PARDISO manual

PARDISO manual
Parallel Sparse Direct And
Multi-Recursive Iterative Linear Solvers
PARDISO
User Guide Version 5.0.0
(Updated February 07, 2014)
Olaf
Schenk
and
Klaus
G ä r t n e r
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
1
If you use the sparse direct solver in PARDISO version 5.0.0, please cite:
References
[1] A. Kuzmin, M. Luisier, and O. Schenk. Fast methods for computing selected elements of the Greens
function in massively parallel nanoelectronic device simulations. In F. Wolf, B. Mohr, and D. Mey,
editors, Euro-Par 2013 Parallel Processing, Vol. 8097, Lecture Notes in Computer Science, pp. 533–
544, Springer Berlin Heidelberg, 2013.
[2] O. Schenk and K. Gärtner. Solving unsymmetric sparse systems of linear equations with PARDISO.
Journal of Future Generation Computer Systems, 20(3):475–487, 2004.
Note: This document briefly describes the interface to the shared-memory and distributed-memory
multiprocessing parallel direct and multi-recursive iterative solvers in PARDISO 5.0.01 . A discussion
of the algorithms used in PARDISO and more information on the solver can be found at http://www.
pardiso-project.org
Revision: 1.321 2014-01-06
1 Please note that this version is a significant extension compared to Intel’s MKL version and that these improvements
and features are not available in Intel’s MKL 9.3 release or later releases.
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
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Contents
1 Introduction
1.1 Supported Matrix Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Structurally Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Nonsymmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Computing Selected Elements of the Inverse A−1 : PARDISO-INV . . . . . . . . . .
1.3 Computing the Schur-complement: PARDISO-SCHUR . . . . . . . . . . . . . . . . .
1.4 Parallel Distributed-Memory Factorization for Symmetric Indefinite Linear Systems
1.5 Multi-Recursive Iterative Sparse Solver for Symmetric Indefinite Linear Systems . .
1.6 Direct-Iterative Preconditioning for Nonsymmetric Linear Systems . . . . . . . . . .
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2 Specification
2.1 Arguments of PARDISOINIT . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Arguments of PARDISO . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 IPARM Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 DPARM Control Parameters . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary of arguments and parameters . . . . . . . . . . . . . . . . . .
2.6 Computing the Schur-complement: Example for nonsymmetric matrices
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3 Linking and Running the Software
3.1 LINUX on IA64 platform . . . . .
3.1.1 Linking the libraries . . . .
3.1.2 Setting environment . . . .
3.2 Mac OSX on 64-bit platforms . . .
3.2.1 Linking the libraries . . . .
3.2.2 Setting environment . . . .
3.3 Windows on IA64 platforms . . . .
3.3.1 Linking the libraries . . . .
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4 Using the MATLAB interface
4.1 An example with a real matrix . .
4.2 A brief note of caution . . . . . . .
4.3 An example with a real, symmetric
4.4 Complex matrices . . . . . . . . .
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5 Installing the MATLAB interface
33
6 Using the PARDISO Matrix Consistency Checker
33
7 Authors
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8 Acknowledgments
34
9 License
34
10 Further comments
36
A Examples for sparse symmetric linear systems
A.1 Example results for symmetric systems . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Example pardiso sym.f for symmetric linear systems (including selected inversion)
A.3 Example pardiso sym.c for symmetric linear systems (including selected inversion)
A.4 Example pardiso sym schur.cpp for Schur-complement computation for symmetric
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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linear
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36
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48
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
A.5 Example iterative solver pardiso iter sym.f for symmetric linear systems . . . . . . . . . .
3
53
B Examples for sparse symmetric linear systems on distributed-memory architectures 56
B.1 Example pardiso sym mpp.c for the solution of symmetric linear systems on distributed
memory architecutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
C Release notes
60
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
1
4
Introduction
The package PARDISO is a high-performance, robust, memory–efficient and easy to use software for
solving large sparse symmetric and nonsymmetric linear systems of equations on shared–memory and
distributed-memory architectures. The solver uses a combination of left- and right-looking Level-3 BLAS
supernode techniques [15]. In order to improve sequential and parallel sparse numerical factorization
performance, the algorithms are based on a Level-3 BLAS update, and pipelining parallelism is exploited
with a combination of left- and right-looking supernode techniques [9, 11, 12, 14]. The parallel pivoting
methods allow complete supernode pivoting in order to balance numerical stability and scalability during
the factorization process. For sufficiently large problem sizes, numerical experiments demonstrate that the
scalability of the parallel algorithm is nearly independent of the shared-memory and distributed-memory
multiprocessing architecture and a speedup of up to seven using eight processors has been observed. The
approach is based on OpenMP [4] directives and MPI parallelization, and has been successfully tested on
almost all shared-memory parallel systems.
PARDISO calculates the solution of a set of sparse linear equations with multiple right-hand sides,
AX = B,
using a parallel LU , LDLT or LLT factorization, where A and X, B are n by n and n by nrhs matrices
respectively. PARDISO supports, as illustrated in Figure 1, a wide range of sparse matrix types and
computes the solution of real or complex, symmetric, structurally symmetric or nonsymmetric, positive
definite, indefinite or Hermitian sparse linear systems of equations on shared-memory multiprocessing
architectures.
Figure 1: Sparse matrices that can be solved with PARDISO Version 5.0.0
PARDISO 5.0.0 has the unique feature among all solvers that it can compute the exact bit-identical
solution on multicores and cluster of multicores (see release notes).
1.1
Supported Matrix Types
PARDISO performs the following analysis steps depending on the structure of the input matrix A.
1.1.1
Symmetric Matrices
The solver first computes a symmetric fill-in reducing permutation P based on either the minimum degree
algorithm [6] or the nested dissection algorithm from the METIS package [2], followed by the parallel
left-right looking numerical Cholesky factorization [15] P AP T = LLT or P AP T = LDLT for symmetric,
indefinite matrices. The solver uses diagonal pivoting or 1 × 1 and 2 × 2 Bunch-Kaufman pivoting for
symmetric indefinite matrices and an approximation of X is found by forward and backward substitution
and iterative refinement.
The coefficient matrix is perturbed whenever numerically acceptable 1 × 1 and 2 × 2 pivots cannot
be found within a diagonal supernode block. One or two passes of iterative refinement may be required
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
5
to correct the effect of the perturbations. This restricting notion of pivoting with iterative refinement is
effective for highly indefinite symmetric systems. Furthermore the accuracy of this method is for a large
set of matrices from different applications areas as accurate as a direct factorization method that uses
complete sparse pivoting techniques [13]. Another possibility to improve the pivoting accuracy is to use
symmetric weighted matchings algorithms. These methods identify large entries in A that, if permuted
close to the diagonal, permit the factorization process to identify more acceptable pivots and proceed
with fewer pivot perturbations. The methods are based on maximum weighted matchings and improve
the quality of the factor in a complementary way to the alternative idea of using more complete pivoting
techniques.
The inertia is also computed for real symmetric indefinite matrices.
1.1.2
Structurally Symmetric Matrices
The solver first computes a symmetric fill-in reducing permutation P followed by the parallel numerical
factorization of P AP T = QLU T . The solver uses partial pivoting in the supernodes and an approximation
of X is found by forward and backward substitution and iterative refinement.
1.1.3
Nonsymmetric Matrices
The solver first computes a non-symmetric permutation PM P S and scaling matrices Dr and Dc with
the aim to place large entries on the diagonal, which enhances greatly the reliability of the numerical
factorization process [1]. In the next step the solver computes a fill-in reducing permutation P based
on the matrix PM P S A + (PM P S A)T followed by the parallel numerical factorization QLU R = A2 , A2 =
P PM P S Dr ADc P with supernode pivoting matrices Q and R and P = P (PM P S ) to keep sufficiently
large parts of the cycles of PM P S in one diagonal block. When the factorization algorithm reaches a
point where it cannot factorize the supernodes with this pivoting strategy, it uses a pivoting perturbation
strategy similar to [5]. The magnitude of the potential pivot is tested against a constant threshold of
α = · ||A2 ||∞ , where is the machine precision, and ||A2 ||∞ is the ∞-norm of the scaled and permuted
matrix A. Any tiny pivots lii encountered during elimination are set to the sign(lii ) · · ||A2 ||∞ — this
trades off some numerical stability for the ability to keep pivots from getting too small. Although many
failures could render the factorization well-defined but essentially useless, in practice it is observed that
the diagonal elements are rarely modified for a large class of matrices. The result of this pivoting approach
is that the factorization is, in general, not exact and iterative refinement may be needed.
1.2
Computing Selected Elements of the Inverse A−1 : PARDISO-INV
The parallel selected inversion method is an efficient way for computing e.g. the diagonal elements of the
inverse of a sparse matrix. For a symmetric matrix A, the selected inversion algorithm first constructs
the LDLT factorization of A, where L is a block lower diagonal matrix called the Cholesky factor, and
D is a block diagonal matrix. In the second step, the selected inversion algorithm only computes all
the elements A−1
ij such that Lij 6= 0 in exact arithmetic. The computational scaling of the selected
inversion algorithm is only proportional to the number of nonzero elements in the Cholesky factor L,
even though A−1 might be a full matrix. The selected inversion algorithm scales as O(N ) for quasi-1D
systems, O(N 1.5 ) for quasi-2D systems, O(N 2 ) for 3D systems. The selected inversion method achieves
significant improvement over the tradtional forward/backward method at all dimensions.
The parallel threaded implementation [3] of the selected inversion algorithm, called PARDISO-INV,
is available for sparse symmetric or unsymmetric matrices, and it is fully integrated into the PARDISO
package. In addition, it includes pivoting to maintain accuracy during both sparse inversion steps.
1.3
Computing the Schur-complement: PARDISO-SCHUR
The Schur-complement framework is a general way to solve linear systems. Much work has been done in
this area; see, for example [8] and the references therein. Let Ax = b be the system of interest. Suppose
A has the form
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
A=
A11
A21
A12
A22
6
.
A traditional approach to compute the exact Schur complement S = A22 − A21 A−1
11 A12 by using offthe-shelf linear solvers such as PARDISO is (i) to factor A11 and (ii) to solve for each right-hand side of
A12 (in fact for small blocks of right-hand sides). However, this approach has two important drawbacks:
(1) the sparsity of the right-hand sides A12 and A21 is not exploited, and (2) the calculations limit the
efficient use of multicore shared-memory environments because it is well known that the triangular solves
do not scale very well with the number of cores. In [7] we developed a fast and parallel reformulation of
the sparse linear algebra calculations that circumvents the issues in the Schur complement computations.
The idea is to leverage the good scalability of parallel factorizations as an alternative method of building
the Schur complement basecd on forward/backward substitutions. We employ an incomplete augmented
factorization technique that solves the sparse linear systems with multiple right-hand sides at once using
an incomplete sparse factorization of the matrix A.
The parallel threaded implementation of the Schur-complement algorithm, called PARDISO-SCHUR,
is available for sparse symmetric or unsymmetric matrices, and it is fully integrated into the PARDISO
package. In addition, it includes pivoting to maintain accuracy during the Schur complement computations.
1.4
Parallel Distributed-Memory Factorization for Symmetric Indefinite Linear Systems
A new and scalable MPI-based numerical factorization and parallel forward/backward substitution on
distributed-memory architectures for symmetric indefinite matrices has been implemented in PARDISO
5.0.0. It has the unique feature among all solvers that it can compute the exact bit-identical solution on
multicores and cluster of multicores (see release notes).
This solver is only available for real symmetric indefinite matrices.
1.5
Multi-Recursive Iterative Sparse Solver for Symmetric Indefinite Linear
Systems
The solver also includes a novel preconditioning solver [10]. Our preconditioning approach for symmetric
indefinite linear system is based on maximum weighted matchings and algebraic multilevel incomplete
LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more
complete pivoting techniques for the highly ill-conditioned symmetric indefinite matrices. In considering
how to solve the linear systems in a manner that exploits sparsity as well as symmetry, we combine
a diverse range of topics that includes preprocessing the matrix with symmetric weighted matchings,
solving the linear system with Krylov subspace methods, and accelerating the linear system solution with
multilevel preconditioners based upon incomplete factorizations.
1.6
Direct-Iterative Preconditioning for Nonsymmetric Linear Systems
The solver also allows a combination of direct and iterative methods [16] in order to accelerate the linear
solution process for transient simulation. Many applications of sparse solvers require solutions of systems
with gradually changing values of the nonzero coefficient matrix, but the same identical sparsity pattern.
In these applications, the analysis phase of the solvers has to be performed only once and the numerical
factorizations are the important time-consuming steps during the simulation. PARDISO uses a numerical
factorization A = LU for the first system and applies these exact factors L and U for the next steps in a
preconditioned Krylov-Subspace iteration. If the iteration does not converge, the solver will automatically
switch back to the numerical factorization. This method can be applied for nonsymmetric matrices in
PARDISO and the user can select the method using only one input parameter. For further details see
the parameter description (IPARM(4), IPARM(20)).
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
2
7
Specification
The library is available on 32-bit and 64-bit architectures. The library is based on Fortran and C source
code and the top level driver routines PARDISO and PARDISOINIT assume the following interface.
On 32-bit architectures:
SUBROUTINE PARDISOINIT(PT, MTYPE, SOLVER, IPARM, DPARM, ERROR)
INTEGER*4 PT(64), MYTPE, SOLVER, IPARM(64), ERROR
REAL*8
DPARM(64)
and
SUBROUTINE PARDISO(PT, MAXFCT, MNUM, MTYPE, PHASE, N, A, IA, JA,
1
PERM, NRHS, IPARM, MSGLVL, B, X, ERROR, DPARM)
INTEGER*4 PT(64)
INTEGER*4 MAXFCT, MNUM, MTYPE, PHASE, N, NRHS, ERROR
1
IA(*), JA(*), PERM(*), IPARM(*)
REAL*8
DPARM(64)
REAL*8
A(*), B(N,NRHS), X(N,NRHS)
On 64-bit architectures:
SUBROUTINE
INTEGER*8
INTEGER*4
REAL*8
PARDISOINIT(PT, MTYPE, SOLVER, IPARM, DPARM, ERROR)
PT(64)
PT(64), MYTPE, SOLVER, IPARM(64), ERROR
DPARM(64)
and
SUBROUTINE PARDISO(PT, MAXFCT, MNUM, MTYPE, PHASE, N, A, IA, JA,
1
PERM, NRHS, IPARM, MSGLVL, B, X, ERROR, DPARM)
INTEGER*8 PT(64)
INTEGER*4 MAXFCT, MNUM, MTYPE, PHASE, N, NRHS, ERROR
1
IA(*), JA(*), PERM(*), IPARM(*)
REAL*8
DPARM(64)
REAL*8
A(*), B(N,NRHS), X(N,NRHS)
The sparse direct solvers use 8-byte integer length internally that allows storage of factors with more
than 232 nonzeros elements. The following are examples for calling PARDISO from a Fortran program
c...Check license of the solver and initialize the solver
CALL PARDISOINIT(PT, MTYPE, SOLVER, IPARM, DPARM, ERROR)
c...Solve matrix sytem
CALL PARDISO(PT, MAXFCT, MNUM, MTYPE, PHASE, N, A, IA, JA,
1
PERM, NRHS, IPARM, MSGLVL, B, X, ERROR, DPARM)
for calling PARDISO from a C program
/* Check license of the solver and initialize the solver */
pardisoinit(pt, &mtype, &solver, iparm, dparm, &error);
/* Solve matrix sytem */
pardiso(pt, &maxfct, &mnum, &mtype, &phase, &n, a, ia, ja,
perm, &nrhs, iparm, &msglvl, b, x, &error, dparm)
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
K
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Nonsymmetric Matrix
IA(K)
JA(K)
A(K)
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5
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18
21
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3
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11.
-3.
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8
Symmetric Matrix
IA(k)
JA(K)
A(K)
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5
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10
12
15
17
18
19
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3
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5.
Figure 2: Illustration of the two input compressed sparse row formats of an 8 × 8 general sparse nonsymmetric linear system and the upper triangular part of a symmetric linear system for the direct solver
PARDISO. The algorithms in PARDISO require JA to be increasingly ordered per row and the presence
of the diagonal element per row for any symmetric or structurally symmetric matrix. The nonsymmetric
matrices need no diagonal elements in the PARDISO solver.
2.1
Arguments of PARDISOINIT
PARDISOINIT checks the current license in the file pardiso.lic and initializes the internal timer and the
address pointer PT. It sets the solver default values according to the matrix type.
