?pbsv

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?pbsv

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

Syntax

call spbsv( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call dpbsv( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call cpbsv( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call zpbsv( uplo, n, kd, nrhs, ab, ldab, b, ldb, info )

call pbsv( ab, 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 band 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 band matrix and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A*X = B.

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.
kd	INTEGER. The number of superdiagonals of the matrix `A` if `uplo = 'U'`, or the number of subdiagonals if `uplo = 'L';kd`≥ 0.
nrhs	INTEGER. The number of right-hand sides, the number of columns in `B; nrhs≥ 0`.
ab, b	REAL for spbsv DOUBLE PRECISION for dpbsv COMPLEX for cpbsv DOUBLE COMPLEX for zpbsv. Arrays: `ab`(size `ldab` by ), `b`(size `ldb` by ). The array `ab` contains the upper or the lower triangular part of the matrix `A` (as specified by `uplo`) in band storage (see Matrix Storage Schemes). The second dimension of `ab` 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)`.
ldab	INTEGER. The leading dimension of the array `ab`; `ldab`≥`kd` +1.
ldb	INTEGER. The leading dimension of `b`; `ldb≥ max(1, n)`.

Output Parameters

The upper or lower triangular part of A (in band storage) is overwritten by the Cholesky factor U or L, as specified by uplo, in the same storage format as A.

Overwritten by the solution matrix X.

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 pbsv interface are as follows:

ab	Holds the array `A` of size (`kd+1,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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