sgbsvx
sgbsvx - 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,
SUBROUTINE SGBSVX( FACT, TRANSA, N, NSUB, NSUPER, NRHS, A, LDA, AF,
* LDAF, IPIVOT, EQUED, ROWSC, COLSC, B, LDB, X, LDX, RCOND, FERR,
* BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, TRANSA, EQUED
INTEGER N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER IPIVOT(*), WORK2(*)
REAL RCOND
REAL A(LDA,*), AF(LDAF,*), ROWSC(*), COLSC(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE SGBSVX_64( FACT, TRANSA, N, NSUB, NSUPER, NRHS, A, LDA,
* AF, LDAF, IPIVOT, EQUED, ROWSC, COLSC, B, LDB, X, LDX, RCOND,
* FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, TRANSA, EQUED
INTEGER*8 N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER*8 IPIVOT(*), WORK2(*)
REAL RCOND
REAL A(LDA,*), AF(LDAF,*), ROWSC(*), COLSC(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE GBSVX( FACT, [TRANSA], [N], NSUB, NSUPER, [NRHS], A, [LDA],
* AF, [LDAF], IPIVOT, EQUED, ROWSC, COLSC, B, [LDB], X, [LDX],
* RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
INTEGER :: N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: IPIVOT, WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: ROWSC, COLSC, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: A, AF, B, X
SUBROUTINE GBSVX_64( FACT, [TRANSA], [N], NSUB, NSUPER, [NRHS], A,
* [LDA], AF, [LDAF], IPIVOT, EQUED, ROWSC, COLSC, B, [LDB], X, [LDX],
* RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
INTEGER(8) :: N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: ROWSC, COLSC, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: A, AF, B, X
#include <sunperf.h>
void sgbsvx(char fact, char transa, int n, int nsub, int nsuper, int nrhs, float *a, int lda, float *af, int ldaf, int *ipivot, char equed, float *rowsc, float *colsc, 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 nsub, long nsuper, long nrhs, float *a, long lda, float *af, long ldaf, long *ipivot, char equed, float *rowsc, float *colsc, float *b, long ldb, float *x, long ldx, float *rcond, float *ferr, float *berr, long *info);
sgbsvx uses 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,
where A is a band matrix of order N with KL subdiagonals and KU
superdiagonals, and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also
provided.
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.
-
* 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.
-
* TRANSA (input)
-
Specifies the form of the system of equations.
-
* N (input)
-
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
-
* NSUB (input)
-
The number of subdiagonals within the band of A. NSUB >= 0.
-
* NSUPER (input)
-
The number of superdiagonals within the band of A. NSUPER >= 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)
-
On entry, the matrix A in band storage, in rows 1 to NSUB+NSUPER+1.
The j-th column of A is stored in the j-th column of the
array A as follows:
A(NSUPER+1+i-j,j) = A(i,j) for max(1,j-NSUPER)<=i<=min(N,j+kl)
If FACT = 'F' and EQUED is not 'N', then A must have been
equilibrated by the scaling factors in ROWSC and/or COLSC. 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 = 'ROWSC': A := diag(ROWSC) * A
EQUED = 'COLSC': A := A * diag(COLSC)
EQUED = 'B': A := diag(ROWSC) * A * diag(COLSC).
-
* LDA (input)
-
The leading dimension of the array A. LDA >= NSUB+NSUPER+1.
-
* AF (input/output)
-
If FACT = 'F', then AF is an input argument and on entry
contains details of the LU factorization of the band matrix
A, as computed by SGBTRF. U is stored as an upper triangular
band matrix with NSUB+NSUPER superdiagonals in rows 1 to NSUB+NSUPER+1,
and the multipliers used during the factorization are stored
in rows NSUB+NSUPER+2 to 2*NSUB+NSUPER+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
equilibrated matrix).
-
* LDAF (input)
-
The leading dimension of the array AF. LDAF >= 2*NSUB+NSUPER+1.
-
* IPIVOT (input)
-
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)
-
Specifies the form of equilibration that was done.
-
* ROWSC (input/output)
-
The row scale factors for A. If EQUED = 'ROWSC' or 'B', A is
multiplied on the left by diag(ROWSC); if EQUED = 'N' or 'COLSC', ROWSC
is not accessed. ROWSC is an input argument if FACT = 'F';
otherwise, ROWSC is an output argument. If FACT = 'F' and
EQUED = 'ROWSC' or 'B', each element of ROWSC must be positive.
-
* COLSC (input/output)
-
The column scale factors for A. If EQUED = 'COLSC' or 'B', A is
multiplied on the right by diag(COLSC); if EQUED = 'N' or 'ROWSC', COLSC
is not accessed. COLSC is an input argument if FACT = 'F';
otherwise, COLSC is an output argument. If FACT = 'F' and
EQUED = 'COLSC' or 'B', each element of COLSC must be positive.
-
* B (input/output)
-
On entry, the right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if TRANSA = 'N' and EQUED = 'ROWSC' or 'B', B is overwritten by
diag(ROWSC)*B;
if TRANSA = 'T' or 'COLSC' and EQUED = 'COLSC' or 'B', B is
overwritten by diag(COLSC)*B.
-
* LDB (input)
-
The leading dimension of the array B. LDB >= max(1,N).
-
* X (output)
-
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(COLSC))*X if TRANSA = 'N' and
EQUED = 'COLSC' or 'B', or inv(diag(ROWSC))*X if TRANSA = 'T' or 'COLSC'
and EQUED = 'ROWSC' 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
indicated 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 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)
-
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)
-
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
estimator 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.
-
* WORK2 (workspace)
-
-
* INFO (output)
-