PT(64) — INTEGER
Input/Output
On entry: This is the solver’s internal data address pointer. These addresses are passed to the
solver and all related internal memory management is organized through this pointer.
On exit: Internal address pointers.
Note: PT is a 32-bit or 64-bit integer array on 32 or 64 bit operating systems, respectively. It
has 64 entries.
Never change PT!
MTYPE — INTEGER
Input
On entry: This scalar value defines the matrix type. PARDISO supports the following matrices
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
TYPE
1
2
-2
3
4
-4
6
11
13
Matrix
real and
real and
real and
complex
complex
complex
complex
real and
complex
9
structurally symmetric
symmetric positive definite
symmetric indefinite
and structurally symmetric
and Hermitian positive definite
and Hermitian indefinite
and symmetric
nonsymmetric
and nonsymmetric
Note: The parameter influences the pivoting method.
SOLVER — INTEGER
On entry: This scalar value defines the solver method that the user would like to use.
SOLVER
0
1
Input
Method
sparse direct solver
multi-recursive iterative solver
IPARM (64) — INTEGER
Input/Output
On entry: IPARM is an integer array of size 64 that is used to pass various parameters to
PARDISO and to return some useful information after the execution of the solver. PARDISOINIT
fills IPARM(1), IPARM(2), and IPARM(4) through IPARM(64) with default values and uses them.
See section 2.3 for a detailed description.
Note: There is no default value for IPARM(3), which reflects the number of processors and
this value must always be supplied by the user. On distributed-memory architectures, the user
must also supply the value for IPARM(52), which reflects the number of compute nodes. Mixed
OpenMP-MPI parallelization is used for the distributed-memory solver.
DPARM (64) — REAL
Input/Output
On entry: DPARM is a double-precision array of size 64 that is used to pass various parameters to
PARDISO and to return some useful information after the execution of the solver. The array will
only be used to initialize the multi-recursive iterative linear solver in PARDISO. PARDISOINIT
fills DPARM(1) through DPARM(64) with default values and uses them.
See section 2.4 for a detailed description.
ERROR — INTEGER
Output
On output: The error indicator
Error
0
-10
-11
-12
Information
No error.
No license file pardiso.lic found
License is expired
Wrong username or hostname
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
2.2
10
Arguments of PARDISO
Solving a linear system is split into four tasks: analysis and symbolic factorization, numerical factorization, forward and backward substitution including iterative refinement, and finally termination to release
all internal solver memory. The calling sequence and description of the PARDISO parameters is as follows. When an input data structure is not accessed in a call, a NULL pointer or any valid address can
be passed as a place holder for that argument.
PT(64) — INTEGER
see PARDISOINIT.
Input/Output
MAXFCT — INTEGER
Input
On entry: Maximal number of factors with identical nonzero sparsity structure that the user would
like to keep at the same time in memory. It is possible to store several different factorizations with
the same nonzero structure at the same time in the internal data management of the solver. In
many applications this value is equal to 1.
Note: Matrices with different sparsity structure can be kept in memory with different memory
address pointers PT.
MNUM — INTEGER
Input
On entry: Actual matrix for the solution phase. With this scalar the user can define the matrix
that he would like to factorize. The value must be: 1 ≤ MNUM ≤ MAXFCT. In many applications
this value is equal to 1.
MTYPE — INTEGER
see PARDISOINIT. Note: The parameter influences the pivoting method.
Input
PHASE — INTEGER
Input
On entry: PHASE controls the execution of the solver. It is a two-digit integer ij (10i + j,
1 ≤ i ≤ 3, i ≤ j ≤ 3 for normal execution modes). The i digit indicates the starting phase of
execution, and j indicates the ending phase. PARDISO has the following phases of execution:
1.
2.
3.
4.
Phase 1: Fill-reduction analysis and symbolic factorization
Phase 2: Numerical factorization
Phase 3: Forward and Backward solve including iterative refinement
Termination and Memory Release Phase (PHASE ≤ 0)
If a previous call to the routine has computed information from previous phases, execution may
start at any phase:
PHASE
11
12
13
22
-22
23
33
0
-1
Solver Execution Steps
Analysis
Analysis, numerical factorization
Analysis, numerical factorization, solve, iterative refinement
Numerical factorization
Selected Inversion
Numerical factorization, solve, iterative refinement
Solve, iterative refinement
Release internal memory for L and U matrix number MNUM
Release all internal memory for all matrices
Note: The analysis phase needs the numerical values of the matrix A in case of scalings (IPARM(11)=1)
and symmetric weighted matching (IPARM(13)=1 (normal matching) or IPARM(13)=2 (advanced
matchings)).
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
N — INTEGER
On entry: Number of equations in the sparse linear systems of equations A × X = B.
11
Input
Constraint: N > 0.
A(∗) — REAL/COMPLEX
Input
On entry: Nonzero values of the coefficient matrix A. The array A contains the values corresponding to the indices in JA. The size and order of A is the same as that of JA. The coefficients
can be either real or complex. The matrix must be stored in compressed sparse row format with
increasing values of JA for each row. Refer to Figure 2 for more details.
Note: For symmetric or structural symmetric matrices it is important that the diagonal elements
are also available and stored in the matrix. If the matrix is symmetric or structural symmetric,
then the array A is only accessed in the factorization phase, in the triangular solution with iterative
refinement. Nonsymmetric matrices are accessed in all phases of the solution process.
IA(N+1) — INTEGER
Input
On entry: IA is an integer array of size N+1. IA(i) points to the first column index of row i in the
array JA in compressed sparse row format. Refer to Figure 2 for more details.
Note: The array is accessed in all phases of the solution process. Note that the row and column
numbers start from 1.
JA(∗) — INTEGER
Input
On entry: The integer array JA contains the column indices of the sparse matrix A stored in
compress sparse row format. The indices in each row must be sorted in increasing order.
Note: The array is accessed in all phases of the solution process. For symmetric and structurally
symmetric matrices it is assumed that zero diagonal elements are also stored in the list of nonzeros
in A and JA. For symmetric matrices the solver needs only the upper triangular part of the system
as shown in Figure 2.
PERM (N) — INTEGER
Input
On entry: The user can supply his own fill-in reducing ordering to the solver. This permutation
vector is only accessed if IPARM(5) = 1.
Note: The permutation PERM must be a vector of size N. Let A be the original matrix and
B = P AP T be the permuted matrix. The array PERM is defined as follows. Row (column) of
A is the PERM(i) row (column) of B. The numbering of the array must start by 1 and it must
describe a permutation.
NRHS — INTEGER
On entry: NRHS is the number of right-hand sides that need to be solved for.
Input
IPARM (64) — INTEGER
Input/Output
On entry: IPARM is an integer array of size 64 that is used to pass various parameters to
PARDISO and to return some useful information after the execution of the solver. If IPARM(1)
is 0, then PARDISOINIT fills IPARM(2) and IPARM(4) through IPARM(64) with default values.
See section 2.3 for a detailed description.
On output. Some IPARM values will contain useful information, e.g. numbers of nonzeros in the
factors, etc. .
Note: There is no default value for IPARM(3) and this value must always be supplied by the
user, regardless of whether IPARM(1) is 0 or 1. On distributed-memory architectures, the user
must also supply the value for IPARM(52), which reflects the number of compute nodes. Mixed
OpenMP-MPI parallelization is used for the distributed-memory solver.
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MSGLVL — INTEGER
Input
On entry: Message level information. If MSGLVL = 0 then PARDISO generates no output. If
MSGLVL = 1 the direct solver prints statistical information to the screen. The multi-recursive
iterative solver prints statistical information in the file pardiso-ml.out.
B (N,NRHS) — REAL/COMPLEX
On entry: Right-hand side vector/matrix.
Input/Output
On output: The array is replaced by the solution if IPARM(6) = 1.
Note: B is only accessed in the solution phase.
X (N,NRHS) — REAL/COMPLEX
On output: Solution if IPARM(6)= 0.
Output
Note: X is only accessed in the solution phase.
ERROR — INTEGER
On output: The error indicator
Error
0
-1
-2
-3
-4
-5
-6
-7
-8
-10
-11
-12
-100
-101
-102
-103
Output
Information
No error.
Input inconsistent.
Not enough memory.
Reordering problem.
Zero pivot, numerical fact. or iterative refinement problem.
Unclassified (internal) error.
Preordering failed (matrix types 11, 13 only).
Diagonal matrix problem.
32-bit integer overflow problem.
No license file pardiso.lic found.
License is expired.
Wrong username or hostname.
Reached maximum number of Krylov-subspace iteration in iterative solver.
No sufficient convergence in Krylov-subspace iteration within 25 iterations.
Error in Krylov-subspace iteration.
Break-Down in Krylov-subspace iteration.
DPARM (64) — REAL
Input/Output
On entry: DPARM is a double-precision array of size 64 that is used to pass various parameters
to PARDISO and to return some useful information after the execution of the iterative solver.
The array will be used in the multi-recursive iterative linear solver in PARDISO. This iterative
solver is used in case of IPARM(32) = 1. In addition, DPARM returns double return values from
the direct solver such as the determinant for symmetric indefinite matrices.
See section 2.4 for a detailed description.
2.3
IPARM Control Parameters
All parameters are integer values.
IPARM (1) — Use default values.
Input
On entry: If IPARM(1) = 0, then IPARM(2) and IPARM(4) through IPARM(64) are filled with
default values, otherwise the user has to supply all values in IPARM.
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13
Note: If IPARM(1) 6= 0 on entry, the user has to supply all values form IPARM(2) to IPARM(64).
IPARM (2) — Fill-In reduction reordering.
Input
On entry: IPARM(2) controls the fill-in reducing ordering for the input matrix. If IPARM(2) is
0 then the minimum degree algorithm is applied [6], if IPARM(2) is 2 the solver uses the nested
dissection algorithm from the METIS package [2].
The default value of IPARM(2) is 2.
IPARM (3) — Number of processors.
Input
On entry: IPARM(3) must contain the number of processors that are available for parallel execution. The number must be equal to the OpenMP environment variable OMP NUM THREADS.
Note: If the user has not explicitly set OMP NUM THREADS, then this value can be set by the
operating system to the maximal numbers of processors on the system. It is therefore always
recommended to control the parallel execution of the solver by explicitly setting OMP NUM THREADS.
If fewer processors are available than specified, the execution may slow down instead of speeding
up.
There is no default value for IPARM(3).
IPARM (4) — Preconditioned CGS.
Input
On entry: This parameter controls preconditioned CGS [16] for nonsymmetric or structural symmetric matrices and Conjugate-Gradients for symmetric matrices. IPARM(4) has the form
IPARM(4) = 10 ∗ L + K ≥ 0.
The values K and L have the following meaning:
Value K:
K
1
2
Description
K can be chosen independently of MTYPE.
LU was computed at least once before call with IPARM (4) 6= 0.
LU preconditioned CGS iteration for A × X = B is executed.
LU preconditioned CG iteration for A × X = B is executed.
Value L: The value L controls the stopping criterion of the Krylov-Subspace iteration:
CGS = 10−L is used in the stopping criterion ||dxi ||/||dx1 || < CGS with ||dxi || = ||(LU )−1 ri ||
and ri is the residual at iteration i of the preconditioned Krylov-Subspace iteration.
Strategy: A maximum number of 150 iterations is set on the expectation that the iteration will
converge before consuming half the factorization time. Intermediate convergence rates and the
norm of the residual are checked and can terminate the iteration process. The basic assumption
is: LU was computed at least once — hence it can be recomputed if the iteration process converges
slowly.
If PHASE=23: the factorization for the supplied A is automatically recomputed in cases where
the Krylov-Subspace iteration fails and the direct solution is returned. Otherwise the solution
from the preconditioned Krylov Subspace iteration is returned.
If PHASE=33: the iteration will either result in success or an error (ERROR=4), but never in a
recomputation of LU. A simple consequence is: (A,IA,JA) may differ from the (A,IA,JA)’ supplied
to compute LU (MTYPE, N have to be identical). More information on the failure can be obtained
from IPARM(20).
Note: The default is IPARM(4)=0 and other values are only recommended for advanced users.
Examples:
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IPARM(4)
31
61
62
14
Examples
LU-preconditioned CGS iteration with a stopping
tolerance of 10−3 for nonsymmetric matrices
LU-preconditioned CGS iteration with a stopping
tolerance of 10−6 for nonsymmetric matrices
LU-preconditioned CG iteration with a stopping
tolerance of 10−6 for symmetric matrices
IPARM (5) — User permutation.
Input
On entry: IPARM(5) controls whether a user-supplied fill-in reducing permutation is used instead
of the integrated multiple-minimum degree or nested dissection algorithm.
Note: This option may be useful for testing reordering algorithms or adapting the code to special
application problems (e.g. to move zero diagonal elements to the end of P AP T ). The permutation
must be a vector of size N. Let A be the original matrix and B = P AP T be the permuted matrix.
The array PERM is defined as follows. Row (column) of A is the PERM(i) row (column) of B.
The numbering of the array must start at 1.
The default value of IPARM(5) is 0.
IPARM (6) — Write solution on X.
Input
On entry: If IPARM(6) is 0, which is the default, then the array X contains the solution and
the value of B is not changed. If IPARM(6) is 1 then the solver will store the solution on the
right-hand side B.
Note: The array X is always changed.
The default value of IPARM(6) is 0.
IPARM (7) — Number of performed iterative refinement steps.
Output
On output: The number of iterative refinement steps performed during the solve step are reported
in IPARM(7).
IPARM (8) — Iterative refinement steps.
Input
On entry: On entry to the solve and iterative refinement step, IPARM(8) should be set to the
maximum number of iterative refinement steps that the solver will perform. Iterative refinement
will stop if a satisfactory level of accuracy of the solution in terms of backward error has been
achieved.
The solver will automatically perform two steps of iterative refinement when perturbed pivots have
occured during the numerical factorization and IPARM(8) was equal to zero.
The number of iterative refinement steps is reported in IPARM(7).
Note: If IPARM(8) < 0 the accumulation of the residual is using complex*32 or real*16 in case
that the compiler supports it (GNU gfortran does not). Perturbed pivots result in iterative refinement (independent of IPARM(8)=0) and the iteration number executed is reported in IPARM(7).
The default value is 0.
IPARM (9) — Unused.
This value is reserved for future use. Value must be set to 0.
Input
IPARM (10) — Pivot perturbation.
Input
On entry: IPARM(10) instructs PARDISO how to handle small pivots or zero pivots for nonsymmetric matrices (MTYPE=11, MTYPE=13) and symmetric matrices (MTYPE=-2, MTYPE=-4,
or MTYPE=6). For these matrices the solver uses a complete supernode pivoting approach.
When the factorization algorithm reaches a point where it cannot factorize the supernodes with
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15
this pivoting strategy, it uses a pivoting perturbation strategy similar to [5, 13], compare ??. The
magnitude of the potential pivot is tested against a constant threshold of α = · ||A2 ||∞ , where
= 10−IP ARM (10) and ||A2 ||∞ is the ∞-norm of the scaled and permuted matrix A. Any tiny
pivots lii encountered during elimination are set to the sign(lii ) · · ||A2 ||∞ .
The default value of IPARM(10) is 13 and therefore = 10−13 for nonsymmetric matrices. The
default value of IPARM(10) is 8 and = 10−8 for symmetric indefinite matrices (MTYPE=-2,
MTYPE=-4, or MTYPE=6)
IPARM (11) — Scaling vectors.
Input
On entry: PARDISO can use a maximum weighted matching algorithm to permute large elements
close the diagonal and to scale the matrix so that the largest elements are equal to 1 and the
absolute value of the off-diagonal entries are less or equal to 1. This scaling method is applied to
nonsymmetric matrices (MTYPE=11 or MTYPE=13). The scaling can also be used for symmetric
indefinite matrices (MTYPE=-2, MTYPE=-4, or MTYPE=6) if symmetric weighted matchings
is applied (IPARM (13)=1 or IPARM (13)=3).
Note: It is recommended to use IPARM(11)=1 (scaling) and IPARM(13)=1 (matchings) for
highly indefinite symmetric matrices e.g. from interior point optimizations or saddle point problems. It is also very important to note that the user must provide in the analysis phase (PHASE=11)
the numerical values of the matrix A if IPARM(11)=1 (scaling) or PARM(13)=1 or 2 (matchings).
The default value of IPARM(11) is 1 for nonsymmetric matrices (MTYPE=11 or MTYPE=13)
and IPARM(11) is 0 for symmetric matrices (MTYPE=-2, MTYPE=-4, or MTYPE=6).
IPARM (12) — Solving with transpose matrix.
