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NAME
ssysvx - use the diagonal pivoting factorization to compute
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, * )
#include <sunperf.h>
void ssysvx(char fact, char uplo, int n, int nrhs, float
*sa, int lda, float *af, int ldaf, int *ipivot,
float *sb, int ldb, float *sx, int ldx, float
*srcond, float *ferr, float *berr, int *info);
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 upper
(lower) triangular matrices, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
2. 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, steps 3 and 4 are
skipped.
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 contain 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 triangular part of the matrix A, and the
strictly lower triangular part of A is not refer-
enced. 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 tri-
angular part of A is not referenced.
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 IPIV(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 structure 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, 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 precision. This
condition is indicated by a return code of INFO >
0, and the solution and error bounds are not com-
puted.
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 ele-
ment 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.
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 has been completed, but the block diagonal
matrix D is exactly singular, so the solution and
error bounds could not be computed. = N+1: the
block diagonal matrix D is nonsingular, but RCOND
is less than machine precision. The factorization
has been completed, but the matrix is singular to
working precision, so the solution and error
bounds have not been computed.
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