pardiso

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 11/07/2023

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pardiso

Calculates the solution of a set of sparse linear equations with single or multiple right-hand sides.

Syntax

call pardiso (pt, maxfct, mnum, mtype, phase, n, a, ia, ja, perm, nrhs, iparm, msglvl, b, x, error)

Include Files

mkl.fi, mkl_pardiso.f90

Description

The pardiso routine calculates the solution of a set of sparse linear equations

A*X = B

with single or multiple right-hand sides, using a parallel LU, LDL, or LL^T factorization, where A is an n-by-n matrix, and X and B are n-by-nrhs vectors or matrices.

Notes:

This routine supports usage of the mkl_progress with OpenMP, TBB, and sequential threading. See mkl_progress for details. The case of iparm(24)=10 does not support this feature.
If iparm(27) is set to 1 (Matrix checker), Intel® oneAPI Math Kernel Library PARDISO uses the auxiliary routine sparse_matrix_checker to check integer arrays ia and ja. sparse_matrix_checker has its own set of error values (from 21 to 24) that are returned in the event of an unsuccessful matrix check. For more details, refer to the sparse_matrix_checker documentation.

Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex. Notice revision #20201201

Product and Performance Information

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.

Notice revision #20201201

Input Parameters

NOTE:

The types given for parameters in this section are specified in FORTRAN 77 notation. See Intel MKL PARDISO Parameters in Tabular Formfor detailed description of types of Intel® oneAPI Math Kernel Library (oneMKL) PARDISO parameters in Fortran 90 notation.

pt

INTEGER for 32-bit or 64-bit architectures

INTEGER*8 for 64-bit architectures

Array with size of 64.

Handle to internal data structure. The entries must be set to zero prior to the first call to pardiso. Unique for factorization.

CAUTION:

After the first call to pardiso do not directly modify pt, as that could cause a serious memory leak.

Use the pardiso_handle_store or pardiso_handle_store_64 routine to store the content of pt to a file. Restore the contents of pt from the file using pardiso_handle_restore or pardiso_handle_restore_64. Use pardiso_handle_store and pardiso_handle_restore with pardiso, and pardiso_handle_store_64 and pardiso_handle_restore_64 with pardiso_64.

maxfct

INTEGER

Maximum number of factors with identical sparsity structure that must be kept in memory at the same time. In most applications this value is equal to 1. It is possible to store several different factorizations with the same nonzero structure at the same time in the internal data structure management of the solver.

pardiso can process several matrices with an identical matrix sparsity pattern and it can store the factors of these matrices at the same time. Matrices with a different sparsity structure can be kept in memory with different memory address pointers pt.

mnum

INTEGER

Indicates the actual matrix for the solution phase. With this scalar you can define which matrix to factorize. The value must be: 1 ≤mnum≤maxfct.

In most applications this value is 1.

mtype

INTEGER

Defines the matrix type, which influences the pivoting method. The Intel® oneAPI Math Kernel Library (oneMKL) PARDISO solver supports the following matrices:

1: real and structurally symmetric
2: real and symmetric positive definite
-2: real and symmetric indefinite
3: complex and structurally symmetric
4: complex and Hermitian positive definite
-4: complex and Hermitian indefinite
6: complex and symmetric
11: real and nonsymmetric
13: complex and nonsymmetric

phase

INTEGER

Controls the execution of the solver. Usually it is a two- or three-digit integer. The first digit indicates the starting phase of execution and the second digit indicates the ending phase. Intel® oneAPI Math Kernel Library (oneMKL) PARDISO has the following phases of execution:

Phase 1: Fill-reduction analysis and symbolic factorization
Phase 2: Numerical factorization
Phase 3: Forward and Backward solve including optional iterative refinement

This phase can be divided into two or three separate substitutions: forward, backward, and diagonal (see Separate Forward and Backward Substitution).
Memory release phase (phase= 0 or phase= -1)

If a previous call to the routine has computed information from previous phases, execution may start at any phase. The phase parameter can have the following values:

phase: Solver Execution Steps
11: Analysis
12: Analysis, numerical factorization
13: Analysis, numerical factorization, solve, iterative refinement
22: Numerical factorization
23: Numerical factorization, solve, iterative refinement
33: Solve, iterative refinement
331: like phase=33, but only forward substitution
332: like phase=33, but only diagonal substitution (if available)
333: like phase=33, but only backward substitution
0: Release internal memory for L and U matrix number mnum
-1: Release all internal memory for all matrices

If iparm(36) = 0, phases 331, 332, and 333 perform this decomposition:

If iparm(36) = 2, phases 331, 332, and 333 perform a different decomposition:

You can supply a custom implementation for phase 332 instead of calling pardiso. For example, it can be implemented with dense LAPACK functionality. Custom implementation also allows you to substitute the matrix S with your own.