Input
On entry: Solve a system AT x = b by using the factorization of A.
The default value of IPARM(12) is 0. IPARM(12)=1 indicates that you would like to solve
transposed system AT x = b.
IPARM (13) — Improved accuracy using (non-)symmetric weighted matchings.
Input
On entry: PARDISO can use a maximum weighted matching algorithm to permute large elements
close the diagonal. This strategy adds an additional level of reliability to our factorization methods
and can be seen as a complement to the alternative idea of using more extensive pivoting techniques
during the numerical factorization.
Note: It is recommended to use IPARM(11) = 1 (scalings) and IPARM(13)=1 (normal matchings)
and IPARM(13)=2 (advanced matchings, higher accuracy) for highly indefinite symmetric matrices
e.g. from interior point optimizations or saddle point problems. Note that the user must provide
in the analysis phase (PHASE=11) the numerical values of the matrix A in both of these cases.
The default value of IPARM(13) is 1 for nonsymmetric matrices (MTYPE=11 or MTYPE=13)
and 0 for symmetric matrices (MTYPE=-2, MTYPE=-4, or MTYPE=6).
IPARM (14) — Number of perturbed pivots.
Output
On output: After factorization, IPARM(14) contains the number of perturbed pivots during the
elimination process for MTYPE=-2, -4, 6, 11, or 13.
IPARM (15) — Peak memory symbolic factorization.
Output
On output: IPARM(15) provides the user with the total peak memory in KBytes that the solver
needed during the analysis and symbolic factorization phase. This value is only computed in phase
1.
IPARM (16) — Permanent memory symbolic factorization.
Output
On output: IPARM(16) provides the user with the permanently needed memory — hence allocated
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16
at the end of phase 1 — in KBytes to execute the factorization and solution phases. This value is
only computed in phase 1 and IPARM (15) > IPARM (16) holds.
IPARM (17) — Memory numerical factorization and solution.
Output
On output: IPARM(17) provides the user with the total double precision memory consumption
(KBytes) of the solver for the factorization and solution phases. This value is only computed in
phase 2.
Note: Total peak memory consumption is
peakmem = max (IPARM(15), IPARM(16)+IPARM(17)).
IPARM (18) — Number nonzeros in factors.
Input/Output
On entry: if IPARM(18) < 0 the solver will report the number of nonzeros in the factors.
On output: The numbers of nonzeros in the factors are returned to the user.
The default value of IPARM(18) is -1.
IPARM (19) — GFlops of factorization.
Input/Output
9
On entry: if IPARM(19) < 0 the solver will compute the number of GFlops (10 ) to factor the
matrix A. This will increase the reordering time.
On output: Number of GFlops (109 operations) needed to factor the matrix A.
The default value of IPARM(19) is 0.
IPARM (20) — CG/CGS diagnostics.
Output
On output: The value is used to give CG/CGS diagnostics (e.g. the number of iterations and
cause of failure):
•
IPARM(20) > 0: CGS succeeded and the number of iterations is reported in IPARM(20).
•
IPARM(20) < 0: Iterations executed, but CG/CGS failed. The error details in IPARM(20)
are of the form:
IPARM(20) = - it cgs*10 - cgs error.
If PHASE was 23, then the factors L, U have been recomputed for the matrix A and the error
flag ERROR should be zero in case of a successful factorization. If PHASE was 33, then ERROR
= -4 will signal the failure.
Description of cgs error:
cgs error
1
2
3
4
5
Description
Too large fluctuations of the residual;
||dxmax it cgs/2 || too large (slow convergence);
Stopping criterion not reached at max it cgs;
Perturbed pivots caused iterative refinement;
Factorization is too fast for this matrix. It is better
to use the factorization method with IPARM(4) = 0.
IPARM (21) — Pivoting for symmetric indefinite matrices.
Input
On entry: IPARM (21) controls the pivoting method for sparse symmetric indefinite matrices. If
IPARM(21) is 0, then 1 × 1 diagonal pivoting is used. If IPARM(21) is 1, then 1 × 1 and 2 × 2
Bunch-Kaufman pivoting will be used within the factorization process.
Note: It is also recommended to use IPARM(11) = 1 (scaling) and IPARM(13) = 1 or 2 (matchings) for highy indefinite symmetric matrices e.g. from interior point optimizations or saddle point
problems.
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Bunch-Kaufman pivoting is available for matrices MTYPE=-2, -4, or 6.
The default value of IPARM(21) is 1.
IPARM (22) — Inertia: Number of positive eigenvalues.
Output
On output:
The number of positive eigenvalues for symmetric indefinite matrices is reported in IPARM(22).
IPARM (23) — Inertia: Number of negative eigenvalues.
Output
On output:
The number of negative eigenvalues for symmetric indefinite matrices is reported in IPARM(23).
IPARM (24) — Parallel Numerical Factorization.
Input
On entry: This parameter selects the scheduling method for the parallel numerical factorization.
If IPARM(24) = 0 then the solver will use a parallel factorization that has been used in the solver
during the last years. IPARM(24) = 1 selects a two-level scheduling algorithms which should
result in a better parallel efficiency.
The default value of IPARM(24) is 1.
IPARM (25) — Parallel Forward/Backward Solve.
Input
On entry: The parameter controls the parallelization for the forward and backward solve. IPARM(25)
= 0 indicates that a sequential forward and backward solve is used, whereas IPARM(25) = 1 selects
a parallel solve.
The default value of IPARM(25) is 1.
IPARM (26) — Splitting of Forward/Backward Solve.
Input
On entry: The user can apply a partial forward and backward substitution with this parameter.
IPARM(25) = 0 indicates that a normal solve step is performed with both the factors L and
U . IPARM(25) = 1 indicates that a forward solve step is performed with the factors L or U T .
IPARM(25) = 2 indicates that a backward solve step is performed with the factors U or LT —
compare IPARM(12).
The default value of IPARM(26) is 0.
IPARM (27) — Unused.
This value is reserved for future use. Value must be set to 0.
Input
IPARM (28) — Parallel Reordering for METIS.
Input
On entry: The parameter controls the parallel execution of the METIS reordering. IPARM(28) = 1
indicates that a parallel reordering is performed, IPARM(28) = 0 indicates a sequential reordering.
The default value of IPARM(28) is 0.
IPARM (29) — Switch between 32-bit and 64-bit factorization.
Input
On entry: The parameter controls the IEEE accuracy of the solver IPARM(29) = 0 indicates that
a 64-bit factorization is performed and IPARM(29) = 1 indicates a 32-bit factorization.
The default value of IPARM(29) is 0. The option is only available for symmetric indefinite matrices
(MTYPE=-2) and for real unsymmetric matrices (MTYPE=11).
IPARM (30) — Control the size of the supernodes.
Input
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On entry: The parameter controls the column size of the supernodes, e.g., IPARM(30) = 40
indicates that 40 columns are used for each supernode.
The default value of IPARM(30) is 80 for symmetric matrices.
IPARM (31) — Partial solve for sparse right-hand sides and sparse solution.
Input
On entry: The parameter controls the solution method in case that the user is interested in only
a few components of the solution vector X. The user has to define the vector components in the
input vector PERM. An entry PERM(i)=1 means that either the right-hand side is nonzero at
this component, and it also defines that the solution i-th component should be computed in the
vector X. The vector PERM must be present in all phases of PARDISO.
The default value of IPARM(31) is 0. An entry equal IPARM(31)=1 selects the sparse right-hand
sides solution method.
IPARM (32) — Use the multi-recursive iterative linear solver.
Input
On entry: The default value of IPARM(32) is 0 (sparse direct solver) and IPARM(32)=1 will use
the iterative method in PARDISO.
IPARM (33) — Determinant of a real symmetric indefinite matrix.
Input
On entry: IPARM(33)=1 will compute the determinant of a real symmetric indefinite matrices
and will return the result in DPARM(33).
On output: The parameter returns the natural logarithm of the determinant of a real symmetric
indefinite matrix A.
The default value of IPARM(33) is 0.
IPARM (34) — Identical solution independent on the number of processors.
Input
On entry: IPARM(34)=1 will always compute the identical solution independent on the number of
processors that PARDISO is using during the parallel factorization and solution phase. This option
is only working with the METIS reordering (IPARM(2)=2), and you also have to select a parallel
two-level factorization (IPARM(24)=1) and a parallel two-level solution phase (IPARM(25)=1).
In addition, the user has define the upper limit of processors in IPARM(3) for which he would like
to have the same results.
Example IPARM(3)=8, IPARM(2)=2, and IPARM(34)=IPARM(24)=IPARM(25)=1: This option means that the user would like to have identical results for up to eight processors. The number
of cores that the user is actually using in PARDISO can be set by the OpenMP environment variable OMP NUM THREADS. In our example, the solution is always the same for OMP NUM THREADS=1
up to OMP NUM THREADS=IPARM(3)=8.
IPARM (35) — Unused.
This value is reserved for future use. Value MUST be set to 0.
Output
IPARM (36) — Selected inversion for A−1
Input
ij .
On entry: IPARM(36) will control the selected inversion process based on the internal L and
U factors. If IPARM(36) = 0 and PHASE = -22, PARDISO will overwrite these factors with
the selected inverse elements of A−1 . If IPARM(36) = 1 and PHASE = -22, PARDISO will
not overwrite these factors. It will instead allocate additional memory of size of the numbers of
elements in L and U to store these inverse elements.
The default value of IPARM(36) is 0.
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The following code fragment shows how to use the selected inversion process in PARDISO from a
Fortran program
c...Reorder and factor a matrix sytem
PHASE=12
CALL PARDISO(PT, MAXFCT, MNUM, MTYPE, PHASE,N, A, IA, JA,
PERM, NRHS, IPARM, MSGLVL, B, X, ERROR, DPARM)
c...Compute selected inversion process and return inverse elements
c...in CSR structure (a,ia,ja)
PHASE=-22
IPARM(36)=0
CALL PARDISO(PT, MAXFCT, MNUM, MTYPE, PHASE,N, A, IA, JA,
PERM, NRHS, IPARM, MSGLVL, B, X, ERROR, DPARM)
IPARM (37) — Selected inversion for A−1
Input
ij .
On entry: IPARM(37) will control the selected inversion process for symmetric matrices. If
IPARM(37) = 0, PARDISO will return the selected inverse elements A−1
ij in upper symmetric
triangular CSR format. If IPARM(37) = 1, PARDISO will return the selected inverse elements
A−1
ij in full symmetric triangular CSR format.
The default value of IPARM(37) is 0.
IPARM (38) — Schur-complement computation.
Input
On entry: IPARM(38) will control the Schur-complement computation
and it will
compute the
A11 A12
−1
partial Schur complement S = A22 − A21 A11 A12 for the matrix A =
. IPARM(38)
A21 A22
indicates the numbers of rows/columns size of the matrix A22 .
The default value of IPARM(38) is 0.
The following code fragment shows how to use the Schur-complement method in PARDISO from
a C++ program to compute a Schur-complement of size 5.
// Reorder and factor a matrix sytem
int phase = 12;
int nrows = 5;
iparm[38-1] = nrows;
phase = 12;
pardiso(pt, &maxfct, &mnum, &mtype, &phase, &n, a, ia, ja, &idum, &nb, &iparm[0],
&msglvl, &ddum, &ddum, &error, dparm);
// allocate memory for the Schur-complement and copy it there.
int nonzeros = iparm[39-1];
int* iS = new int[nrows+1];
int* jS = new int[nonzeros];
double* S = new double[nonzeros];
pardiso get schur(pt, &maxfct, &mnum, &mtype, S, iS, jS);
IPARM (39) — Nonzeros in Schur-complement S = A22 − A21 A−1
Output
11 A12 .
−1
On Output: The numbers of nonzeros in the Schur-complement S = A22 − A21 A11 A12 .
IPARM (40) to IPARM(50) — Unused.
These values are reserved for future use. Value MUST be set to 0.
IPARM (51) — Use parallel distributed-memory solver.
Output
Input
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On Input: IPARM(51)=1 will always use the distributed MPI-based parallel numerical factorization and parallel forward-backward substitution.
The default value of IPARM(51) is 0.
IPARM (52) — Number of compute nodes for the distributed-memory parallel solver. Input
On Input: IPARM(52) defines the number of compute nodes (e.g. an eight core Intel Nehalem
node) that are used with the distributed MPI-based parallel numerical factorization and parallel
forward-backward substitution.
The default value of IPARM(52) is 1.
Example IPARM(52)=2 (number of compute nodes), IPARM(3)=8 (number of cores on each
node): This option means that the user would like to use 16 cores in total for the distributedmemory parallel factorization.
IPARM (53) to IPARM(65) — Unused.
These values are reserved for future use. Value MUST be set to 0.
2.4
Output
DPARM Control Parameters
All parameters are 64-bit double values. These parameters are only important for the multirecursive
iterative solver (except DPARM(33) which returns the determinant of symmetric real indefinite matrices).
A log-file with detailed information of the convergence process is written in pardiso-ml.out. The initial
guess for the solution has to be provided in the solution vector x. A vector x initialized with zeros is
recommended.
DPARM (1) — Maximum number of Krylov-Subspace Iteration.
Input
On entry: This values sets the maximum number of SQMR iteration for the iterative solver.
The default value of DPARM(1) is 300.
DPARM (2) — Relative Residual Reduction.
Input
On entry: This value sets the residual reduction for the SQMR Krylov-Subspace Iteration.
The default value of DPARM(2) is 1e-6.
DPARM (3) — Coarse Grid Matrix Dimension .
Input
On entry: This value sets the matrix size at which the multirecursive method switches to the
sparse direct method.
The default value of DPARM(3) is 5000.
DPARM (4) — Maximum Number of Grid Levels .
Input
On entry: This value sets the maximum number of algebraic grid levels for the multirecursive
iterative solver,
The default value of DPARM(4) is 10.
DPARM (5) — Dropping value for the incomplete factor .
On entry: This value sets the dropping value for the incomplete factorization method.
Input
The default value of DPARM(5) is 1e-2.
DPARM (6) — Dropping value for the schurcomplement .
On entry: This value sets the dropping value for the schurcomplement.
The default value of DPARM(4) is 5e-3.
Input
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
21
DPARM (7) — Maximum number of Fill-in in each column in the factor.
Input
On entry: This value sets the maximum number of fill-in in each column in the factor compared
to the original matrix.
The default value of DPARM(7) is 10.
DPARM (8) — Bound for the inverse of the incomplete factor L.
On entry: This value sets a numerical bound κ for the inverse of the incomplete factor.
Input
The default value of DPARM(8) is 500.
DPARM (9) — Maximum number of stagnation steps in Krylov-Subspace method.
Input
On entry: This value sets the maximum number of non-improvement steps in Krylov-Subspace
method.
The default value of DPARM(9) is 25.
DPARM (10) to DPARM(32) — Unused.
These values are reserved for future use. Value MUST be set to 0.
Output
DPARM (33) — Determinant for real symmetric indefinite matrices.
Output
On output: This value returns the determinant for real symmetric indefinite matrices. The is only
computed in case of IPARM (33)=1
DPARM (34) — Relative residual after Krylov-Subspace convergence.
Output
On output: This value returns the relative residual after Krylov-Subspace convergence. The is
only computed in case of IPARM (32)=1
DPARM (35) — Number of Krylov-Subspace iterations.
Output
On output: This value returns the number of Krylov-Subspace iterations. The is only computed
in case of IPARM (32)=1
DPARM (36) to DPARM(64) — Unused.
These values are reserved for future use. Value MUST be set to 0.
2.5
Summary of arguments and parameters
Output
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
22
CALL PARDISOINIT(PT, MTYPE, SOLVER, IPARM, DPARM, ERROR)
CALL PARDISO(PT, MAXFCT, MNUM, MTYPE, PHASE, N, A, IA, JA,
1
PERM, NRHS, IPARM, MSGLVL, B, X, ERROR, DPARM)
CALL PARDISO GET SCHUR(PT, MAXFCT, MNUM, MTYPE, S, IS, JS)
CALL PARDISO_RESIDUAL(MTYPE, N, A, IA, JA, B, X, Y, NORM B, NORM RES)
pardisoinit (void *pt[64], int *mtype, int *solver, int iparm[64],
double dparm[64], int *error);
pardiso (void *pt[64], int *maxfct, int *mnum, int *mtype,
int *phase, int *n, double a[], int ia[], int ja[],
int perm[], int *nrhs, int iparm[64], int *msglvl,
double b[], double x[], int *error, double dparm[64] );
pardiso get schur(pt, &maxfct, &mnum, &mtype, S, iS, jS);
pardiso_residual (&mtype, &n, a, ia, ja, b, x, y, &norm b, &norm res);
2.6
Computing the Schur-complement: Example for nonsymmetric matrices
Suppose the system of linear equations has the form
A11 A12
x1
b1
=
.
