ssysvx(3)

NAME

SSYSVX - use the diagonal pivoting factorization to com

pute the solution to a real system of linear equations A * X = B,

SYNOPSIS

SUBROUTINE SSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF,  LDAF,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, IWORK, INFO
)
    CHARACTER      FACT, UPLO
    INTEGER        INFO, LDA, LDAF, LDB,  LDX,  LWORK,  N,
NRHS
    REAL           RCOND
    INTEGER        IPIV( * ), IWORK( * )
    REAL            A(  LDA, * ), AF( LDAF, * ), B( LDB, *
), BERR( * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE

SSYSVX uses the diagonal pivoting factorization to compute

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

where A is an N-by-N symmetric matrix and X and B are N-by-NRHS

matrices.

Error bounds on the solution and a condition estimate are

also provided.

DESCRIPTION

The following steps are performed:

1. If FACT = 'N', the diagonal pivoting method is used to

factor A.
The form of the factorization is
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',

where U (or L) is a product of permutation and unit up

per (lower)
triangular matrices, and D is symmetric and block diag

onal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then

the routine
returns with INFO = i. Otherwise, the factored form of

A is used
to estimate the condition number of the matrix A. If

the
reciprocal of the condition number is less than machine

precision,
INFO = N+1 is returned as a warning, but the routine

still goes on
to solve for X and compute error bounds as described

below.

3. The system of equations is solved for X using the fac

tored form
of A.

4. Iterative refinement is applied to improve the computed

solution
matrix and calculate error bounds and backward error

estimates
for it.

ARGUMENTS

FACT (input) CHARACTER*1

Specifies whether or not the factored form of A

has been supplied on entry. = 'F': On entry, AF and IPIV con

tain the factored form of A. AF and IPIV will not be modified.

= 'N': The matrix A will be copied to AF and factored.

UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': 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 matrices B and X. NRHS >= 0.

A (input) REAL array, dimension (LDA,N)The symmetric matrix A. If UPLO = 'U', the lead

ing N-by-N upper triangular part of A contains the upper triangu

lar part of the matrix A, and the strictly lower triangular part

of A is not referenced. If UPLO = 'L', 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 ref

erenced.

LDA (input) INTEGER
The leading dimension of the array A. LDA >=

max(1,N).

AF (input or output) REAL array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on

entry contains the block diagonal matrix D and the multipliers

used to obtain the factor U or L from the factorization A =

U*D*U**T or A = L*D*L**T as computed by SSYTRF.

If FACT = 'N', then AF is an output argument and

on exit returns the block diagonal matrix D and the multipliers

used to obtain the factor U or L from the factorization A =

U*D*U**T or A = L*D*L**T.

LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >=

max(1,N).

IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and

on entry contains details of the interchanges and the block

structure of D, as determined by SSYTRF. If IPIV(k) > 0, then

rows and columns k and IPIV(k) were interchanged and D(k,k) is a

1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) <

0, then rows and columns k-1 and -IPIV(k) were interchanged and

D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IP

IV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k)

were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = 'N', then IPIV is an output argument and

on exit contains details of the interchanges and the block struc

ture of D, as determined by SSYTRF.

B (input) REAL 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) REAL array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution

matrix X.

LDX (input) INTEGER
The leading dimension of the array X. LDX >=

max(1,N).

RCOND (output) REAL
The estimate of the reciprocal condition number of

the matrix A. If RCOND is less than the machine precision (in

particular, if RCOND = 0), the matrix is singular to working pre

cision. This condition is indicated by a return code of INFO >

0.

FERR (output) REAL array, dimension (NRHS)The estimated forward error bound for each solu

tion vector X(j) (the j-th column of the solution matrix X). If

XTRUE is the true solution corresponding to X(j), FERR(j) is an

estimated upper bound for the magnitude of the largest element in

(X(j) - XTRUE) divided by the magnitude of the largest element in

X(j). The estimate is as reliable as the estimate for RCOND, and

is almost always a slight overestimate of the true error.

BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each

solution vector X(j) (i.e., the smallest relative change in any

element of A or B that makes X(j) an exact solution).

WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal

LWORK.

LWORK (input) INTEGER
The length of WORK. LWORK >= 3*N, and for best

performance LWORK >= N*NB, where NB is the optimal blocksize for

SSYTRF.

If LWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the WORK array,

returns this value as the first entry of the WORK array, and no

error message related to LWORK is issued by XERBLA.

IWORK (workspace) INTEGER array, dimension (N)

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization

has been completed but the factor D is exactly singular, so the

solution and error bounds could not be computed. RCOND = 0 is re

turned. = N+1: D is nonsingular, but RCOND is less than machine

precision, meaning that the matrix is singular to working preci

sion. Nevertheless, the solution and error bounds are computed

because there are a number of situations where the computed solu

tion can be more accurate than the value of RCOND would suggest.

LAPACK version 3.0 15 June 2000

SSYSVX(3)