zgbsvx(3P) Sun Performance Library zgbsvx(3P)NAMEzgbsvx - use the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
SYNOPSIS
SUBROUTINE ZGBSVX(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
DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER IPIVOT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION R(*), C(*), FERR(*), BERR(*), WORK2(*)
SUBROUTINE ZGBSVX_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
DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER*8 N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER*8 IPIVOT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION R(*), C(*), FERR(*), BERR(*), WORK2(*)
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
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
INTEGER :: N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: IPIVOT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: R, C, FERR, BERR, WORK2
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
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
INTEGER(8) :: N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: IPIVOT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: R, C, FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void zgbsvx(char fact, char transa, int n, int kl, int ku, int nrhs,
doublecomplex *a, int lda, doublecomplex *af, int ldaf, int
*ipivot, char *equed, double *r, double *c, doublecomplex *b,
int ldb, doublecomplex *x, int ldx, double *rcond, double
*ferr, double *berr, int *info);
void zgbsvx_64(char fact, char transa, long n, long kl, long ku, long
nrhs, doublecomplex *a, long lda, doublecomplex *af, long
ldaf, long *ipivot, char *equed, double *r, double *c, dou‐
blecomplex *b, long ldb, doublecomplex *x, long ldx, double
*rcond, double *ferr, double *berr, long *info);
PURPOSEzgbsvx uses the LU factorization to compute the solution to a complex
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 super‐
diagonals, 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 (Conjugate 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)
DOUBLE COMPLEX 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)
DOUBLE COMPLEX array, dimension (LDAF,N) 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 ZGBTRF.
U is stored as an upper triangular band matrix with KL+KU
superdiagonals in rows 1 to KL+KU+1, and the multipliers used
during 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 ZGBTRF; 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)
DOUBLE PRECISION 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 argu‐
ment. If FACT = 'F' and EQUED = 'R' or 'B', each element of
R must be positive.
C (input or output)
DOUBLE PRECISION array, dimension (N) The column scale fac‐
tors 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 out‐
put argument. If FACT = 'F' and EQUED = 'C' or 'B', each
element of C must be positive.
B (input/output)
DOUBLE COMPLEX 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 over‐
written 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)
DOUBLE COMPLEX 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 sys‐
tem 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(2*N)
WORK2 (workspace)
dimension(N) On exit, WORK2(1) contains the reciprocal pivot
growth factor norm(A)/norm(U). The "max absolute element"
norm is used. If WORK2(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 WORK2(1)
contains the reciprocal pivot growth factor for the leading
INFO columns of A.
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 zgbsvx(3P)