A21 A22
x2
b2
The Schur complement matrix S = A22 − A21 A−1
11 A12 can be returned to the user (see IPARM (39) and
IPARM (39)). PARDISO provides a partial factorization of the complete matrix and returns the Schur
matrix in user memory. The Schur matrix is considered as a sparse matrix and PARDISO will return
matrix in sparse CSR format to the user. The partial factorization that builds the Schur matrix can also
be used to solve linear systems A11 x1 = b1 associated with the A11 block. Entries in the right-hand side
corresponding to indices from the Schur matrix need not be set on entry and they are not modified on
output. The following code fragment shows how to use the Schur-complement method in PARDISO from
a C++ program to compute a Schur-complement.
// Reorder and partial factorization of the matrix sytem
int phase
= 12;
iparm[38-1] = xxx; // size of the A_22 block
pardiso(pt, &maxfct, &mnum, &mtype, &phase, &n, a, ia, ja,
&idum, &nb, &iparm[0], &msglvl, &ddum, &ddum, &error,
dparm);
// Schur-complement matrix S=(S, is, jS) in CSR format.
int nonzeros = iparm[39-1];
int* iS
= new int[nrows+1];
int* jS
= new int[nonzeros];
double* S
= new double[nonzeros];
pardiso get schur(pt, &maxfct, &mnum, &mtype, S, iS, jS);
// solve with A_11x1= b1;
int phase
= 33;
pardiso(pt, &maxfct, &mnum, &mtype, &phase, &n, a, ia, ja,
&idum, &nb, &iparm[0], &msglvl, &b1, &x1, &error, dparm);
Table 1: Overview of subroutine arguments.
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Name
Type
PT (64)
MAXFCT
MNUM
MTYPE
INT
INT
INT
INT
1
2
-2
11
3
4
-4
6
13
INT
11
12
13
22
-22
23
33
-1
0
INT
R/C
INT
INT
INT
INT
INT
INT
0
1
R/C
R/C
INT
REAL
PHASE
N
A (∗)
IA (N+1)
JA (∗)
PERM (N)
NRHS
IPARM (64)
MSGLVL
B (N, NRHS)
X (N, NRHS)
ERROR
DPARM (64)
Description
23
Input/Output
Internal memory address pointer.
I/O
Number of numerical factorizations in memory.
I
Actual matrix to factorize.
I
Matrix type.
I
real and structurally symmetric, supernode pivoting
real and symmetric positive definite
real and symmetric indefinite, diagonal or Bunch-Kaufman pivoting
real and nonsymmetric, complete supernode pivoting
complex and structurally symmetric, supernode pivoting
complex and hermitian positive definite
complex and hermitian indefinite, diagonal or Bunch-Kaufman pivoting
complex and symmetric
complex and nonsymmetric, supernode pivoting
Solver Execution Phase.
I
Analysis
Analysis, Numerical Factorization
Analysis, Numerical Factorization, Solve, Iterative Refinement
Numerical Factorization
Selected Inversion
Numerical Factorization, Solve, Iterative Refinement
Solve, Iterative Refinement
Release all internal memory for all matrices
Release memory for matrix number MNUM
Number of equations.
I
Matrix values.
I
Beginning of each row.
I
Column indices.
I
User permutation.
I
Number of right-hand sides.
I
Control parameters.
I/O
Message level.
I
No output.
Output statistical information
Right-hand sides.
I/O
Solution vectors (see IPARM(6)).
O
Error indicator.
O
Control parameters for iterative solver
I/O
Table 2: Overview of subroutine arguments.
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Name
Description
IPARM(1)
Use default options.
0
Set all entries to their default values except IPARM(3).
Use METIS reordering.
0
Do not use METIS.
2*
Use METIS nested dissection reordering.
Number of processors.
p
Number of OpenMP threads. This must be identical or slightly larger to
the environment variable OMP NUM THREADS.
Do preconditioned CGS iterations (see description). Default is 0.
Use user permutation.
0*
Do not use user perm.
1
Use the permutation provided in argument PERM.
Solution on X / B
0*
Write solution to X
1
Write solution to B
Max. numbers of iterative refinement steps.
k=0* Do at most k steps of iterative refinement for all matrices.
eps pivot (perturbation 10−k ).
13*
Default for nonsymmetric matrices
8*
Default for symmetric indefinite matrices
Use (non-) symmetric scaling vectors.
0
Do not use .
1*
Use (nonsymmetric matrices).
0*
Do not use (symmetric matrices).
Solve transposed matrix.
0*
Do normal solve.
1
Do solve with transposed matrix.
Improved accuracy using (non-)symmetric matchings
0
Do not use.
1*
Use (nonsymmetric matrices).
2
Use a very robust method for symmetric indefinite matrices.
0*
Do not use (symmetric matrices).
Number of nonzeros in LU.
0
Do not determine.
−1 *
Will only be determined if −1 on entry.
Gflops for LU factorization.
0*
Do not determine.
−1
Will only be determined if −1 on entry. Increases ordering time.
IPARM(2)
IPARM(3)
IPARM(4)
IPARM(5)
IPARM(6)
IPARM(8)
IPARM(10)
IPARM(11)
IPARM(12)
IPARM(13)
IPARM(18)
IPARM(19)
Table 3: Overview of input IPARM control parameters. An asterisk (*) indicates the default value.
24
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Name
Description
IPARM(21)
Pivoting for symmetric indefinite matrices. Default is 1.
0
1 × 1 Diagonal Pivoting
1* 1 × 1 and 2 × 2 Bunch-Kaufman pivoting.
Parallel Numerical Factorization
0
Do one-level parallel scheduling.
1* Do two-level parallel scheduling.
Parallel Forward/Backward Solve
0
Do sequential solve.
1* Do parallel solve.
IPARM(24)
IPARM(25)
IPARM(26)
Partial Forward/Backward Solve
0* Do forward/backward solve with L and U .
1
Do forward solve with L or U T .
2
Do backward solve with U or LT .
IPARM(28)
Parallel METIS reordering
0* Do sequential METIS reordering.
1
Do parallel METIS reordering.
IPARM(29)
32-bit/64-bit IEEE accuracy
0* Use 64-bit IEEE accuracy.
1
Use 32-bit IEEE accuracy.
IPARM(30)
Control size of supernodes
0* Use default configuration.
1
Use use configuration.
IPARM(31)
Partial solve for sparse right-hand sides and sparse solutions
0* Compute all components in the solution vector.
1
Compute only a few selected in the solution vector.
IPARM(32)
Use the multi-recursive iterative linear solver
0* Use sparse direct solver.
1
Use multi-recursive iterative solver.
IPARM(33)
Determinant for real symmetric matrices
0* Do not compute determinant.
1
Compute determinant.
IPARM(34)
Identical solution independent on the number of processors
0* No identical parallel results.
1
Identical parallel results
Table 4: Overview of input IPARM control parameters. An asterisk (*) indicates the default value.
25
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Name
Description
IPARM(36)
Selected inversion for A−1
ij
0* Overwrite internal factor with inverse elements.
1
Do not overwrite internal factor with inverse elements.
IPARM(37)
Selected inversion for A−1
ij for symmetric matrices
0* Return inverse elements in upper triangular symmetric compressed CSR
format (1-index based).
1
Return inverse elements in full symmetric compressed CSR format (1index based).
IPARM(38)
Schur-complement computation
0* Indicates the numbers of rows/columns in S.
k
Schur-complement matrix S is a k× matrix.
IPARM(51)
Use parallel distributed-memory solver
0* Use OpenMP-threaded solver
1
Use Mixed OpenMP-MPI solver
IPARM(52)
Number of nodes for distributed-memory solver
1* For OpenMP-threaded solver
p
Use p compute nodes
26
Table 5: Overview of input IPARM control parameters. An asterisk (*) indicates the default value.
Name
Description
IPARM(7)
IPARM(14)
IPARM(15)
IPARM(16)
IPARM(17)
IPARM(18)
Number of iterative refinement steps.
Number of perturbed pivots.
Peak Memory in KBytes during analysis.
Permanent Memory in KBytes from analysis that is used in phases 2 and 3.
Peak Double Precision Memory in KBytes including one LU Factor.
Number of nonzeros in LU.
0
Do not determine.
−1 * Will only be determined if −1 on entry.
Gflops for LU factorization.
0*
Do not determine.
−1
Will only be determined if −1 on entry. Increases ordering time.
Numbers of CG Iterations.
Number of positive eigenvalues for symmetric indefinite systems.
Number of negative eigenvalues for symmetric indefinite systems.
Number of nonzeros in Schur-complement matrix S
Determinant for real symmetric indefinite matrices.
Relative residual after Krylov-Subspace convergence.
Number of Krylov-Subspace iterations.
IPARM(19)
IPARM(20)
IPARM(22)
IPARM(23)
IPARM(39)
DPARM(33)
DPARM(34)
DPARM(35)
Table 6: Overview of output IPARM/DPARM control parameters. An asterisk (*) indicates the default
value.
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Name
Description
IPARM(1)
IPARM(2)
IPARM(3)
IPARM(4)
IPARM(5)
IPARM(6)
IPARM(7)
IPARM(8)
IPARM(9)
IPARM(10)
IPARM(11)
IPARM(12)
IPARM(13)
IPARM(14)
IPARM(15)
IPARM(16)
IPARM(17)
IPARM(18)
IPARM(19)
IPARM(20)
IPARM(21)
IPARM(22)
IPARM(23)
IPARM(24)
IPARM(25)
IPARM(26)
IPARM(27)
IPARM(28)
IPARM(29)
IPARM(30)
IPARM(31)
IPARM(32)
IPARM(33)
IPARM(34)
IPARM(35)
IPARM(36)
IPARM(37)
IPARM(38)
IPARM(39)
IPARM(51)
IPARM(52)
Use default options.
Fill-in reduction reordering (METIS).
Number of processors.
Preconditioned CGS / CG iterations.
Use user permutation.
Solution on X / B.
Executed number of iterative refinement steps.
Max. numbers of iterative refinement steps.
For future use.
Size of Pivot perturbation
Use (non-) symmetric scaling vectors.
Solve transposed matrix.
Use (non-)symmetric matchings
Number of perturbed pivots.
Peak memory [kB] phase 1.
Permanent integer memory [kb].
Peak real memory [kB] including one LU.
Number of nonzeros in LU.
Gflops for LU factorization.
Numbers of CG Iterations.
Pivoting for symmetric indefinite matrices.
Number of positive eigenvalues for symmetric indefinite systems
Number of negative eigenvalues for symmetric indefinite systems
Scheduling type for numerical factorization.
Parallel forward/backward solve.
Partial forward/backward solve.
For future use.
Parallel METIS reordering.
32-bit/64-bit IEEE accuracy.
Size of a panel in supernodes.
Partial solve for sparse right-hand sides and sparse solutions.
Use the multi-recursive iterative linear solver.
Compute determinant of a real symmetric indefinite systems.
Identical solution independent on the number of processors.
Internal use.
Selected inversion for A−1
ij .
Selected inversion for A−1
ij for symmetric matrices.
Schur-complement computation.
Number of nonzeros in Schur-complement matrix S.
Use parallel distributed-memory solver.
Number of nodes for distributed-memory solver.
Table 7: Overview IPARM control parameters.
27
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Name
Description
DPARM(1)
Maximum number of Krylov-Subspace Iteration.
300∗
Default.
1 ≤ DP ARM (1) ≤ 500
Relative Residual Reduction.
1e − 6∗ Default.
1e − 1 ≥ DP ARM (2) ≥ 1e − 12
Coarse Grid Matrix Dimension.
1e − 6∗ Default.
1e − 1 ≥ DP ARM (3) ≤ ge − 12
Maximum Number of Grid Levels.
10*
Default.
1 ≤ DP ARM (4) ≤ 1000
Dropping value for the incomplete factor.
1e-2*
Default.
1e − 16 ≤ DP ARM (5) ≤ 1
Dropping value for the schurcomplement.
5e-5*
Default.
1e − 16 ≤ DP ARM (6) ≤ 1
Maximum number of fill-in in each column in the factor.
10*
Default.
1 ≤ DP ARM (7) ≤ 1000000000
Bound for the inverse of the incomplete factor L.
500*
Default.
1 ≤ DP ARM (9) ≤ 1000000000
Maximum number of non-improvement steps in Krylov-Subspace method
25*
Default.
1 ≤ DP ARM (9) ≤ 1000000000
Returns the determinant fo a real symmetric indefinite matrix.
Relative residual after Krylov-Subspace convergence.
Number of Krylov-Subspace iterations.
DPARM(2)
DPARM(3)
DPARM(4)
DPARM(5)
DPARM(6)
DPARM(7)
DPARM(8)
DPARM(9)
DPARM(33)
DPARM(34)
DPARM(35)
Table 8: Overview of DPARM control parameters. An asterisk (*) indicates the default value.
28
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
3
29
Linking and Running the Software
A general remark for Unix operating systems is that the environment variable LD LIBRARY PATH must be
pointing to the PARDISO library. The environment variable OMP NUM THREADS must be set in all cases
of using the multi-threaded library version.
The software can be downloaded from the PARDISO page at http://www.pardiso-project.org. A
license file pardiso.lic is required for the execution of the solver and it will be generated at the download
area of the web page. The user must supply the user identification (Unix: “whoami”, Windows: username
exactly as Windows has it). This license file must always be present either in the home directory of the user
or in the directory from which the user runs a program linked with the PARDISO library. Alternatively,
this file can be placed in a fixed location and the environment variable PARDISO LIC PATH must be set to
the path of its location.
3.1
LINUX on IA64 platform
The current version of PARDISO requires that either the following GNU or Intel compilers are installed
on your system:
1. GNU Fortran gfortran compiler version 4.6.3 or higher for 64-bit mode
2. GNU C compiler gcc version 4.6.3 or higher for 64-bit mode
or
1. INTEL Fortran compiler ifort version 13.0.1 for 64-bit mode
2. INTEL C compiler icc version 13.0.1 for 64-bit mode
3.1.1
Linking the libraries
The libraries contain C and Fortran routines and can be linked with GNU gfortran or gcc, or Intel ifort
or icc. Currently, the following libraries are available for LINUX IA32 and IA64 platforms:
• libpardiso500-GNU463-X86-64.so for linking with GNU compilers version 4.6.3 on IA64 in
64-bit mode.
• libpardiso500-INTEL1301-X86-64.so for linking with Intel compilers version 13.0.1 on IA64 in
64-bit mode.
Here are some examples of linking user code with the PARDISO solver on 64-bit LINUX architectures.
The user may need to add additional flags and options as required by the user’s code.
1. GNU version 4.6.1, 64-bit mode:
gfortran <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-GNU461-X86-64
-L <Path to directory of LAPACK/BLAS>
-l <fast LAPACK and BLAS libraries> -fopenmp -lpthread -lm
gcc <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-GNU461-X86-64
-L <Path to directory of LAPACK/BLAS>
-l <Fast LAPACK and BLAS libraries> -lgfortran -fopenmp -lpthread -lm
2. INTEL version 13.0.1, 64-bit mode:
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30
ifort <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-INTEL1301-X86-64
-L <Path to directory of LAPACK/BLAS>
-l <fast LAPACK and BLAS libraries>
-L/opt/intel/mkl/lib/intel64 -lmkl gf lp64 -lmkl sequential
-lmkl core -lm -lgfortran -fopenmp
icc <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-INTEL1301-X86-64
-L <Path to directory of LAPACK/BLAS>
-l <Fast LAPACK and BLAS libraries>
-L <Path to INTEL System libraries>
-L/opt/intel/mkl/lib/intel64 -lmkl gf lp64 -lmkl sequential
-lmkl core -lm -lgfortran -fopenmp
3.1.2
Setting environment
The user can control the parallel execution of the PARDISO solver with the OpenMP environment variable
OMP NUM THREADS. It is very important for parallel execution that the input parameter IPARM(3) be set
to the corresponding correct value of OMP NUM THREADS. The environment variable LD LIBRARY PATH must
point to the PARDISO library. It is also recommended to set the Intel OpenMP environment variable to
MKL SERIAL=YES.
3.2
Mac OSX on 64-bit platforms
The current version of PARDISO requires that the following GNU compilers be installed on your system:
1. GNU C compiler gcc version 4.7.1 or higher for 64-bit mode
3.2.1
Linking the libraries
The libraries contain C and Fortran routines and can be linked with the GNU gfortran or gcc compilers.
