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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, * )
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 triangular
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 equilibration.
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 factors
given by R and C. AB, AFB, and IPIV are not modi-
fied. = 'N': The matrix A will be copied to AFB
and factored.
= 'E': The matrix A will be equilibrated if neces-
sary, 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 follows:
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.
(LDAFB,N)
AFB (input or output) DOUBLE PRECISION array, dimension
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 multi-
pliers 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 factoriza-
tion 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 factori-
zation 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 factori-
zation A = L*U of the equilibrated matrix A.
EQUED (input/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 premul-
tiplied by diag(R). = 'C': Column equilibration,
i.e., A has been postmultiplied by diag(C). = 'B':
Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C). EQUED is an
input variable if FACT = 'F'; otherwise, it is an
output variable.
R (input/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 vari-
able if FACT = 'F'; otherwise, R is an output vari-
able. If FACT = 'F' and EQUED = 'R' or 'B', each
element of R must be positive.
C (input/output) DOUBLE PRECISION 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 variable if FACT = 'F'; otherwise, C is an
output variable. If FACT = 'F' and EQUED = 'C' or
'B', each element of C must be positive.
(LDB,NRHS)
B (input/output) DOUBLE PRECISION array, dimension
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 'B', or
inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R'
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 partic-
ular, if RCOND = 0), the matrix is singular to work-
ing 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 bounds for each solution
vector X(j) (the j-th column of the solution matrix
X). If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j). The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.
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 entry of A or B that makes X(j) an
exact solution).
WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 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 solution and error bounds could not
be computed. = N+1: RCOND is less than machine pre-
cision. The factorization has been completed, but
the matrix A is singular to working precision, and
the solution and error bounds have not been com-
puted.