# dsposv (l) - Linux Man Pages

## NAME

DSPOSV - computes the solution to a real system of linear equations A * X = B,

## SYNOPSIS

SUBROUTINE DSPOSV(
UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,

+ SWORK, ITER, INFO )

CHARACTER UPLO

INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS

REAL SWORK( * )

DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),

+ X( LDX, * )

## PURPOSE

DSPOSV computes the solution to a real system of linear equations
B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices.
DSPOSV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below). If the approach fails the method switches to a DOUBLE PRECISION factorization and solve.
The iterative refinement is not going to be a winning strategy if the ratio SINGLE PRECISION performance over DOUBLE PRECISION performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement.
The iterative refinement process is stopped if

ITER ITERMAX
or for all the RHS we have:

RNRM SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where

o ITER is the number of the current iteration in the iterative
refinement process

o RNRM is the infinity-norm of the residual

o XNRM is the infinity-norm of the solution

o ANRM is the infinity-operator-norm of the matrix A

o EPS is the machine epsilon returned by DLAMCH(aqEpsilonaq) The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
respectively.

## ARGUMENTS

UPLO (input) CHARACTER
= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A (input or input/ouptut) DOUBLE PRECISION array,
dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = aqUaq, 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 = aqLaq, 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. On exit, if iterative refinement has been successfully used (INFO.EQ.0 and ITER.GE.0, see description below), then A is unchanged, if double precision factorization has been used (INFO.EQ.0 and ITER.LT.0, see description below), then the array A contains the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (N*NRHS)
This array is used to hold the residual vectors.
SWORK (workspace) REAL array, dimension (N*(N+NRHS))
This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
ITER (output) INTEGER
< 0: iterative refinement has failed, double precision factorization has been performed -1 : the routine fell back to full precision for implementation- or machine-specific reasons -2 : narrowing the precision induced an overflow, the routine fell back to full precision -3 : failure of SPOTRF
-31: stop the iterative refinement after the 30th iterations > 0: iterative refinement has been sucessfully used. Returns the number of iterations
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i of (DOUBLE PRECISION) A is not positive definite, so the factorization could not be completed, and the solution has not been computed. =========