Currently, the following library is available:
• Parallel libpardiso500-MACOS-X86-64.dylib for linking with GNU compilers in 64-bit mode.
Here are some examples of linking user code with the PARDISO solver on Mac OSX. The user may need
to add additional flags and options as required by the user’s code.
1. GNU 64-bit mode, sequential:
gfortran <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-MACOS-X86-64.dylib
-L <Path to directory of LAPACK/BLAS>
-l <fast LAPACK and BLAS libraries> -liomp5.dylib
gcc <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-MACOS-X86-64.dylib
-L <Path to directory of LAPACK/BLAS>
-l <Fast LAPACK and BLAS libraries> -liomp5.dylib
3.2.2
Setting environment
The user can control parallel execution of the PARDISO solver with the OpenMP environment variable
OMP NUM THREADS. It is very important for the parallel execution that the input parameter IPARM(3) be
set to the corresponding correct value of OMP NUM THREADS. The environment variable LD LIBRARY PATH
must point to the PARDISO library.
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3.3
31
Windows on IA64 platforms
The current version of PARDISO requires the Intel compilers and Microsoft Visual Studio 2012 compilers
are installed on your system. This might require that the machine PARDISO runs on has VS2K8 installed
(or the Windows SDK for Windows Server 2012).
3.3.1
Linking the libraries
The Windows libraries comes with a library and associated import library (DLL, LIB), contain C and
Fortran routines and the following libraries are available for Windows IA64 platforms:
• Parallel libpardiso500-WIN-X86-64.[dll, lib] for linking with Microsoft cl compilers on Intel64
in 64-bit mode.
Here are some examples of linking user code with the PARDISO solver on 64-bit Windows architectures. The user may need to add additional flags and options as required by the user’s code.
1. This 64-bit Windows version comes internally with optimized BLAS/LAPACK objects from the
Intel MKL library Version 10.2
cl <source/objects files> -o <executable>
-L <Path to directory of PARDISO> -lpardiso500-WIN-X86-64.lib
4
Using the MATLAB interface
This section illustrates how to use the MATLAB interface.2 Its usage very much follows the conventions
described in previous sections, so I assume the reader has read these sections prior to using the MATLAB
interface. For instructions on how to install this package, see the section below.
4.1
An example with a real matrix
Suppose you have a real, sparse, matrix A in MATLAB, as well as a real, dense matrix B, and you would
like to use PARDISO to find the solutions X to the systems of linear equations
A * X = B
The first step is to initialize PARDISO by running the following command in the MATLAB prompt:
info = pardisoinit(11,0);
This tells PARDISO that we would like to solve systems involving real, sparse non-symmetric matrices,
and using the sparse direct solver. The general call looks like
info = pardisoinit(mtype,solver);
where mtype and solver correspond to the arguments to PARDISOINIT defined in Sec. 2.1. You will
notice that the variable info contains the following fields: the matrix type (mtype), the arrays iparm
and dparm storing PARDISO control parameters (see Sections 2.3 and 2.4), and a field which keeps track
the location in memory where the PARDISO data structures are stored. Do not modify this value!
The second step is to create a symbolic factorization of the matrix via
info = pardisoreorder(A,info,verbose);
where verbose is either true or false. Then we can ask PARDISO to factorize the matrix:
info = pardisofactor(A,info,verbose);
2 This section and the next section were contributed by Peter Carbonetto, a Ph.D. student at the Dept. of Computer
Science, University of British Columbia.
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32
Once we’ve factorized the sparse matrix A, the last step is to compute the solution with the following
command:
[X, info] = pardisosolve(A,B,info,verbose);
The return value X should now contain the solutions to the systems of equations A * X = B. You can
now use the same factorization to compute the solution to linear systems with different right-hand sides.
Once you do not need the factorization any longer, you should deallocate the PARDISO data structures
in memory with the commands
pardisofree(info);
clear info
4.2
A brief note of caution
In a sparse matrix, MATLAB only stores the values that are not zero. However, in PARDISO it is
important that you also provide it with the diagonal entries of a matrix, even if those entries are exactly
zero. This is an issue that commonly arises when using PARDISO to solve the linear systems that arise
in primal-dual interior-point methods. The solution is then to add a very small number to the diagonal
entries of the matrix like so:
A = A + eps * speye(size(A));
4.3
An example with a real, symmetric matrix
Now suppose that your sparse matrix A is symmetric, so that A == A’. You must now initialize PARDISO
with a different matrix type like so:
info = pardisoinit(-2,0);
Let’s further suppose that you think you can obtain a factorization more efficiently with your own
customized reordering of the rows and columns of of the n × n matrix A. This reordering must be stored
in a vector p of length n, and this vector must be a permutation of the numbers 1 through n. To complete
the numeric factorization, you would then issue the following commands in the MATLAB prompt:
info = pardisoreorder(tril(A),info,verbose,p);
info = pardisofactor(tril(A),info,verbose);
Notice that you must only supply the lower triangular portion of the matrix. (Type help tril in
MATLAB for more information on this command.) If you supply the entire matrix, MATLAB will report
an error.
4.4
Complex matrices
The MATLAB interface also works with complex matrices (either symmetric, Hermitian or non-symmetric).
In all these cases, it is important that you initialize PARDISO to the correct mtype. Another thing you
must do is make sure that both the sparse matrix A and the dense matrix B are complex. You can check
this with the iscomplex command. If all the entries of B are real, you can can convert B to a complex
matrix by typing
B = complex(B);
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33
Installing the MATLAB interface
The MATLAB interface for PARDISO was created using the MATLAB external (“MEX”) interface. For
more information on the MEX interface, consult the MathWorks website. In order to be able to use this
interface in MATLAB, you will first have to compile the MEX files on your system.
For those who have the program make, the simplest thing to do is edit the Makefile and type make
all in the command prompt. (Note that the Makefile was written for those who are running Linux. For
other operating systems, you may have to modify the Makefile to suit your setup.) For those of you who
do not have make, you will have to call the mex command directly to compile the MEX object files.
At the top of the Makefile, there are four variables you will need to change to suit your system setup.
The variable MEXSUFFIX must be the suffix appended to the MEX files on your system. Again, see the
MathWorks website for details. The variable MEX must point to the mex executable located somewhere
within your MATLAB installation directory. The variable CXX must point to your C++ compiler. And
the variable PARDISOHOME must point to the directory where the PARDISO library is stored.
Note that it is crucial that your C++ compiler be compatible with your MATLAB installation. In
particular, the compiler must also be of the correct version. The MathWorks website keeps a detailed
list of MATLAB software package versions and the compilers that are compatible with them.
If you are not using the Makefile provided, to compile, say, the MEX File for the command pardisoinit,
you would need to make a call to the mex program that looks something like this:
mex -cxx CXX=g++ CC=g++ LD=g++ -L\home\lib\pardiso − lpardiso
-lmwlapack -lmwblas -lgfortran -lpthread -lm -output pardisoinit
common.cpp matlabmatrix.cpp sparsematrix.cpp pardisoinfo.cpp
pardisoinit.cpp
The flag -lgfortran links to a special library needed to handle the Fortran code for those that use the
gfortran compiler. For those that use a different compiler, you may need to link to a different library.
The option -lpthread may or may not be necessary. And, of course, depending on your system setup
you may need to include other flags such as -largeArrayDims.
6
Using the PARDISO Matrix Consistency Checker
The PARDISO 5.0.0 library comes with new routines that can be used to check the input data for
consistency. Given the input matrix A and the right-hand-side vecor or matrix B, the following tests are
carried out:
• the arrays (IA, JA), that describe the nonzero structure of the matrix A, is a valid CSR format with
index base 1
• The numerical values of A and B, given in A and B, are finite non-NaN values
All routines come in two flavors, one for real valued matrices and another one for complex valued
matrices, which carry the suffix Z. All arguments are exactly the same as for the main routine PARDISO,
see section 2 for detailed explanation. The only output argument for all routines is ERROR. If it carries
a non-zero value after calling any of the following routines, an error was found and an error message is
printed to STDERR.
For testing the input matrix A as described above, use the function PARDISO CHKMATRIX:
SUBROUTINE PARDISO_CHKMATRIX(MTYPE, N, A, IA, JA, ERROR)
SUBROUTINE PARDISO_CHKMATRIX_Z(MTYPE, N, A, IA, JA, ERROR)
For testing the RHS matrix B for infinite and NaN values, use the routine PARDISO CHKVEC:
SUBROUTINE PARDISO_CHKVEC(N, NRHS, B, ERROR)
SUBROUTINE PARDISO_CHKVEC_Z(N, NRHS, B, ERROR)
A routine that automatically invokes PARDISO CHKMATRIX and PARDISO CHKVEC is the following:
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
34
SUBROUTINE PARDISO_PRINTSTATS(MTYPE, N, A, IA, JA, NRHS, B, ERROR)
SUBROUTINE PARDISO_PRINTSTATS_Z(MTYPE, N, A, IA, JA, NRHS, B, ERROR)
In addition to performing the above mentioned tests, this routines prints the range of absolute coefficient
values to STDOUT. In many applications certain expectations for these ranges exist, so it may be useful to
compare them with the printed values.
The derived C API prototypes are as follows. To represent complex valued matrices in C, please refer
to the example pardiso unsym complex.c. In the following, we assume a structure
typedef struct {double re; double im;} doublecomplex;
void pardiso_chkmatrix
(int *mtype, int *n, double *a,
int *ia, int *ja, int *error);
void pardiso_chkmatrix_z (int *mtype, int *n, doublecomplex *a,
int *ia, int *ja, int *error);
void pardiso_chkvec
(int *n, int *nrhs, double *b,
int *error);
void pardiso_chkvec_z (int *n, int *nrhs, doublecomplex *b, int *error);
void pardiso_printstats (int *mtype, int *n, double *a,
int *ia,
int *ja, int *nrhs, double *b,
int *error);
void pardiso_printstats_z (int *mtype, int *n, doublecomplex *a, int *ia,
int *ja, int *nrhs, doublecomplex *b, int *error);
We recommend to execute the checking functionality prior to every call to PARDISO with new or
modified input data. Since only a few passes through the input data structures are needed in order to
carry out all the tests, the induced overhead will be negligible for most applications.
7
Authors
The authors Olaf Schenk (Department of Computer Science, University of Basel, Switzerland) and Klaus
Gärtner (Weierstrass Institute of Applied Analysis and Stochastics, Berlin, Germany) have a list of future
features that they would like to integrate into the solver. In case of errors, wishes, big surprises — it may
be sometimes possible to help.
8
Acknowledgments
Functionalities related to nonsymmetric maximal matching algorithms have been implemented by Stefan
Röllin (Integrated Systems Laboratory, Swiss Federal Institute of Technology, Zurich, Switzerland). We
are also grateful to Esmond Ng (Lawrence Berkeley National Laboratory, Berkeley, USA) for several
discussions on efficient sequential BLAS-3 supernodal factorization algorithms. We are also grateful to
George Karypis (Department of Computer Science, University of Minnesota, Minnesota, USA) for his fillin reducing METIS reordering package. Arno Liegmann (Integrated Systems Laboratory, Swiss Federal
Institute of Technology, Zurich, Switzerland) provided useful help at the start of the project.
We wish to thank Stefan Röllin (ETHZ), Michael Saunders (Stanford) for a providing very useful
contributions to this project. The Matlab interface to PARDISO has been developed by Peter Carbonetto
(UBC Vancouver). The PARDISO matrix consistency checking interface has been developed by Robert
Luce (TU Berlin). Thank you all for these contributions.
9
License
You shall acknowledge using references [12] and [15] the contribution of this package in any publication of material dependent upon the use of the package. You shall use reasonable endeavors to notify the authors of the package of this publication. More information can be obtained from http:
//www.pardiso-project.org
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
35
The software can be downloaded from the PARDISO page3 . A license file pardiso.lic is required for
the execution of the solver and it will be generated at the download area of the web page. The user must
supply information such as hostname (Unix: “hostname”) and the user identification (Unix: “whoami”).
This license file can be in the home directory of the user, or in the directory from which the user runs a
program linked with the PARDISO library. Alternatively, this file can be placed in a fixed location and
the environment variable PARDISO LIC PATH set to the path of its location.
References
[1] I. S. Duff and J. Koster. The design and use of algorithms for permuting large entries to the diagonal
of sparse matrices. SIAM J. Matrix Analysis and Applications, 20(4):889–901, 1999.
[2] G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular
graphs. SIAM Journal on Scientific Computing, 20(1):359–392, 1998.
[3] Andrey Kuzmin, Mathieu Luisier, and Olaf Schenk. Fast methods for computing selected elements
of the green’s function in massively parallel nanoelectronic device simulations. In Felix Wolf, Bernd
Mohr, and Dieter Mey, editors, Euro-Par 2013 Parallel Processing, volume 8097 of Lecture Notes in
Computer Science, pages 533–544. Springer Berlin Heidelberg, 2013.
[4] R. Menon L. Dagnum. OpenMP: An Industry-Standard API for Shared-Memory Programming.
IEEE Computational Science & Engineering, 1:46–55, 1998. http://www.openmp.org.
[5] X.S. Li and J.W. Demmel. A scalable sparse direct solver using static pivoting. In Proceeding of
the 9th SIAM conference on Parallel Processing for Scientic Computing, San Antonio, Texas, March
22-34,1999.
[6] J.W.H. Liu. Modification of the Minimum-Degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11(2):141–153, 1985.
[7] Cosmin G. Petra, Olaf Schenk, Miles Lubin, and Klaus Gärtner. An augmented incomplete factorization approach for computing the Schur complement in stochastic optimization. Technical report,
Preprint ANL/MCS-P4030-0113, Argonne National Laboratory, 2013.
[8] Y. Saad. Iterative Methods for Sparse Linear Systems. SIAMm 2nd edition, 2003.
[9] O. Schenk. Scalable Parallel Sparse LU Factorization Methods on Shared Memory Multiprocessors.
PhD thesis, ETH Zürich, 2000.
[10] O. Schenk, M. Bollhöfer, and R. A. Römer. On large scale diagonalization techniques for the Anderson model of localization. SIAM Review, 50(1):91–112, 2008. SIGEST Paper.
[11] O. Schenk and K. Gärtner. Two-level scheduling in PARDISO: Improved scalability on shared
memory multiprocessing systems. Parallel Computing, 28:187–197, 2002.
[12] O. Schenk and K. Gärtner. Solving unsymmetric sparse systems of linear equations with PARDISO.
Journal of Future Generation Computer Systems, 20(3):475–487, 2004.
[13] O. Schenk and K. Gärtner. On fast factorization pivoting methods for symmetric indefinite systems.
Elec. Trans. Numer. Anal, 23:158–179, 2006.
[14] O. Schenk and K. Gärtner. Sparse factorization with two-level scheduling in PARDISO. In Proceeding
of the 10th SIAM conference on Parallel Processing for Scientic Computing, Portsmouth, Virginia,
March 12-14, 2001.
[15] O. Schenk, K. Gärtner, and W. Fichtner. Efficient sparse LU factorization with left-right looking
strategy on shared memory multiprocessors. BIT, 40(1):158–176, 2000.
3 The
solver is available at http://www.pardiso-project.org
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
36
[16] P. Sonneveld. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM Journal on
Scientific and Statistical Computing, 10:36–52, 1989.
10
Further comments
Note 1: PARDISO does support since version 4.1 (January 2011) the solution of sparse symmetric
indefinite systems in a message-passing parallel environment. In addition, it is a thread-safe library and
the calling program can execute several parallel factorization simultaneously.
Note 2: PARDISO has no out-of-core capabilities.
Note 2: PARDISO uses a modified version of METIS-4.01 [2] and this version is included in the
libraries.
Note 3: Note that PARDISO needs the BLAS and LAPACK routines for the computational kernels.
On IBM platforms these routines are part of the IBM Engineering and Scientific Subroutine Library
(ESSL) and it is highly recommended that the user link with this thread-safe library. If this library
is not available on your machine, then the ATLAS BLAS (http://math-atlas.sourceforge.net) or
the GOTO BLAS (http://www.cs.utexas.edu/users/flame/goto) can be used. However, it is always
important that the serial and thread-safe BLAS and LAPACK library be linked because PARDISO
performs all the parallelization.
Note 4: Some technical papers related to the software and the example programs in the appendices
can be obtained from the site http://www.pardiso-project.org.
