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NAME
dgbsvx - 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 DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB,
AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
RCOND, FERR, BERR, WORK, IWORK, INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
), BERR( * ), C( * ), FERR( * ), R( * ), WORK( *
), X( LDX, * )
#include <sunperf.h>
void dgbsvx(char fact, char trans, int n, int kl, int ku,
int nrhs, double *dab, int ldab, double *afb, int
ldafb, int *ipivot, char *equed, double *r, double
*dc, double *db, int ldb, double *dx, int ldx,
double *drcond, double *ferr, double *berr, int
*info);
PURPOSE
DGBSVX 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.
DESCRIPTION
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 tri-
angular matrices with KL subdiagonals, and U is upper tri-
angular with KL+KU superdiagonals.
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, AFB 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. AB, AFB, and IPIV are not
modified. = 'N': The matrix A will be copied to
AFB and factored.
= 'E': The matrix A will be equilibrated if
necessary, then copied to AFB 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.
KL (input) INTEGER
The number of subdiagonals within the band of A.
KL >= 0.
KU (input) INTEGER
The number of superdiagonals within the band of A.
KU >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number
of columns of the matrices B and X. NRHS >= 0.
AB (input/output) DOUBLE PRECISION array, dimension
(LDAB,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 AB as follows:
AB(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. AB 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 fol-
lows: EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >=
KL+KU+1.
AFB (input or output) DOUBLE PRECISION array, dimen-
sion (LDAFB,N)
If FACT = 'F', then AFB is an input argument and
on entry contains details of the LU factorization
of the band matrix A, as computed by DGBTRF. 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 AFB is the factored form of the
equilibrated matrix A.
If FACT = 'N', then AFB is an output argument and
on exit returns details of the LU factorization of
A.
If FACT = 'E', then AFB is an output argument and
on exit returns details of the LU factorization of
the equilibrated matrix A (see the description of
AB for the form of the equilibrated matrix).
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >=
2*KL+KU+1.
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 = L*U as computed by DGBTRF; 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 = 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 fac-
torization A = 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) DOUBLE PRECISION array, dimen-
sion (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) DOUBLE PRECISION array, dimen-
sion (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) DOUBLE PRECISION array, dimension
(LDB,NRHS)
On entry, the right hand side matrix B. On exit,
if EQUED = 'N', B is not modified; if TRANS = 'N'
and EQUED = 'R' or 'B', B is overwritten 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) DOUBLE PRECISION 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) DOUBLE PRECISION
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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimen-
sion (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 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 A is singular to working
precision, and the solution and error bounds have
not been computed.
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