CGBSVX(l) ) CGBSVX(l)
NAME
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 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) 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 equili-
brated 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 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 (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 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.
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 com-
puted by CGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipli-
ers 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 factorization 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 factorization A = L*U of the origi-
nal matrix A.
If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equili-
brated 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 equilibration, i.e., A has been postmulti-
plied 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 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) 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 particular, 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 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) 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).
WORK (workspace) COMPLEX array, dimension (2*N)
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 estimator 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 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. 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.
LAPACK version 3.0 15 June 2000 CGBSVX(l)