Note 5: Some remarks on the memory management of the solver. All memory is allocated in a few
steps based on the true demand and at the time of the first demand. The data that is used in the symbolic
analysis phase only is released at the termination of the symbolic analysis phase. The data controlling
the numeric phases is reallocated. Any memory, potentially used to solve a similar linear system again,
will be released only on explicit user demand by PHASE=( 0, or -1). Different pointer arrays PT can
store data related to different structures. A small fraction of the memory is growing proportional to
IPARM(3).
PARDISO memory is competing with later – with respect to the call tree – newly allocated memory
in total memory size used. For large linear systems memory is likely to be the limiting issue, hence the
user code should follow the strategy ’allocate as late and free as soon as possible’, too.
A
Examples for sparse symmetric linear systems
Appendices A.2 and A.3 give two examples (Fortran, C) for
PARDISO. They solve the system of equations Ax = b, where

7.0 0.0 1.0 0.0 0.0
2.0
7.0
 0.0 −4.0 8.0 0.0 2.0
0.0
0.0

 1.0 8.0 1.0 0.0 0.0
0.0
0.0

 0.0 0.0 0.0 7.0 0.0
0.0
9.0

A=
 0.0 2.0 0.0 0.0 5.0 −1.0 5.0
 2.0 0.0 0.0 0.0 −1.0 0.0
0.0

 7.0 0.0 0.0 9.0 5.0
0.0 11.0
0.0 0.0 5.0 0.0 0.0
5.0
0.0
A.1
solving symmetric linear systems with
0.0
0.0
5.0
0.0
0.0
5.0
0.0
5.0












 and b = 










Example results for symmetric systems
Upon successful execution of the solver, the result of the solution X is as follows:
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0






.





Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
[PARDISO]:
[PARDISO]:
[PARDISO]:
[PARDISO]:
[PARDISO]:
[PARDISO]:
37
License check was successful ...
Matrix type
: real symmetric
Matrix dimension : 8
Matrix non-zeros : 18
Abs. coeff. range: min 0.00e+00 max 1.10e+01
RHS no. 1: min 0.00e+00 max 7.00e+00
================
PARDISO: solving a symmetric indef.
system
================
Summary PARDISO 5.0.0: ( reorder to reorder )
=======================
Times:
======
Time
Time
Time
Time
Time
Time
fulladj:
reorder:
symbfct:
parlist:
malloc :
total :
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
s
s
s
s
s
s total - sum: 0.000000 s
Statistics:
===========
< Parallel Direct Factorization with #cores: >
<
and #nodes: >
< Numerical Factorization with Level-3 BLAS performance >
< Linear system Ax = b>
#equations:
#non-zeros in A:
non-zeros in A (%):
#right-hand sides:
8
18
28.125000
1
< Factors L and U >
#columns for each panel:
# of independent subgraphs:
< preprocessing with state of the art partitioning metis>
#supernodes:
size of largest supernode:
number of nonzeros in L
number of nonzeros in U
number of nonzeros in L+U
number of perturbed pivots
number of nodes in solve
mflop
for the numerical factorization:
Reordering completed ...
Number of nonzeros in factors = 30
Number of factorization MFLOPS = 0
================ PARDISO: solving a symmetric indef.
1
1
80
0
5
4
29
1
30
0
8
0.000075
system
================
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
38
Summary PARDISO 5.0.0: ( factorize to factorize )
=======================
Times:
======
Time
Time
Time
Time
A to LU:
numfct :
malloc :
total :
0.000000
0.000000
0.000000
0.000000
s
s
s
s total - sum: 0.000000 s
Statistics:
===========
< Parallel Direct Factorization with #cores: >
<
and #nodes: >
< Numerical Factorization with Level-3 BLAS performance >
< Linear system Ax = b>
#equations:
#non-zeros in A:
non-zeros in A (%):
#right-hand sides:
1
1
8
18
28.125000
1
< Factors L and U >
#columns for each panel:
# of independent subgraphs:
< preprocessing with state of the art partitioning metis>
#supernodes:
size of largest supernode:
number of nonzeros in L
number of nonzeros in U
number of nonzeros in L+U
number of perturbed pivots
number of nodes in solve
mflop
for the numerical factorization:
80
0
5
4
29
1
30
0
8
0.000075
Factorization completed ...
================
PARDISO: solving a symmetric indef.
system
Summary PARDISO 5.0.0: ( solve to solve )
=======================
Times:
======
Time solve
Time total
: 0.000000 s
: 0.000000 s total - sum: 0.000000 s
Statistics:
===========
< Parallel Direct Factorization with #cores: >
<
and #nodes: >
1
1
================
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
39
< Numerical Factorization with Level-3 BLAS performance >
< Linear system Ax = b>
#equations:
#non-zeros in A:
non-zeros in A (%):
#right-hand sides:
< Factors L and U >
#columns for each panel:
# of independent subgraphs:
< preprocessing with state of the art partitioning metis>
#supernodes:
size of largest supernode:
number of nonzeros in L
number of nonzeros in U
number of nonzeros in L+U
number of perturbed pivots
number of nodes in solve
mflop
for the numerical factorization:
8
18
28.125000
1
80
0
5
4
29
1
30
0
8
0.000075
Solve completed ...
The solution of the system is:
x(1) = -0.3168031420208231
x(2) = -0.4955685649709140
x(3) = -0.2129608358961057
x(4) =
5.6704583348771778E-002
x(5) =
0.8607062136425950
x(6) =
0.3140983363592574
x(7) =
0.4003408796176218
x(8) =
1.4988624995368485
A.2
Example pardiso sym.f for symmetric linear systems (including selected
inversion)
C ---------------------------------------------------------------------C
Example program to show the use of the " PARDISO " routine
C
for symmetric linear systems
C -------------------------------------------------------------------C
This program can be downloaded from the following site :
C
http :// www . pardiso - project . org
C
C ( C ) Olaf Schenk , Institute of Computational Science
C
Universita della Svizzera italiana , Lugano , Switzerland .
C
Email : olaf . [email protected] . ch
C -------------------------------------------------------------------PROGRAM pardiso_sym
IMPLICIT NONE
C ..
Internal solver memory pointer
INTEGER *8 pt (64)
C ..
All other variables
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
INTEGER
INTEGER
INTEGER
INTEGER
REAL *8
REAL *8
REAL *8
REAL *8
REAL *8
40
maxfct , mnum , mtype , phase , n , nrhs , error , msglvl
iparm (64)
ia (9)
ja (18)
dparm (64)
a (18)
b (8)
x (8)
y (8)
INTEGER i , j , idum , solver
REAL *8 waltime1 , waltime2 , ddum , normb , normr
C .. Fill all arrays containing matrix data .
DATA n /8/ , nrhs /1/ , maxfct /1/ , mnum /1/
DATA ia /1 ,5 ,8 ,10 ,12 ,15 ,17 ,18 ,19/
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
C
C
C
C
C
C
DATA ja
/1 ,
2,
6,
7,
5,
8,
4,
5,
6,
6,
7,
7,
8,
7,
8/
DATA a
/7. d0 ,
1. d0 ,
-4. d0 , 8. d0 ,
1. d0 ,
2. d0 , 7. d0 ,
2. d0 ,
7. d0 ,
5. d0 ,
9. d0 ,
5. d0 , -1. d0 , 5. d0 ,
0. d0 ,
5. d0 ,
11. d0 ,
5. d0 /
.. set right hand side
do i = 1 , n
b(i) = i
end do
.. Setup Pardiso control parameters und initialize the solvers
internal adress pointers . This is only necessary for the FIRST
call of the PARDISO solver .
mtype
solver
C
3,
3,
3,
= -2 ! unsymmetric matrix symmetric , indefinite
= 10 ! use sparse direct method
.. PARDISO license check and initialize solver
call pardisoinit ( pt , mtype , solver , iparm , dparm , error )
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
C
41
.. Numbers of Processors ( value of OMP_NUM_THREADS )
iparm (3) = 1
IF ( error . NE . 0) THEN
IF ( error . EQ . -10 ) WRITE (* ,*) ’ No license file found ’
IF ( error . EQ . -11 ) WRITE (* ,*) ’ License is expired ’
IF ( error . EQ . -12 ) WRITE (* ,*) ’ Wrong username or hostname ’
STOP
ELSE
WRITE (* ,*) ’[ PARDISO ]: License check was successful ... ’
END IF
c
c
c
.. pa rd is o_ ch k_ ma tr ix (...)
Checks the consistency of the given matrix .
Use this functionality only for debugging purposes
CALL par diso_c hkmatr ix ( mtype , n , a , ia , ja , error );
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
c ..
c
c
c
pardiso_chkvec (...)
Checks the given vectors for infinite and NaN values
Input parameters ( see PARDISO user manual for a description ):
Use this functionality only for debugging purposes
CALL pardiso_chkvec (n , nrhs , b , error );
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
c ..
c
c
pa rd is o_ pr in ts ta ts (...)
prints information on the matrix to STDOUT .
Use this functionality only for debugging purposes
CALL p ar di so _p ri nt st at s ( mtype , n , a , ia , ja , nrhs , b , error );
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
C ..
C
Reordering and Symbolic Factorization , This step also allocates
all memory that is necessary for the factorization
phase
= 11
msglvl
= 1
iparm (33) = 1
1
! only reordering and symbolic factorization
! with statistical information
! compute determinant
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , a , ia , ja ,
idum , nrhs , iparm , msglvl , ddum , ddum , error , dparm )
WRITE (* ,*) ’ Reordering completed ... ’
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
42
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
END IF
WRITE (* ,*) ’ Number of nonzeros in factors
WRITE (* ,*) ’ Number of factorization MFLOPS
= ’ , iparm (18)
= ’ , iparm (19)
C .. Factorization .
phase
= 22 ! only factorization
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , a , ia , ja ,
1
idum , nrhs , iparm , msglvl , ddum , ddum , error , dparm )
IF ( iparm (33). EQ .1) THEN
write (* ,*) ’ Log of determinant is
ENDIF
’,
dparm (33)
WRITE (* ,*) ’ Factorization completed ... ’
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
C .. Back substitution and iterative refinement
iparm (8) = 1
! max numbers of iterative refinement steps
phase
= 33 ! only solve
1
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , a , ia , ja ,
idum , nrhs , iparm , msglvl , b , x , error , dparm )
WRITE (* ,*) ’ Solve completed ...
’
WRITE (* ,*) ’ The solution of the system is ’
DO i = 1 , n
WRITE (* ,*) ’ x ( ’ ,i , ’) = ’ , x ( i )
END DO
C .. Residual test
normb = 0
normr = 0
CALL pardiso_residual ( mtype , n , a , ia , ja , b , x , y , normb , normr )
WRITE (* ,*) ’ The norm of the residual is ’ , normr / normb
C .. Selected Inversion
phase = -22
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , a , ia , ja ,
1
idum , nrhs , iparm , msglvl , b , x , error , dparm )
c ..
Print diagonal elements of the inverse of A = (a , ia , ja ) */
do i = 1 , n
j = ia ( i )
write (* ,*) ’ Diagonal ’ ,i , ’ - element of A ^{ -1} = ’ , a ( j )
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
43
end do
C .. Termination and release of memory
phase
= -1
! release internal memory
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , ddum , idum , idum ,
1
idum , nrhs , iparm , msglvl , ddum , ddum , error , dparm )
END
A.3
Example pardiso sym.c for symmetric linear systems (including selected
inversion)
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /*
Example program to show the use of the " PARDISO " routine
/*
on symmetric linear systems
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /*
This program can be downloaded from the following site :
/*
http :// www . pardiso - project . org
/*
/* ( C ) Olaf Schenk , Institute of Computational Science
/*
Universita della Svizzera italiana , Lugano , Switzerland .
/*
Email : olaf . [email protected] . ch
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # include < stdio .h >
# include < stdlib .h >
# include < math .h >
/* PARDISO prototype . */
void pardisoinit ( void
* , int
*,
int * , int * , double * , int *);
void pardiso
( void
* , int
*,
int * , int * ,
int * , int * ,
double * , int
*,
int * , int * ,
int * , int * ,
int * , double * , double * , int * , double *);
void par diso_ chkmat rix ( int * , int * , double * , int * , int * , int *);
void pardiso_chkvec
( int * , int * , double * , int *);
void p ar di so _p ri nt st at s ( int * , int * , double * , int * , int * , int * ,
double * , int *);
int main ( void )
{
/* Matrix data . */
int
n = 8;
int
ia [ 9] = { 0 , 4 , 7 , 9 , 11 , 14 , 16 , 17 , 18 };
int
ja [18] = { 0 ,
2,
5, 6,
1, 2,
4,
2,
7,
3,
6,
4, 5, 6,
5,
7,
6,
7 };
double a [18] = { 7.0 ,
1.0 ,
2.0 , 7.0 ,
*/
*/
*/
*/
*/
*/
*/
*/
*/
*/
*/
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
-4.0 , 8.0 ,
1.0 ,
2.0 ,
7.0 ,
int
int
nnz = ia [ n ];
mtype = -2;
44
5.0 ,
9.0 ,
5.0 , -1.0 , 5.0 ,
0.0 ,
5.0 ,
11.0 ,
5.0 };
/* Real symmetric matrix */
/* RHS and solution vectors . */
double
b [8] , x [8];
int
nrhs = 1;
/* Number of right hand sides . */
/* Internal solver memory pointer pt ,
/* 32 - bit : int pt [64]; 64 - bit : long int pt [64]
/* or void * pt [64] should be OK on both architectures
void
* pt [64];
*/
*/
*/
/* Pardiso control parameters . */
int
iparm [64];
double
dparm [64];
int
maxfct , mnum , phase , error , msglvl , solver ;
/* Number of processors . */
int
num_procs ;
/* Auxiliary variables . */
char
* var ;
int
i, k;
double
int
ddum ;
idum ;
/* Double dummy */
/* Integer dummy . */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. Setup Pardiso control parameters .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
error = 0;
solver =0; /* use sparse direct solver */
pardisoinit ( pt , & mtype , & solver , iparm , dparm , & error );
if ( error != 0)
{
if ( error == -10 )
printf ( " No license file found \ n " );
if ( error == -11 )
printf ( " License is expired \ n " );
if ( error == -12 )
printf ( " Wrong username or hostname \ n " );
return 1;
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
45
}
else
printf ( " [ PARDISO ]: License check was successful ... \ n " );
/* Numbers of processors , value of OMP_NUM_THREADS */
var = getenv ( " OMP_NUM_THREADS " );
if ( var != NULL )
sscanf ( var , " % d " , & num_procs );
else {
printf ( " Set environment OMP_NUM_THREADS to 1 " );
exit (1);
}
iparm [2] = num_procs ;
maxfct = 1;
mnum
= 1;
/* Maximum number of numerical factorizations .
/* Which factorization to use . */
msglvl = 1;
error = 0;
/* Print statistical information
/* Initialize error flag */
*/
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. Convert matrix from 0 - based C - notation to Fortran 1 - based
/*
notation .
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - for ( i = 0; i < n +1; i ++) {
ia [ i ] += 1;
}
for ( i = 0; i < nnz ; i ++) {
ja [ i ] += 1;
}
*/
*/
*/
*/
/* Set right hand side to i . */
for ( i = 0; i < n ; i ++) {
b [ i ] = i + 1;
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. par di so _c hk _m at ri x (...)
/*
Checks the consistency of the given matrix .
/*
Use this functionality only for debugging purposes
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
*/
*/
*/
*/
pard iso_ch kmatr ix (& mtype , &n , a , ia , ja , & error );
if ( error != 0) {
printf ( " \ nERROR in consistency of matrix : % d " , error );
exit (1);
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. pardiso_chkvec (...)
/*
Checks the given vectors for infinite and NaN values
/*
Input parameters ( see PARDISO user manual for a description ):
/*
Use this functionality only for debugging purposes
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
*/
*/
*/
*/
*/
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
46
pardiso_chkvec (& n , & nrhs , b , & error );
if ( error != 0) {
printf ( " \ nERROR in right hand side : % d " , error );
exit (1);
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. par di so _p ri nt st at s (...)
/*
prints information on the matrix to STDOUT .
/*
Use this functionality only for debugging purposes
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
*/
*/
*/
*/
pa rd is o_ pr in ts ta ts (& mtype , &n , a , ia , ja , & nrhs , b , & error );
if ( error != 0) {
printf ( " \ nERROR right hand side : % d " , error );
exit (1);
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. Reordering and Symbolic Factorization . This step also allocates
/*
all memory that is necessary for the factorization .
