M{ŖS bj'%VV`M[wo^@j/xڞ3 61X?F2z! 0PlmSC'wNB& <|l*IszLp~@@%Zg! < {MLs5'.? ?*P`،zh!W~7;ЍNkbP~VUԳ.H foU ~y ߆w#$ 9 pVjYxɵw=^_TV>GoOI-oǁNlkwY_ Wc$ 2?'f#" ucn똛 0Bh:sP' * tӑɼ .t\Kl}e-[ v /͍ ]UƊϽɝfB֝e U*Q1SG3.kHDTfx-L [W͉uB Ԥ/mzo`r%DQ[' 9J}/m 3QJE|LLA sdGҷy9-ۑ b w'͏ilu~m .U"~N-m\ݩuUL+^ |5.2|"].Mcs%'5͓Ucu 2" + ӗp^Ub{VQ]rV [ =g9=_!kr" _u&hꢵNZ˗cX]ϟnYiVT +_ xX1X8{ S=@U ɂ'qPqU]7a[xnPז LˇKdP؍ +vR| WDbi磣"ytR@q(ʚjbT߭5VL-GpF>zr:KhMA| |!w>D>6R +aW] +Д\Wn e5'}2%)Q,4? v,қ;AeE: ybI/B{>1r1c& kQ5^$'XgZ o%#s;ƍ$=AL-GuJ`| (iOi ncnGȞApC@k<In9V6cн5tBfD&GWPE&<<74P([pyR /-n _tGg}(z) #d7:W! a%;LA7 #Lc#]L0 \վcAɈQ$3cc2gɃ+? ^Sۦ ^(}5PLo^a +!, KUQTc'g\ +U?dcvZBWb: +-٫DFOń)u4!\7H/lZ'51kb/2G 4 UBl} eDM!# dnզpt] vV%4Mtj6#]W$ 3]u/+([U 9GmAv2!}SO] 3]e 3`̛ jײǻ풕KnN Xʔq { +MonDˎq<#)ꋜU#{(ûDcM$h++.MwM=uHb9yQ\;Qi Js6Bw+/75;_M@ +OAR b37Pr ^RQ.vU! DhAxecy6O7:LLuqBn!xh1ow4S^"R t/e+!2`5}#NM2*.#!ѺQ 굝e*.&>B =8P. a`/آ@D]
DS0zM4)JZcb}`kxSu 浺c$Vk|f+Sۋ\4SY)L<`⻢? \xu* +OofJ% [7̕[qio+GvQdþ հ窲AT@LWC>^M90s ;Y%d҃r<[4s$̂o] B 38< yi_=@X Y]CiRcj> 3x<~b&y%hĥp?=Nz9V j UկBNEF4/#7{@? phV2Śs*z v|Wq/({ Apːz۔LEXJg @D1cL:*|o&M snr6NŁ0T8a}__EW+tWpgi b3%A9)Ỷt8}+>nqõE;(gjܺ~\#<&%-p&"ֿwD xhO ΨV= tu gjU} ?ѩ`< - Subroutine init
+ HREF="node18.html">User Interface
+
- Examples
+ HREF="node16.html">Getting Started
+
- Subroutine init
+ HREF="node18.html">User Interface
+
- Examples
+ HREF="node16.html">Getting Started
+
![\framebox{
\begin{minipage}{.85\textwidth}
\begin{tabbing}
\quad \=\quad \=\quad...
-...mm]
\>endif \ [1mm]
\>return $u^k$ \ [1mm]
end
\end{tabbing}
\end{minipage}
}](img22.png)
+