?posv

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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?posv

Computes the solution to the system of linear equations with a symmetric or Hermitian positive-definite coefficient matrix A and multiple right-hand sides.

Syntax

call sposv( uplo, n, nrhs, a, lda, b, ldb, info )

call dposv( uplo, n, nrhs, a, lda, b, ldb, info )

call cposv( uplo, n, nrhs, a, lda, b, ldb, info )

call zposv( uplo, n, nrhs, a, lda, b, ldb, info )

call dsposv( uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, iter, info )

call zcposv( uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, rwork, iter, info )

call posv( a, b [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n symmetric/Hermitian positive-definite matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

The Cholesky decomposition is used to factor A as

A = U^T*U (real flavors) and A = U^H*U (complex flavors), if uplo = 'U'

or A = L*L^T (real flavors) and A = L*L^H (complex flavors), if uplo = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A*X = B.

The dsposv and zcposv are mixed precision iterative refinement subroutines for exploiting fast single precision hardware. They first attempt to factorize the matrix in single precision (dsposv) or single complex precision (zcposv) and use this factorization within an iterative refinement procedure to produce a solution with double precision (dsposv) / double complex precision (zcposv) normwise backward error quality (see below). If the approach fails, the method switches to a double precision or double complex precision factorization respectively and computes the solution.

The iterative refinement is not going to be a winning strategy if the ratio single precision/complex performance over double precision/double complex performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ilaenv in the future. At present, iterative refinement is implemented.

The iterative refinement process is stopped if
iter > itermax
or for all the right-hand sides:
rnmr < sqrt(n)*xnrm*anrm*eps*bwdmax,
where

iter is the number of the current iteration in the iterative refinement process
rnmr is the infinity-norm of the residual
xnrm is the infinity-norm of the solution
anrm is the infinity-operator-norm of the matrix A
eps is the machine epsilon returned by dlamch (‘Epsilon’).

The values itermax and bwdmax are fixed to 30 and 1.0d+00 respectively.

Input Parameters

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored: If `uplo = 'U'`, the upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
n	INTEGER. The order of matrix `A`; n≥ 0.
nrhs	INTEGER. The number of right-hand sides, the number of columns in `B; nrhs≥ 0`.
a, b	REAL for sposv DOUBLE PRECISION for dposv and dsposv. COMPLEX for cposv DOUBLE COMPLEX for zposv and zcposv. Arrays: a(size lda,), b(ldb, ). The array a contains the upper or the lower triangular part of the matrix `A` (see uplo). The second dimension of a must be at least `max(1, n)`. Note that in the case of zcposv the imaginary parts of the diagonal elements need not be set and are assumed to be zero. The array b contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least `max(1,nrhs)`.
lda	INTEGER. The leading dimension of a; `lda≥ max(1, n)`.
ldb	INTEGER. The leading dimension of b; `ldb≥ max(1, n)`.
ldx	INTEGER. The leading dimension of the array `x`; `ldx≥ max(1, n)`.
work	DOUBLE PRECISION for dsposv DOUBLE COMPLEX for zcposv. Workspace array, size `(n*nrhs)`. This array is used to hold the residual vectors.
swork	REAL for dsgesv COMPLEX for zcgesv. Workspace array, size `(n*(n+nrhs))`. This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
rwork	DOUBLE PRECISION. Workspace array, size (`n`).

Output Parameters

a	If `info` = 0, the upper or lower triangular part of `a` is overwritten by the Cholesky factor `U` or `L`, as specified by `uplo`. If iterative refinement has been successfully used (info= 0 and iter≥ 0), then `A` is unchanged. If double precision factorization has been used (info= 0 and iter < 0), then the array `A` contains the factors `L` or `U` from the Cholesky factorization.
b	Overwritten by the solution matrix `X`.
x	DOUBLE PRECISION for dsposv DOUBLE COMPLEX for zcposv. Array, size `ldx` by `nrhs`. If info = 0, contains the `n`-by-`nrhs` solution matrix `X`.
iter	INTEGER. If `iter < 0`: iterative refinement has failed, double precision factorization has been performed If `iter = -1`: the routine fell back to full precision for implementation- or machine-specific reason If `iter = -2`: narrowing the precision induced an overflow, the routine fell back to full precision If `iter = -3`: failure of spotrf for dsposv, or cpotrf for zcposv If `iter = -31`: stop the iterative refinement after the 30th iteration. If `iter > 0`: iterative refinement has been successfully used. Returns the number of iterations.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, the leading minor of order `i` (and therefore the matrix `A` itself) is not positive definite, so the factorization could not be completed, and the solution has not been computed.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine posv interface are as follows:

a	Holds the matrix `A` of size (`n,n`).
b	Holds the matrix `B` of size (`n,nrhs`).
uplo	Must be 'U' or 'L'. The default value is 'U'.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?posv

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

See Also