# ?pbsv

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

## Syntax

FORTRAN 77:

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 )

Fortran 95:

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

## Description

This 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*LT` (real flavors) and `A = L*LH` (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(ldab, *)`, `b(ldb,*)`. 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 first dimension of the array ab; ldab ≥ kd +1.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

## Output Parameters

ab

  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.

b

  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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine pbsv interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. 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: Driver Routines