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NAME
sgesvx - use the LU factorization to compute the solution to
a real system of linear equations A * X = B,
SYNOPSIS
SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF,
IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR,
BERR, WORK, IWORK, INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * ), IWORK( * )
REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR( * ), C(
* ), FERR( * ), R( * ), WORK( * ), X( LDX, * )
#include <sunperf.h>
void sgesvx(char fact, char trans, int n, int nrhs, float
*sa, int lda, float *af, int ldaf, int *ipivot,
char *equed, float *r, float *sc, float *sb, int
ldb, float *sx, int ldx, float *srcond, float
*ferr, float *berr, int *info);
PURPOSE
SGESVX uses the LU factorization to compute the solution to
a real system of linear equations A * X = B, where A is an
N-by-N 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 = 'E', real scaling factors are computed to
equilibrate the system:
TRANS = 'N':
diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T':
(diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C':
(diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether or not the system will be equilibrated depends on
the scaling of the matrix A, but if equilibration is used, A
is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if
TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangu-
lar matrix, and U is upper triangular.
3. 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 4-6 are
skipped.
4. The system of equations is solved for X using the fac-
tored form of A.
5. Iterative refinement is applied to improve the computed
solution matrix and calculate error bounds and backward
error estimates for it.
6. If equilibration was used, the matrix X is premultiplied
by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or
'C') so that it solves the original system before equilibra-
tion.
ARGUMENTS
FACT (input) CHARACTER*1
Specifies whether or not the factored form of the
matrix A is supplied on entry, and if not, whether
the matrix A should be equilibrated before it is
factored. = 'F': On entry, AF and IPIV contain
the factored form of A. If EQUED is not 'N', the
matrix A has been equilibrated with scaling fac-
tors given by R and C. A, AF, and IPIV are not
modified. = 'N': The matrix A will be copied to
AF and factored.
= 'E': The matrix A will be equilibrated if
necessary, then copied to AF and factored.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Transpose)
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/output) REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. If FACT = 'F' and
EQUED is not 'N', then A must have been equili-
brated by the scaling factors in R and/or C. A is
not modified if FACT = 'F' or
On exit, if EQUED .ne. 'N', A is scaled as fol-
lows: EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
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 factors L and U from the fac-
torization A = P*L*U as computed by SGETRF. If
EQUED .ne. 'N', then AF is the factored form of
the equilibrated matrix A.
If FACT = 'N', then AF is an output argument and
on exit returns the factors L and U from the fac-
torization A = P*L*U of the original matrix A.
If FACT = 'E', then AF is an output argument and
on exit returns the factors L and U from the fac-
torization A = P*L*U of the equilibrated matrix A
(see the description of A for the form of the
equilibrated matrix).
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 the pivot indices from the fac-
torization A = P*L*U as computed by SGETRF; row i
of the matrix was interchanged with row IPIV(i).
If FACT = 'N', then IPIV is an output argument and
on exit contains the pivot indices from the fac-
torization A = P*L*U of the original matrix A.
If FACT = 'E', then IPIV is an output argument and
on exit contains the pivot indices from the
factorization A = P*L*U of the equilibrated matrix
A.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT =
'N').
= 'R': Row equilibration, i.e., A has been
premultiplied by diag(R). = 'C': Column equili-
bration, i.e., A has been postmultiplied by
diag(C). = 'B': Both row and column equilibra-
tion, i.e., A has been replaced by diag(R) * A *
diag(C). EQUED is an input argument if FACT =
'F'; otherwise, it is an output argument.
R (input or output) REAL array, dimension (N)
The row scale factors for A. If EQUED = 'R' or
'B', A is multiplied on the left by diag(R); if
EQUED = 'N' or 'C', R is not accessed. R is an
input argument if FACT = 'F'; otherwise, R is an
output argument. If FACT = 'F' and EQUED = 'R' or
'B', each element of R must be positive.
C (input or output) REAL array, dimension (N)
The column scale factors for A. If EQUED = 'C' or
'B', A is multiplied on the right by diag(C); if
EQUED = 'N' or 'R', C is not accessed. C is an
input argument if FACT = 'F'; otherwise, C is an
output argument. If FACT = 'F' and EQUED = 'C' or
'B', each element of C must be positive.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if EQUED = 'N', B is not modified; if
TRANS = 'N' and EQUED = 'R' or 'B', B is overwrit-
ten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED
= 'C' or 'B', B is overwritten by diag(C)*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 to
the original system of equations. Note that A and
B are modified on exit if EQUED .ne. 'N', and the
solution to the equilibrated system is
inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or
or 'B'.
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 after equilibration (if done). If
RCOND is less than the machine precision (in par-
ticular, 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 computed.
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 (4*N)
On exit, WORK(1) contains the reciprocal pivot
growth factor norm(A)/norm(U). The "max absolute
element" norm is used. If WORK(1) is much less
than 1, then the stability of the LU factorization
of the (equilibrated) matrix A could be poor. This
also means that the solution X, condition estima-
tor RCOND, and forward error bound FERR could be
unreliable. If factorization fails with 0<INFO<=N,
then WORK(1) contains the reciprocal pivot growth
factor for the leading INFO columns of A.
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: U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, so the solution and error bounds could
not be computed. = N+1: RCOND is less than
machine precision. The factorization has been
completed, but the matrix is singular to working
precision, and the solution and error bounds have
not been computed.
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