?hesv

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?hesv

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

Syntax

call chesv( uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info )

call zhesv( uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info )

call hesv( a, b [,uplo] [,ipiv] [,info] )

Include Files

mkl.fi, lapack.f90

Description

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

The diagonal pivoting method is used to factor A as A = U*D*U^H or A = L*D*L^H, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

The factored form of A is then used to solve the system of equations A*X = B.

Input Parameters

uplo	CHARACTER1. Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored and how `A` is factored: If `uplo = 'U'`, the array `a` stores the upper triangular part of the matrix `A`, and `A` is factored as `UDU^H`. If `uplo = 'L'`, the array `a` stores the lower triangular part of the matrix `A`, and `A` is factored as `LD*L^H`.
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, work	COMPLEX for chesv DOUBLE COMPLEX for zhesv. Arrays: `a`(size `lda` by ), `bb(size ldb by )`, `work(*)`. The array `a` contains the upper or the lower triangular part of the Hermitian matrix `A` (see `uplo`). The second dimension of `a` must be at least `max(1, n)`. 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)`. `work` is a workspace array, dimension at least `max(1,lwork)`.
lda	INTEGER. The leading dimension of `a`; `lda≥ max(1, n)`.
ldb	INTEGER. The leading dimension of `b`; `ldb≥ max(1, n)`.
lwork	INTEGER. The size of the `work` array (`lwork`≥ 1). If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the `work` array, returns this value as the first entry of the `work` array, and no error message related to `lwork` is issued by xerbla. See Application Notes below for details and for the suggested value of `lwork`.

Output Parameters

a	If `info = 0`, `a` is overwritten by the block-diagonal matrix `D` and the multipliers used to obtain the factor `U` (or `L`) from the factorization of `A` as computed by ?hetrf.
b	If `info = 0`, `b` is overwritten by the solution matrix `X`.
ipiv	INTEGER. Array, size at least `max(1, n)`. Contains details of the interchanges and the block structure of `D`, as determined by ?hetrf. If `ipiv(i) = k > 0`, then `d_ii` is a 1-by-1 diagonal block, and the `i`-th row and column of `A` was interchanged with the `k`-th row and column. If `uplo = 'U'`and `ipiv`(`i`) =`ipiv`(`i`-1) = -`m` < 0, then `D` has a 2-by-2 block in rows/columns `i` and `i-1`, and `(i-1)`-th row and column of `A` was interchanged with the `m`-th row and column. If `uplo = 'L'`and `ipiv`(`i`) =`ipiv`(`i`+1) = -`m` < 0, then `D` has a 2-by-2 block in rows/columns `i` and `i+1`, and `(i+1)-`th row and column of `A` was interchanged with the `m`-th row and column.
`work(1)`	If `info` = 0, on exit `work(1)` contains the minimum value of `lwork` required for optimum performance. Use this `lwork` for subsequent runs.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, `d_ii` is 0. The factorization has been completed, but `D` is exactly singular, so the solution could not be 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 hesv interface are as follows:

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

Application Notes

For better performance, try using lwork = n*blocksize, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: LAPACK Linear Equation Driver Routines

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