NOTE:

For very large Schur complement matrices use LAPACK functionality to compute the Schur complement vector instead of the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO phase 332 implementation.

n

INTEGER

Number of equations in the sparse linear systems of equations A*X = B. Constraint: n > 0.

a

DOUBLE PRECISION - for real types of matrices (mtype=1, 2, -2 and 11) and for double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=0)

REAL - for real types of matrices (mtype=1, 2, -2 and 11) and for single precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=1)

DOUBLE COMPLEX - for complex types of matrices (mtype=3, 6, 13, 14 and -4) and for double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=0)

COMPLEX - for complex types of matrices (mtype=3, 6, 13, 14 and -4) and for single precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=1)

Array. Contains the non-zero elements of the coefficient matrix A corresponding to the indices in ja. The coefficient matrix can be either real or complex. The matrix must be stored in the three-array variant of the compressed sparse row (CSR3) or in the three-array variant of the block compressed sparse row (BSR3) format, and the matrix must be stored with increasing values of ja for each row.

For CSR3 format, the size of a is the same as that of ja. Refer to the values array description in Three Array Variation of CSR Format for more details.

For BSR3 format the size of a is the size of ja multiplied by the square of the block size. Refer to the values array description in Three Array Variation of BSR Format for more details.

NOTE:

If you set iparm(37)to a negative value, Intel® oneAPI Math Kernel Library (oneMKL) PARDISO converts the data from CSR3 format to an internal variable BSR (VBSR) format. SeeSparse Data Storage.

ia

INTEGER

Array, size (n+1).

For CSR3 format, ia(i) (i≤n) points to the first column index of row i in the array ja. That is, ia(i) gives the index of the element in array a that contains the first non-zero element from row i of A. The last element ia(n+1) is taken to be equal to the number of non-zero elements in A, plus one. Refer to rowIndex array description in Three Array Variation of CSR Format for more details.

For BSR3 format, ia(i) (i≤n) points to the first column index of row i in the array ja. That is, ia(i) gives the index of the element in array a that contains the first non-zero block from row i of A. The last element ia(n+1) is taken to be equal to the number of non-zero blcoks in A, plus one. Refer to rowIndex array description in Three Array Variation of BSR Format for more details.

The array ia is accessed in all phases of the solution process.

Indexing of ia is one-based by default, but it can be changed to zero-based by setting the appropriate value to the parameter iparm(35).

ja

INTEGER

For CSR3 format, array ja contains column indices of the sparse matrix A. It is important that the indices are in increasing order per row. For structurally symmetric matrices it is assumed that all diagonal elements are stored (even if they are zeros) in the list of non-zero elements in a and ja. For symmetric matrices, the solver needs only the upper triangular part of the system as is shown for columns array in Three Array Variation of CSR Format.

For BSR3 format, array ja contains column indices of the sparse matrix A. It is important that the indices are in increasing order per row. For structurally symmetric matrices it is assumed that all diagonal blocks are stored (even if they are zeros) in the list of non-zero blocks in a and ja. For symmetric matrices, the solver needs only the upper triangular part of the system as is shown for columns array in Three Array Variation of BSR Format.

The array ja is accessed in all phases of the solution process.

Indexing of ja is one-based by default, but it can be changed to zero-based by setting the appropriate value to the parameter iparm(35).

perm

INTEGER

Array, size (n). Depending on the value of iparm(5) and iparm(31), holds the permutation vector of size n, specifies elements used for computing a partial solution, or specifies differing values of the input matrices for low rank update.

If iparm(5) = 1, iparm(31) = 0, and iparm(36) = 0, perm specifies the fill-in reducing ordering to the solver. Let A be the original matrix and C = P*A*P^T be the permuted matrix. Row (column) i of C is the perm(i) row (column) of A. The array perm is also used to return the permutation vector calculated during fill-in reducing ordering stage.