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - phase = 11;
*/
*/
*/
*/
pardiso ( pt , & maxfct , & mnum , & mtype , & phase ,
&n , a , ia , ja , & idum , & nrhs ,
iparm , & msglvl , & ddum , & ddum , & error , dparm );
if ( error != 0) {
printf ( " \ nERROR during symbolic factorization : % d " , error );
exit (1);
}
printf ( " \ nReordering completed ... " );
printf ( " \ nNumber of nonzeros in factors = % d " , iparm [17]);
printf ( " \ nNumber of factorization MFLOPS = % d " , iparm [18]);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. Numerical factorization .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
phase = 22;
iparm [32] = 1; /* compute determinant */
pardiso ( pt , & maxfct , & mnum , & mtype , & phase ,
&n , a , ia , ja , & idum , & nrhs ,
iparm , & msglvl , & ddum , & ddum , & error ,
dparm );
if ( error != 0) {
printf ( " \ nERROR during numerical factorization : % d " , error );
exit (2);
}
printf ( " \ nFactorization completed ...\ n " );
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
47
/* .. Back substitution and iterative refinement .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
phase = 33;
iparm [7] = 1;
/* Max numbers of iterative refinement steps . */
pardiso ( pt , & maxfct , & mnum , & mtype , & phase ,
&n , a , ia , ja , & idum , & nrhs ,
iparm , & msglvl , b , x , & error , dparm );
if ( error != 0) {
printf ( " \ nERROR during solution : % d " , error );
exit (3);
}
printf ( " \ nSolve completed ... " );
printf ( " \ nThe solution of the system is : " );
for ( i = 0; i < n ; i ++) {
printf ( " \ n x [% d ] = % f " , i , x [ i ] );
}
printf ( " \ n \ n " );
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* ... Inverse factorization .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
if ( solver == 0)
{
printf ( " \ nCompute Diagonal Elements of the inverse of A ... \ n " );
phase = -22;
iparm [35] = 1; /* no not overwrite internal factor L */
pardiso ( pt , & maxfct , & mnum , & mtype , & phase , &n , a , ia , ja , & idum , & nrhs ,
iparm , & msglvl , b , x , & error , dparm );
/* print diagonal elements */
for ( k = 0; k < n ; k ++)
{
int j = ia [ k ] -1;
printf ( " Diagonal element of A ^{ -1} = % d % d %32.24 e \ n " , k , ja [ j ] -1 , a [ j ]);
}
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. Convert matrix back to 0 - based C - notation .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
for ( i = 0; i < n +1; i ++) {
ia [ i ] -= 1;
}
for ( i = 0; i < nnz ; i ++) {
ja [ i ] -= 1;
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
48
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. Termination and release of memory .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
phase = -1;
/* Release internal memory . */
pardiso ( pt , & maxfct , & mnum , & mtype , & phase ,
&n , & ddum , ia , ja , & idum , & nrhs ,
iparm , & msglvl , & ddum , & ddum , & error ,
dparm );
return 0;
}
A.4
Example pardiso sym schur.cpp for Schur-complement computation for
symmetric linear systems
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /*
Example program to show the use of the " PARDISO " routine
/*
for evaluating the Schur - complement
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /*
This program can be downloaded from the following site :
/*
http :// www . pardiso - project . org
/*
/* ( C ) Drosos Kourounis , Institute of Computational Science
/*
Universita della Svizzera Italiana .
/*
Email : drosos . [email protected] . ch
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
*/
*/
*/
*/
*/
*/
*/
*/
*/
*/
// C ++ compatible
# include
# include
# include
# include
# include
< fstream >
< iostream >
< iomanip >
< cmath >
< stdlib .h >
using namespace std ;
/* PARDISO prototype . */
extern " C " void pardisoinit_d ( void
* , int
*,
int * , int * , double * , int *);
extern " C " void pardiso_d ( void
* , int
*,
int * , int * ,
int * , int * ,
double * , int
*,
int * , int * ,
int * , int * ,
int * , double * , double * , int * , double *);
extern " C " void pa r d i so _ c hk m a tr i x _d ( int * , int * , double * , int * , int * , int *);
extern " C " void pardiso_chkvec_d ( int * , int * , double * , int *);
extern " C " void p a r d i s o _ p r i n t s t a t s _ d ( int * , int * , double * , int * , int * , int * , doub
extern " C " void pa r d i so _ g et _ s ch u r _d ( void * , int * , int * , int * , double * , int * , int *);
void dumpCSR ( const char * filename , int n , int * ia , int * ja , double * a )
{
fstream fout ( filename , ios :: out );
fout << n << endl ;
fout << n << endl ;
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
fout << ia [ n ] << endl ;
for ( int i = 0; i <= n ; i ++)
{
fout << ia [ i ] << endl ;
}
for ( int i = 0; i < ia [ n ]; i ++)
{
fout << ja [ i ] << endl ;
}
for ( int i = 0; i < ia [ n ]; i ++)
{
fout << a [ i ] << endl ;
}
fout . close ();
}
void printCSR ( int n , int nnz , int * ia , int * ja , double * a )
{
cout << " rows : " << setw (10) << n << endl ;
cout << " nnz : " << setw (10) << nnz << endl ;
if ( nnz == n * n )
{
for ( int i = 0; i < n ; i ++)
{
for ( int j = 0; j < n ; j ++)
{
cout << setw (10) << a [ i * n + j ];
}
cout << endl ;
}
}
else
{
for ( int i = 0; i < n ; i ++)
{
for ( int index = ia [ i ]; index < ia [ i +1]; index ++)
{
int j = ja [ index ];
cout << setw (10) << " ( " << i << " , " << j << " ) " << a [ index ];
}
cout << endl ;
}
}
}
void shiftIndices ( int n , int nonzeros , int * ia , int * ja , int value )
{
49
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
50
int i ;
for ( i = 0; i < n +1; i ++)
{
ia [ i ] += value ;
}
for ( i = 0; i < nonzeros ; i ++)
{
ja [ i ] += value ;
}
}
int main ( void )
{
/* Matrix data .
*/
int
int
int
double
int
int
n =10;
ia [11] = { 0 ,
ja [34] = { 0 ,
9,
9,
6,
2,
3,
7,
a [34] = { 7.0 ,
nnz = ia [ n ];
mtype = -2;
11 , 15 , 19 , 22 , 26 , 29 , 32 , 33 , 34};
5, 6, 8, 9, 1, 2, 4, 8, 9, 2, 7, 8,
6, 8, 9, 4, 8, 9, 5, 7, 8, 9, 6, 8,
8 , 9 , 8 , 9};
1.0 ,
-4.0 , 8.0 ,
1.0 ,
2.0 , 7.0 ,
17.0 , 8.5 ,
6.0 , 3.0 ,
5.0 , 6.0 , 3.0 ,
7.0 ,
9.0 ,
16.0 , 8.0 ,
0.0 ,
-4.0 , -2.0 ,
3.0 ,
8.0 ,18.0 , 9.0 ,
11.0 ,
12.0 , 6.0 ,
5.0 , 4.0 , 2.0 ,
0.0 ,
0.0};
2.0 ,
/* Real unsymmetric matrix */
dumpCSR ( " m a t r i x _ s a m p l e _ m t y p e _ 2 . csr " , n , ia , ja , a );
printCSR (n , nnz , ia , ja , a );
/* Internal solver memory pointer pt ,
/* 32 - bit : int pt [64]; 64 - bit : long int pt [64]
/* or void * pt [64] should be OK on both architectures
void
* pt [64];
/* Pardiso control parameters . */
int
iparm [65];
double
dparm [64];
int
solver ;
int
maxfct , mnum , phase , error , msglvl ;
*/
*/
*/
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
51
/* Number of processors . */
int
num_procs ;
/* Auxiliary variables . */
char
* var ;
int
i;
double
int
ddum ;
idum ;
/* Double dummy */
/* Integer dummy . */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. Setup Pardiso control parameters and initialize the solvers
*/
/*
internal adress pointers . This is only necessary for the FIRST
*/
/*
call of the PARDISO solver .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
error = 0;
solver = 0; /* use sparse direct solver */
pardisoinit_d ( pt , & mtype , & solver , & iparm [1] , dparm , & error );
if ( error ! = 0)
{
if ( error == -10 )
printf ( " No license file found \ n " );
if ( error == -11 )
printf ( " License is expired \ n " );
if ( error == -12 )
printf ( " Wrong username or hostname \ n " );
return 1;
}
else
printf ( " [ PARDISO ]: License check was successful ... \ n " );
/* Numbers of processors , value of OMP_NUM_THREADS */
var = getenv ( " OMP_NUM_THREADS " );
if ( var ! = NULL )
sscanf ( var , " % d " , & num_procs );
else
{
printf ( " Set environment OMP_NUM_THREADS to 1 " );
exit (1);
}
iparm [3] = num_procs ;
iparm [11] = 1;
iparm [13] = 0;
maxfct = 1;
mnum
= 1;
/* Maximum number of numerical factorizations .
/* Which factorization to use . */
msglvl = 1;
error = 0;
/* Print statistical information
/* Initialize error flag */
*/
*/
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
52
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. Convert matrix from 0 - based C - notation to Fortran 1 - based
/*
notation .
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - shiftIndices (n , nnz , ia , ja , 1);
*/
*/
*/
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. par di so _c hk _m at ri x (...)
/*
Checks the consistency of the given matrix .
/*
Use this functionality only for debugging purposes
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - p ar d i so _ c hk m a tr i x _ d (& mtype , &n , a , ia , ja , & error );
if ( error ! = 0)
{
printf ( " \ nERROR in consistency of matrix : % d " , error );
exit (1);
}
*/
*/
*/
*/
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. Reordering and Symbolic Factorization . This step also allocates
/*
all memory that is necessary for the factorization .
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
*/
*/
*/
int nrows_S = 2;
phase
= 12;
iparm [38]
= nrows_S ;
int nb = 0;
pardiso_d ( pt , & maxfct , & mnum , & mtype , & phase ,
&n , a , ia , ja , & idum , & nb ,
& iparm [1] , & msglvl , & ddum , & ddum , & error ,
dparm );
if ( error ! = 0)
{
printf ( " \ nERROR during symbolic factorization : % d " , error );
exit (1);
}
printf ( " \ nReordering completed ...\ n " );
printf ( " Number of nonzeros in factors = % d \ n " , iparm [18]);
printf ( " Number of factorization MFLOPS = % d \ n " , iparm [19]);
printf ( " Number of nonzeros is
S
= % d \ n " , iparm [39]);
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. allocate memory for the Schur - complement and copy it there .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
int nonzeros_S = iparm [39];
int * iS
int * jS
= new int [ nrows_S +1];
= new int [ nonzeros_S ];
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
double * S
53
= new double [ nonzeros_S ];
p ar d i so _ g et _ s ch u r _ d ( pt , & maxfct , & mnum , & mtype , S , iS , jS );
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. Convert matrix from 1 - based Fortan notation to 0 - based C
/*
notation .
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - shiftIndices (n , nnz , ia , ja , -1);
*/
*/
*/
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /* .. Convert Schur complement from Fortran notation held internally
/*
to 0 - based C notation
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - shiftIndices ( nrows_S , nonzeros_S , iS , jS , -1);
*/
*/
*/
*/
printCSR ( nrows_S , nonzeros_S , iS , jS , S );
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* .. Termination and release of memory .
*/
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
phase = -1;
/* Release internal memory . */
pardiso_d ( pt , & maxfct , & mnum , & mtype , & phase ,
&n , & ddum , ia , ja , & idum , & idum ,
& iparm [1] , & msglvl , & ddum , & ddum , & error ,
dparm );
delete [] iS ;
delete [] jS ;
delete [] S ;
return 0;
}
A.5
Example iterative solver pardiso iter sym.f for symmetric linear systems
C ---------------------------------------------------------------------C
Example program to show the use of the " PARDISO " routine
C
for symmetric linear systems
C -------------------------------------------------------------------C
This program can be downloaded from the following site :
C
http :// www . pardiso - project . org
C
C ( C ) Olaf Schenk , Institute of Computational Science
C
Universita della Svizzera italiana , Lugano , Switzerland .
C
Email : olaf . [email protected] . ch
C -------------------------------------------------------------------PROGRAM pardiso_sym
IMPLICIT NONE
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
54
C ..
C ..
C ..
C ..
C ..
Internal solver memory pointer for 64 - bit architectures
INTEGER *8 pt (64)
Internal solver memory pointer for 32 - bit architectures
INTEGER *4 pt (64)
This is OK in both cases
INTEGER *8 pt (64)
C ..
All other variables
INTEGER maxfct , mnum , mtype , phase , n , nrhs , error , msglvl
INTEGER iparm (64)
INTEGER ia (9)
INTEGER ja (18)
REAL *8 dparm (64)
REAL *8 a (18)
REAL *8 b (8)
REAL *8 x (8)
INTEGER i , idum , solver
REAL *8 waltime1 , waltime2 , ddum
C .. Fill all arrays containing matrix data .
DATA n /8/ , nrhs /1/ , maxfct /1/ , mnum /1/
DATA ia /1 ,5 ,8 ,10 ,12 ,15 ,17 ,18 ,19/
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
C
C
C
C
C
DATA ja
/1 ,
2,
3,
3,
3,
6,
7,
5,
8,
4,
5,
6,
6,
7,
7,
8,
7,
8/
DATA a
/7. d0 ,
1. d0 ,
-4. d0 , 8. d0 ,
1. d0 ,
2. d0 , 7. d0 ,
2. d0 ,
7. d0 ,
5. d0 ,
9. d0 ,
5. d0 , 1. d0 , 5. d0 ,
0. d0 ,
5. d0 ,
11. d0 ,
5. d0 /
.. set right hand side
do i = 1 , n
b ( i ) = 1. d0
end do
.. Setup Pardiso control parameters und initialize the solvers
internal adress pointers . This is only necessary for the FIRST
call of the PARDISO solver .
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
55
C
mtype
solver
c
c
c
= -2
= 1
! unsymmetric matrix symmetric , indefinite
! use sparse direct method
.. pa rd is o_ ch k_ ma tr ix (...)
Checks the consistency of the given matrix .
Use this functionality only for debugging purposes
CALL par diso_c hkmatr ix ( mtype , n , a , ia , ja , error );
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
c ..
c
c
c
pardiso_chkvec (...)
Checks the given vectors for infinite and NaN values
Input parameters ( see PARDISO user manual for a description ):
Use this functionality only for debugging purposes
CALL pardiso_chkvec (n , nrhs , b , error );
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
c ..
c
c
pa rd is o_ pr in ts ta ts (...)
prints information on the matrix to STDOUT .
Use this functionality only for debugging purposes
CALL p ar di so _p ri nt st at s ( mtype , n , a , ia , ja , nrhs , b , error );
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
C
.. PARDISO license check and initialize solver
call pardisoinit ( pt , mtype , solver , iparm , dparm , error )
IF ( error . NE . 0) THEN
IF ( error . EQ . -10 ) WRITE (* ,*) ’ No license file found ’
IF ( error . EQ . -11 ) WRITE (* ,*) ’ License is expired ’
IF ( error . EQ . -12 ) WRITE (* ,*) ’ Wrong username or hostname ’
STOP
ELSE
WRITE (* ,*) ’[ PARDISO ]: License check was successful ... ’
END IF
C
.. Numbers of Processors ( value of OMP_NUM_THREADS )
iparm (3) = 1
phase
msglvl
1
= 12
= 1
! set up of the preconditioner
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , a , ia , ja ,
idum , nrhs , iparm , msglvl , ddum , ddum , error , dparm )
WRITE (* ,*) ’ Set up completed ... ’
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
56
IF ( error . NE . 0) THEN
WRITE (* ,*) ’ The following ERROR was detected : ’ , error
STOP
ENDIF
C
.. Solve with SQMR
phase
= 33 ! only solve
c
.. Set initial solution to zero
DO i = 1 , n
x(i) = 0
END DO
1
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , a , ia , ja ,
idum , nrhs , iparm , msglvl , b , x , error , dparm )
WRITE (* ,*) ’ Solve completed ...
’
WRITE (* ,*) ’ The solution of the system is ’
DO i = 1 , n
WRITE (* ,*) ’ x ( ’ ,i , ’) = ’ , x ( i )
END DO
C ..
B
Termination and release of memory
phase
= -1
! release internal memory
CALL pardiso ( pt , maxfct , mnum , mtype , phase , n , ddum , idum , idum ,
1
idum , nrhs , iparm , msglvl , ddum , ddum , error , dparm )
END
Examples for sparse symmetric linear systems on distributedmemory architectures
Appendices B.1 gives an example in C for solving symmetric indefinite linear systems with PARDISO on
distributed-memory architectures. Other example are available on the PARDISO web page.
