sgbsvx(3P) Sun Performance Library sgbsvx(3P)NAMEsgbsvx - use the LU factorization to compute the solution to a real
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
SYNOPSIS
SUBROUTINE SGBSVX(FACT, TRANSA, N, KL, KU, NRHS, A, LDA, AF,
LDAF, IPIVOT, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR,
BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, TRANSA, EQUED
INTEGER N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER IPIVOT(*), WORK2(*)
REAL RCOND
REAL A(LDA,*), AF(LDAF,*), R(*), C(*), B(LDB,*), X(LDX,*), FERR(*),
BERR(*), WORK(*)
SUBROUTINE SGBSVX_64(FACT, TRANSA, N, KL, KU, NRHS, A, LDA, AF,
LDAF, IPIVOT, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR,
BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, TRANSA, EQUED
INTEGER*8 N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER*8 IPIVOT(*), WORK2(*)
REAL RCOND
REAL A(LDA,*), AF(LDAF,*), R(*), C(*), B(LDB,*), X(LDX,*), FERR(*),
BERR(*), WORK(*)
F95 INTERFACE
SUBROUTINE GBSVX(FACT, [TRANSA], [N], KL, KU, [NRHS], A, [LDA],
AF, [LDAF], IPIVOT, EQUED, R, C, B, [LDB], X, [LDX],
RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
INTEGER :: N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: IPIVOT, WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: R, C, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: A, AF, B, X
SUBROUTINE GBSVX_64(FACT, [TRANSA], [N], KL, KU, [NRHS], A,
[LDA], AF, [LDAF], IPIVOT, EQUED, R, C, B, [LDB], X, [LDX],
RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
INTEGER(8) :: N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: R, C, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: A, AF, B, X
C INTERFACE
#include <sunperf.h>
void sgbsvx(char fact, char transa, int n, int kl, int ku, int nrhs,
float *a, int lda, float *af, int ldaf, int *ipivot, char
*equed, float *r, float *c, float *b, int ldb, float *x, int
ldx, float *rcond, float *ferr, float *berr, int *info);
void sgbsvx_64(char fact, char transa, long n, long kl, long ku, long
nrhs, float *a, long lda, float *af, long ldaf, long *ipivot,
char *equed, float *r, float *c, float *b, long ldb, float
*x, long ldx, float *rcond, float *ferr, float *berr, long
*info);
PURPOSEsgbsvx uses the LU factorization to compute the solution to a real sys‐
tem of linear equations A * X = B, A**T * X = B, or A**H * X = B, where
A is a band matrix of order N with KL subdiagonals and KU superdiago‐
nals, and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also pro‐
vided.
The following steps are performed by this subroutine:
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 = L * U,
where L is a product of permutation and unit lower triangular
matrices with KL subdiagonals, and U is upper triangular with
KL+KU superdiagonals.
3. If some U(i,i)=0, so that U 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.
4. The system of equations is solved for X using the factored 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 equilibration.
ARGUMENTS
FACT (input)
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
IPIVOT contain the factored form of A. If EQUED is not 'N',
the matrix A has been equilibrated with scaling factors given
by R and C. A, AF, and IPIVOT 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.
TRANSA (input)
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)
TRANSA is defaulted to 'N' for F95 INTERFACE.
N (input) The number of linear equations, i.e., the order of the matrix
A. N >= 0.
KL (input)
The number of subdiagonals within the band of A. KL >= 0.
KU (input)
The number of superdiagonals within the band of A. KU >= 0.
NRHS (input)
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 matrix A in band
storage, in rows 1 to KL+KU+1. The j-th column of A is
stored in the j-th column of the array A as follows:
A(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
If FACT = 'F' and EQUED is not 'N', then A must have been
equilibrated by the scaling factors in R and/or C. A is not
modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
'N' on exit.
On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED =
'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
LDA (input)
The leading dimension of the array A. LDA >= KL+KU+1.
AF (input or output)
REAL array, dimension (LDAF,N) If FACT = 'F', then AF is an
input argument and on entry contains details of the LU fac‐
torization of the band matrix A, as computed by SGBTRF. U is
stored as an upper triangular band matrix with KL+KU super‐
diagonals in rows 1 to KL+KU+1, and the multipliers used dur‐
ing the factorization are stored in rows KL+KU+2 to
2*KL+KU+1. 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 details of the LU factorization of A.
If FACT = 'E', then AF is an output argument and on exit
returns details of the LU factorization of the equilibrated
matrix A (see the description of A for the form of the equi‐
librated matrix).
LDAF (input)
The leading dimension of the array AF. LDAF >= 2*KL+KU+1.
IPIVOT (input or output)
INTEGER array, dimension (N) If FACT = 'F', then IPIVOT is an
input argument and on entry contains the pivot indices from
the factorization A = L*U as computed by SGBTRF; row i of the
matrix was interchanged with row IPIVOT(i).
If FACT = 'N', then IPIVOT is an output argument and on exit
contains the pivot indices from the factorization A = L*U of
the original matrix A.
If FACT = 'E', then IPIVOT is an output argument and on exit
contains the pivot indices from the factorization A = L*U of
the equilibrated matrix A.
EQUED (input or output)
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 equilibration, i.e., A has been
postmultiplied by diag(C). = 'B': Both row and column equi‐
libration, i.e., A has been replaced by diag(R) * A *
diag(C). EQUED is an input argument if FACT = 'F'; other‐
wise, 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 right hand
side matrix B. On exit, if EQUED = 'N', B is not modified;
if TRANSA = 'N' and EQUED = 'R' or 'B', B is overwritten by
diag(R)*B; if TRANSA = 'T' or 'C' and EQUED = 'C' or 'B', B
is overwritten by diag(C)*B.
LDB (input)
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 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 TRANSA = 'N' and EQUED = 'C' or 'B', or
inv(diag(R))*X if TRANSA = 'T' or 'C' and EQUED = 'R' or 'B'.
LDX (input)
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output)
The estimate of the reciprocal condition number of the matrix
A after equilibration (if done). If RCOND is less than the
machine precision (in particular, if RCOND = 0), the matrix
is singular to working precision. This condition is indi‐
cated by a return code of INFO > 0.
FERR (output)
The estimated forward error bound for each solution 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 esti‐
mated upper bound for the magnitude of the largest element in
(X(j) - XTRUE) divided by the magnitude of the largest ele‐
ment 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)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in any ele‐
ment of A or B that makes X(j) an exact solution).
WORK (workspace)
dimension(3*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 sta‐
bility of the LU factorization of the (equilibrated) matrix A
could be poor. This also means that the solution X, condition
estimator RCOND, and forward error bound FERR could be unre‐
liable. If factorization fails with 0<INFO<=N, then WORK(1)
contains the reciprocal pivot growth factor for the leading
INFO columns of A.
WORK2 (workspace)
dimension(N)
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal 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 solu‐
tion and error bounds could not be computed. RCOND = 0 is
returned. = N+1: U is nonsingular, but RCOND is less than
machine precision, meaning that the matrix is singular to
working precision. Nevertheless, the solution and error
bounds are computed because there are a number of situations
where the computed solution can be more accurate than the
value of RCOND would suggest.
6 Mar 2009 sgbsvx(3P)
