?hesv_rook

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 3/22/2024

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?hesv_rook

Computes the solution to the system of linear equations for Hermitian matrices using the bounded Bunch-Kaufman diagonal pivoting method.

Syntax

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

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

call hesv_rook( 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 Hermitian matrix, and X and B are n-by-nrhs matrices.

The bounded Bunch-Kaufman ("rook") diagonal pivoting method is used to factor A as

A = U*D*U^H if uplo = 'U', or

A = L*D*L^H if uplo = 'L',

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.

hetrf_rook is called to compute the factorization of a complex Hermition matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.

The factored form of A is then used to solve the system of equations A*X = B by calling ?HETRS_ROOK, which uses BLAS level 2 routines.

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 array `a` stores the upper triangular part of the matrix `A`. If `uplo = 'L'`, the array `a` stores the lower triangular part of the matrix `A`.
n	INTEGER. The number of linear equations, which is 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_rook COMPLEX16 for zhesv_rook. Arrays: `a`(size `lda` by ), `b`(size `ldb` by ), `work()`. The array a contains the Hermitian matrix `A`. If uplo = `'U'`, the leading n-by-n upper triangular part of a contains the upper triangular part of the matrix `A`, and the strictly lower triangular part of a is not referenced. If uplo = 'L', the leading n-by-n lower triangular part of a contains the lower triangular part of the matrix `A`, and the strictly upper triangular part of a is not referenced. The second dimension of a must be at least max(1, n). The array `b` contains the n-by-nrhs right hand side matrix `B`. 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_rook.
b	If `info = 0`, `b` is overwritten by the n-by-nrhs 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_rook. If uplo = 'U': If `ipiv(k) > 0`, rows and columns `k` and `ipiv(k)` were interchanged and `D`_{k, k} is a 1-by-1 diagonal block. If `ipiv(k) < 0` and `ipiv(k - 1) < 0`, rows and columns `k` and `-ipiv(k)` were interchanged, rows and columns `k` - 1 and `-ipiv(k - 1)` were interchanged, and `D`_{k - 1:k, k - 1:k} is a 2-by-2 diagonal block. If uplo = 'L': If `ipiv(k) > 0`, rows and columns `k` and `ipiv(k)` were interchanged and `D`_{k, k} is a 1-by-1 diagonal block. If `ipiv(k) < 0` and `ipiv(k + 1) < 0`, rows and columns `k` and `-ipiv(k)` were interchanged, rows and columns `k` + 1 and `-ipiv(k + 1)` were interchanged, and `D`_{k:k + 1, k:k + 1} is a 2-by-2 diagonal block.
`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_rook 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'.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?hesv_rook

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

See Also