# ?hpsv

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

## Syntax

FORTRAN 77:

call chpsv( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call zhpsv( uplo, n, nrhs, ap, ipiv, b, ldb, info )

Fortran 95:

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

## Description

This routine solves for X the system of linear equations `A*X = B`, where A is an n-by-n Hermitian matrix stored in packed format, 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*UH` or `A = L*D*LH`, 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

  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`.

ap, b

  COMPLEX for chpsv

  DOUBLE COMPLEX for zhpsv.

  Arrays: `ap(*)`, `b(ldb,*)`.

  The dimension of ap must be at least `max(1,n(n+1)/2)`. The array ap contains the factor U or L, as specified by uplo, in packed storage (see Matrix Storage Schemes).

  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)`.

ldb

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

## Output Parameters

ap

  The block-diagonal matrix D and the multipliers used to obtain the factor U (or L) from the factorization of A as computed by ?hptrf, stored as a packed triangular matrix in the same storage format as A.

b

  If info = 0, b is overwritten by the solution matrix X.

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`. Contains details of the interchanges and the block structure of D, as determined by ?hptrf.

  If `ipiv(i) = k > 0`, then `dii` is a 1-by-1 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.

info

  INTEGER. If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

  If `info = i`, `dii` is 0. The factorization has been completed, but D is exactly singular, so the solution could not be 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 hpsv interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size (`n*(n+1)/2`).

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: Driver Routines