NOTE:

Be aware that setting iparm(5) = 1 prevents use of a parallel algorithm for the solve step.
If iparm(5) = 2, iparm(31) = 0, and iparm(36) = 0, the permutation vector computed in phase 11 is returned in the perm array.
If iparm(5) = 0, iparm(31) > 0, and iparm(36) = 0, perm specifies elements of the right-hand side to use or of the solution to compute for a partial solution.
If iparm(5) = 0, iparm(31) = 0, and iparm(36) > 0, perm specifies elements for a Schur complement.
If iparm(39) = 1, perm specifies values that differ in A for low rank update (see Low Rank Update). The size of the array must be at least 2*ndiff + 1, where ndiff is the number of values of A that are different. The values of perm should be:
perm = {ndiff, row_index1, column_index1, row_index2, column_index2, ...., row_index_ndiff, column_index_ndiff}
where row_index_m and column_index_m are the row and column indices of the m-th differing non-zero value in matrix A. The row and column index pairs can be in any order, but must use zero-based indexing regardless of the value of iparm(35).

See iparm(5), iparm(31), and iparm(39) for more details.

Indexing of perm is one-based by default, but unless iparm(39) = 1 it can be changed to zero-based by setting the appropriate value to the parameter iparm(35).

nrhs

INTEGER

Number of right-hand sides that need to be solved for.

iparm

INTEGER

Array, size (64). This array is used to pass various parameters to Intel® oneAPI Math Kernel Library (oneMKL) PARDISO and to return some useful information after execution of the solver.

See pardiso iparm Parameter for more details about the iparm parameters.

msglvl

INTEGER

Message level information. If msglvl = 0 then pardiso generates no output, if msglvl = 1 the solver prints statistical information to the screen.

b

DOUBLE PRECISION - for real types of matrices (mtype=1, 2, -2 and 11) and for double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=0)

REAL - for real types of matrices (mtype=1, 2, -2 and 11) and for single precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=1)

DOUBLE COMPLEX - for complex types of matrices (mtype=3, 6, 13, 14 and -4) and for double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=0)

COMPLEX - for complex types of matrices (mtype=3, 6, 13, 14 and -4) and for single precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=1)

Array, size (n, nrhs). On entry, contains the right-hand side vector/matrix B, which is placed in memory contiguously. The b(i+(k-1)×nrhs) element must hold the i-th component of k-th right-hand side vector. Note that b is only accessed in the solution phase.

Output Parameters

pt

Handle to internal data structure.

perm

See the Input Parameter description of the perm array.

iparm

On output, some iparm values report information such as the numbers of non-zero elements in the factors.

See pardiso iparm Parameter for more details about the iparm parameters.

b

On output, the array is replaced with the solution if iparm(6) = 1.

x

DOUBLE PRECISION - for real types of matrices (mtype=1, 2, -2 and 11) and for double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=0)

REAL - for real types of matrices (mtype=1, 2, -2 and 11) and for single precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=1)

DOUBLE COMPLEX - for complex types of matrices (mtype=3, 6, 13, 14 and -4) and for double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=0)

COMPLEX - for complex types of matrices (mtype=3, 6, 13, 14 and -4) and for single precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO (iparm(28)=1)

Array, size (n, nrhs). If iparm(6)=0 it contains solution vector/matrix X, which is placed contiguously in memory. The x(i+(k-1)× n) element must hold the i-th component of the k-th solution vector. Note that x is only accessed in the solution phase.

error

INTEGER

The error indicator according to the below table:

error: Information
0: no error
-1: input inconsistent
-2: not enough memory
-3: reordering problem
-4: Zero pivot, numerical factorization or iterative refinement problem. If the error appears during the solution phase, try to change the pivoting perturbation (iparm(10)) and also increase the number of iterative refinement steps. If it does not help, consider changing the scaling, matching and pivoting options (iparm(11), iparm(13), iparm(21))
-5: unclassified (internal) error
-6: reordering failed (matrix types 11 and 13 only)
-7: diagonal matrix is singular
-8: 32-bit integer overflow problem
-9: not enough memory for OOC
-10: error opening OOC files
-11: read/write error with OOC files
-12: (pardiso_64 only) pardiso_64 called from 32-bit library
-13: interrupted by the (user-defined) mkl_progress function
-15: internal error which can appear for iparm(24)=10 and iparm(13)=1. Try switch matching off (set iparm(13)=0 and rerun.)

Parent topic: oneMKL PARDISO - Parallel Direct Sparse Solver Interface

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

pardiso

Syntax

Include Files

Description

Input Parameters

Output Parameters