CGBSVX - 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 CGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB,
AFB, LDAFB, IPIV, EQUED, R, C, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N,
NRHS
REAL RCOND
INTEGER IPIV( * )
REAL BERR( * ), C( * ), FERR( * ), R( * ),
RWORK( * )
COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,
* ), WORK( * ), X( LDX, * )
PURPOSE
CGBSVX 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 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
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 triangu
lar 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 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 premulti
plied 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
= 'E': The matrix A will be equilibrated if nec
essary, 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 (Conjugate 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) COMPLEX 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) COMPLEX array, dimension
(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 CGBTRF. 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
.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 CGBTRF; 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 pre
multiplied by diag(R). = 'C': Column equilibra
tion, 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 argument if FACT = 'F'; otherwise, 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
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) COMPLEX 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) 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
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) REAL
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.
FERR (output) REAL 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 corre
sponding 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 reli
able as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) REAL 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).
RWORK (workspace/output) REAL array, dimension (N)
On exit, RWORK(1) contains the reciprocal pivot
growth factor norm(A)/norm(U). The "max absolute
element" norm is used. If RWORK(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 RWORK(1) contains the reciprocal pivot growth
factor for the leading INFO columns of A.
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. 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 num
ber of situations where the computed solution can
be more accurate than the value of RCOND would
suggest.
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