# ?posv

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

## Syntax

FORTRAN 77:

call sposv( uplo, n, nrhs, a, lda, b, ldb, info )

call dposv( uplo, n, nrhs, a, lda, b, ldb, info )

call cposv( uplo, n, nrhs, a, lda, b, ldb, info )

call zposv( uplo, n, nrhs, a, lda, b, ldb, info )

Fortran 95:

call posv( 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 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 matrix and L is a lower triangular matrix. 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`.

a, b

  REAL for sposv

  DOUBLE PRECISION for dposv

  COMPLEX for cposv

  DOUBLE COMPLEX for zposv.

  Arrays: `a(lda,*)`, `b(ldb,*)`. The array a contains the upper or the lower triangular part of the 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)`.

lda

  INTEGER. The first dimension of a; `lda ≥ max(1, n)`.

ldb

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

## Output Parameters

a

  If info = 0, the upper or lower triangular part of a is overwritten by the Cholesky factor U or L, as specified by uplo.

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 posv interface are as follows:

a

  Holds the matrix A of size (`n,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