B.1
Example pardiso sym mpp.c for the solution of symmetric linear systems
on distributed memory architecutes
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - /*
This program can be downloaded from the following site :
/*
http :// www . pardiso - project . org
/*
/* ( C ) Olaf Schenk , Institute of Computational Science
/*
Universita della Svizzera italiana , Lugano , Switzerland .
/*
Email : olaf . [email protected] . ch
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - static int mpi pardi so_dri ver (
int n ,
int nnz ,
int * ia ,
int * ja ,
double * va ,
*/
*/
*/
*/
*/
*/
*/
*/
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
57
double *b ,
MPI_Comm comm )
{
int mpi_rank ;
int mpi_size ;
MPI_Comm_rank ( comm , & mpi_rank );
MPI_Comm_size ( comm , & mpi_size );
if ( mpi_rank == 0)
{
/*
* Master process
*/
double
double
double
double
*x;
time_analyze = 0.0;
time_factorize = 0.0;
time_solve = 0.0;
void * handle [64];
int
iparam [64];
double dparam [64];
int ido ;
int nb = 1; /* Only one right hand side .
*/
int Nmax_system = 1;
int matrix_number = 1;
int matrix_type = -2; /* Sparse symmetric matrix is indefinite . */
int solver = 0;
/* Use sparse direct method */
int perm_user = 0;
/* Dummy pointer for user permutation .
*/
int msglvl = 1;
int ierror = 0;
/* Initialize the Pardiso error flag
*/
/*
* Setup Pardiso control parameters und initialize the solver ’s
* internal address pointers . This is only necessary for the FIRST
* call of the PARDISO solver .
*
* PARDISO license check and initialize solver
*/
memset ( dparam , 0 , sizeof ( dparam ));
memset ( handle , 0 , sizeof ( handle ));
pardisoinit_ ( handle , & matrix_type , & solver , iparam , dparam , & ierror );
if ( ierror )
return pardiso_error ( ierror , iparam );
IPARAM (3) = 8;
IPARAM (8) = 0;
IPARAM (51) = 1;
IPARAM (52) = mpi_size ;
/*
/*
/*
/*
maximal number of OpenMP tasks */
no iterative refinement */
use MPP factorization kernel */
number of nodes that we would like to use */
printf ( FMT_PARDISO_IPARAM , " IPARAM (3) " , IPARAM (3) ,
" Number of OpenMP tasks per host " );
printf ( FMT_PARDISO_IPARAM , " IPARAM (11) " , IPARAM (11) , " Scaling " );
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
58
printf ( FMT_PARDISO_IPARAM , " IPARAM (13) " , IPARAM (13) , " Matching " );
printf ( FMT_PARDISO_IPARAM , " IPARAM (52) " , IPARAM (52) , " Number of hosts " );
/* Additional initialization for PARDISO MPI solver */
p a rd i s o_ m p i_ i n it _ c _ ( handle , comm , & ierror );
if ( ierror )
return pardiso_error ( ierror , iparam );
/* Allocate work arrays */
x = malloc ( n * sizeof ( double ));
if ( x == NULL )
malloc_error ();
/* Analysis */
time_analyze -= seconds ();
/*
* Start symbolic factorization of the PARDISO solver .
*/
ido = 11; /* perform only symbolic factorization */
pardiso_ ( handle , & Nmax_system , & matrix_number , & matrix_type , & ido , &n ,
va , ia , ja , & perm_user , & nb , iparam , & msglvl , b , x , & ierror , dparam );
if ( ierror )
return pardiso_error ( ierror , iparam );
time_analyze += seconds ();
printf ( TIME_FMT , " time analyze " , time_analyze );
printf ( FMT_PARDISO_IPARAM , " IPARAM (18) " ,
" Number of LU nonzeros " );
printf ( FMT_PARDISO_IPARAM , " IPARAM (19) " ,
printf ( FMT_PARDISO_IPARAM , " IPARAM (15) " ,
printf ( FMT_PARDISO_IPARAM , " IPARAM (16) " ,
printf ( FMT_PARDISO_IPARAM , " IPARAM (17) " ,
IPARAM (18) ,
IPARAM (19) ,
IPARAM (15) ,
IPARAM (16) ,
IPARAM (17) ,
" Number of FLOPS " );
" Analysis memory " );
" Structure memory " );
" Factor memory " );
/*
* Start numerical factorization of the PARDISO solver .
*/
time_factorize = - seconds ();
ido = 22;
pardiso_ ( handle , & Nmax_system , & matrix_number , & matrix_type , & ido , &n ,
va , ia , ja , & perm_user , & nb , iparam , & msglvl , b , x , & ierror ,
dparam );
if ( ierror )
return pardiso_error ( ierror , iparam );
time_factorize += seconds ();
printf ( TIME_FMT , " time_factorize " , time_factorize );
printf ( FMT_PARDISO_IPARAM , " IPARAM (14) " , IPARAM (14) , " Perturbed pivots " );
printf ( FMT_PARDISO_IPARAM , " IPARAM (22) " , IPARAM (22) , " Positive pivots " );
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
59
printf ( FMT_PARDISO_IPARAM , " IPARAM (23) " , IPARAM (23) , " Negative pivots " );
printf ( FMT_PARDISO_DPARAM , " DPARAM (33) " , DPARAM (33) , " Determinant " );
/*
* Start forward / backward substitution of the PARDISO solver .
*/
time_solve = - seconds ();
ido = 33;
pardiso_ ( handle , & Nmax_system , & matrix_number , & matrix_type , & ido , &n ,
va , ia , ja , & perm_user , & nb , iparam , & msglvl , b , x , & ierror ,
dparam );
if ( ierror )
return pardiso_error ( ierror , iparam );
time_solve += seconds ();
printf ( TIME_FMT , " time_solve " , time_solve );
/*
* Compute relative residual
*/
SS P_ pr in t_ ac cu ra cy (n , ia , ja , va , b , x );
{
double norm_b = 0.0;
int i ;
for ( i = 0; i < n ; ++ i )
norm_b += b [ i ] * b [ i ];
norm_b = sqrt ( norm_b );
printf ( FMT_NORM , " norm ( b ) " , norm_b );
double norm_x = 0.0;
for ( i = 0; i < n ; ++ i )
norm_x += x [ i ] * x [ i ];
norm_x = sqrt ( norm_x );
printf ( FMT_NORM , " norm ( x ) " , norm_x );
}
/*
* Free PARDISO internal memory
*/
ido = -1;
pardiso_ ( handle , & Nmax_system , & matrix_number , & matrix_type , & ido , &n ,
va , ia , ja , & perm_user , & nb , iparam , & msglvl , b , x , & ierror ,
dparam );
if ( ierror )
return pardiso_error ( ierror , iparam );
/* Additional finalization for MPI solver */
p a r d i s o _ m p i _ f i n a l i z e _ ( handle , & ierror );
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
if ( ierror )
return pardiso_error ( ierror , iparam );
free ( x );
}
else /* mpi_rank != 0 */
{
/*
* Worker process
*/
void * handle [64];
int isolve ;
int ierror = 0;
/* Initialization for PARDISO MPI solver */
p a rd i s o_ m p i_ i n it _ c _ ( handle , comm , & ierror );
if ( ierror )
return 1;
/* Symbolic factorization */
p a rd i s o_ m p i_ w o rk e r _ ( handle , & ierror );
if ( ierror )
return 1;
/* Numerical factorization */
p a rd i s o_ m p i_ w o rk e r _ ( handle , & ierror );
if ( ierror )
return 1;
/* Triangular solve */
p a rd i s o_ m p i_ w o rk e r _ ( handle , & ierror );
if ( ierror )
return 1;
/* Free solver memory */
p a rd i s o_ m p i_ w o rk e r _ ( handle , & ierror );
/* Finalization for MPI solver */
p a r d i s o _ m p i _ f i n a l i z e _ ( handle , & ierror );
}
return 0;
}
C
Release notes
PARDIS0 - Release Notes - version 5.0.0
--------------------------------------PARDISO Contents
---------------PARDISO - a direct and iterative sparse linear solver library - is a
tuned math solver library designed for high performance on homogeneous
60
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
multicore machines and cluster of multicores including Intel Xeon, Intel
Itanium, AMD Opteron, and IBM Power processors, and includes both 32-bit
and 64-bit library versions. Different versions are available for Linux
MAC OSX, and Windows 64-bit operating systems.
A full suite of sparse direct linear solver routines for all kind of
sparse matrices is provided, and key datastructures have been designed
for high performance on various multicore processors and cluster of multicore
processors in 64-bit modes.
A selected suite of iterative linear solvers for real and complex symmetric
indefinite matrices is provided that takes advantage of a new algebraic
multilevel incomplete factorization that is especially designed for good
performance in very large-scale applications.
New Features
-----------New features of release PARDISO 5.0.0:
(o) Switch to host-free license for all 64-bit libraries. This allows all users
to use the PARDISO software within a cluster environment.
(o) Full support of multi-threaded Schur-complement computations for all
kind of matrices.
(o) Full support of multi-threaded parallel selected inversion to compute selected
entries of the inverse of A.
(o) Faster multi-threaded code for the solution of multiple right hand sides.
(o) Full support of 32-bit sequential and parallel factorizations for all kind
of matrices.
New features of release PARDISO 4.1.3:
(o) Bug fix in the computation of the log of the determinant.
New features of release PARDISO 4.1.2:
(o) Support of different licensing types
- Evaluation license for academic and commercial use.
- Academic license (host-unlocked, user-locked, 1 year).
- Commercial single-user license (host-unlocked, user-locked, 1 year).
- Commercial license (host-unlocked, user-unlocked, redistributable, 1 year).
New features of release PARDISO 4.1.0:
(o) New MPI-based numerical factorization and parallel forward/backward
substitution on
distributed-memory architectures for symmetric
indefinite matrices. PARDISO 4.1.0 has the unique feature among all
solvers that it can compute the exact bit-identical solution
on multicores and cluster of multicores. Here are some results for
a nonlinear FE model with 800’000 elements from automobile sheet
metal forming simulations.
CPUs per node:
4 x Intel(R) Xeon(R) CPU E5620 @ 2.40GHz (8 cores in total)
Memory per node:
12 GiB, Interconnect: Infiniband 4xQDR
61
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
(t_fact: factorization in
seconds, t_solve= solve in seconds)
PARDISO 4.1.0 (deterministic) :
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
t_fact | 1 core
4 cores
8 cores
---------+--------+---------+--------1 host | 92.032
23.312
11.966
2 hosts | 49.051
12.516
7.325
4 hosts | 31.646
8.478
5.018
t_solve | 1 core
4 cores
8 cores
---------+--------+---------+--------1 host | 2.188
0.767
0.545
2 hosts | 1.205
0.462
0.358
4 hosts | 0.856
0.513
0.487
Intel MKL 10.2 (non-deterministic):
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
t_fact | 1 core
4 cores
8 cores
--------+--------+---------+--------1 host | 94.566
27.266
14.018
t_solve | 1 core
4 cores
8 cores
---------+--------+---------+--------1 host | 2.223
2.183
2.207
- The MPI version is only available for academic research purposes.
(o) New host-unlimited licensing meachnism integrated into PARDISO 4.1.0.
We now have two different options available:
- a time-limited user-host locked license, and
- a time-limited user-locked (host-free) license.
(o) 32-bit sequential and parallel factorization and solve routines for
real unsymmetric matrices (matrix_type = 11). Mixed-precision refinement
can used for these 32-bit sparse direct factorizations.
(o) Additional routines that can check the input data (matrix, right-hand-side)
(contribution from Robert Luce, TU Berlin)
New features of release 4.0.0 of PARDISO since version 3.3.0:
(o) Due to the new features the interface to PARDISO and PARDISOINIT has
changed! This version is not backward compatible!
(o) Reproducibility of exact numerical results on multi-core architectures.
The solver is now able to compute the exact bit identical solution
independent on the number of cores without effecting the scalability.
Here are some results for a nonlinear FE model with 500’000 elements.
Intel MKL
1 core 2 cores 4 cores -
PARDISO 10.2
factor: 17.980 sec., solve: 1.13 sec.
factor: 9.790 sec., solve: 1.13 sec.
factor: 6.120 sec., solve: 1.05 sec.
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Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
8 cores - factor:
U
1
2
4
8
Basel
core
cores
cores
cores
3.830 sec., solve: 1.05 sec.
PARDISO 4.0.0:
- factor: 16.820
- factor: 9.021
- factor: 5.186
- factor: 3.170
sec.,
sec.,
sec.,
sec.,
solve:
solve:
solve:
solve:
1.09
0.67
0.53
0.43
sec.
sec.
sec.
sec.
This method is currently only working for symmetric indefinite matrices.
(o) 32-bit sequential and parallel factorization and solve routines for
real symmetric indefinite matrices, for symmetric complex matrices
and for structurally symmetric matrices. Mixed-precision refinement
is used for these 32-bit sparse direct factorizations.
(o) Internal 64-bit integer datastructures for the numerical factors allow
to solve very large sparse matrices with over 2^32 nonzeros in the sparse
direct factors.
(o) Work has been done to significantly improve the parallel performance of
the sparse direct solver which results in a much better scalability for
the numerical factorization and solve on multicore machines. At the same
time, the workspace memory requirements have been substantially reduced,
making the PARDISO direct routine better able to deal with large problem
sizes.
(o) Integration of a parallel multi-threaded METIS reordering that helps to
accelerate the reordering phase (Done by to Stefan Roellin, ETH Zurich)
(o) Integration of a highly efficient preconditioning method that is
based on a multi-recursive incomplete factorization scheme and
stabilized with a new graph-pivoting algorithm. The method have been
selected by the SIAM Journal of Scientific Computing as a very important
milestone in the area of new solvers for symmetric indefinite matrices
and the related paper appeared as a SIGEST SIAM Paper in 2008.
This preconditioner is highly effective for large-scale matrices with
millions of equations.
[1] O. Schenk, M. Bollhoefer, and R. Roemer, On large-scale
diagonalization techniques for the Anderson model of localization.
Featured SIGEST paper in the SIAM Review selected "on the basis of its
exceptional interest to the entire SIAM community".
SIAM Review 50 (2008), pp. 91--112.
(o) Support of 32-bit and 64-bit Windows operating systems (based on
Intel Professional Compiler Suite and the Intel MKL Performance
Library)
(o) A new extended interface to direct and iterative solver.
Double-precision parameters are passed by a dparm array to the solver.
The interface allow for greater flexibility in the storage of input
and output data within supplied arrays through the setting of increment
63
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
64
arguments.
(o) Note that the interface to PARDISO and PARDISOINIT has changed and that
this version is not backward compatible.
(o) Computation of the determinant for symmetric indefinite matrices.
(o) Solve A^Tx=b using the factorization of A.
(o) The solution process e.g. LUx=b can be performed in several phases that
the user can control.
(o) This version of PARDISO is compatible with the interior-point optimization
package IPOPT version 3.7.0
(o) A new matlab interface has been added that allows a flexible use of all
direct and iterative solvers.
Contributions
------------The following colleagues have contributed to the solver (in alphabetical order):
Peter Carbonetto
George Karypis
Arno Liegmann
Esmond Ng
Stefan Roellin
Michael Saunders
(UBC Vancouver, Canada).
(U Minnesota, US)
(ETHZ, Switzerland)
(LBNL, US)
(ETHZ, Switzerland)
(Stanford, US)
Applicability
------------Different PARDISO versions are provided for use with the GNU compilers
gfortran/gcc, with the Intel ifort compiler, and (for use under Solaris)
with the Sun f95/cc compilers.
Required runtime libraries under Microsoft Windows
-------------------------------------------------PARDISO version 4.0.0 and later link with the standard runtime library
provided by the Microsoft Visual Studio 2008 compilers. This requires
that the machine PARDISO runs on either has VS2K8 installed (or the Windows
SDK for Windows Server 2008), or the runtime libraries can be separately
downloaded from the appropriate Microsoft platform links provided below:
Visual Studio 2K8 Redist:
x86
<http://www.microsoft.com/downloads/details.aspx?FamilyID=9B2DA534-3E03-4391-8A4D-074B9F2BC1BF&displayla
x64
<http://www.microsoft.com/downloads/details.aspx?familyid=BD2A6171-E2D6-4230-B809-9A8D7548C1B6&displayla
Parallel Sparse Direct Solver PARDISO — User Guide Version 5.0.0
Bug Reports
----------Bugs should be reported to [email protected] with the string
"PARDISO-Support" in the subject line.
65
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