ZGESVX(l) LAPACK driver routine (version 1.1) ZGESVX(l)
NAME
ZGESVX - use the LU factorization to compute the solution to a complex sys-
tem of linear equations A * X = B,
SYNOPSIS
SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R,
C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO
)
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK( *
)
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
PURPOSE
ZGESVX uses the LU factorization to compute the solution to a complex sys-
tem of linear equations
A * X = B, where A is an N-by-N matrix 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:
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 = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
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 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 FACT = 'E' and 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 sup-
plied on entry, and if not, whether the matrix A should be equili-
brated before it is factored. = 'F': On entry, AF and IPIV con-
tain 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 IPIV 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.
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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the
matrices B and X. NRHS >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A. 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
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) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry contains
the factors L and U from the factorization A = P*L*U as computed by
ZGETRF. 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
the factors L and U from the factorization A = P*L*U of the origi-
nal matrix A.
If FACT = 'E', then AF is an output argument and on exit returns
the factors L and U from the factorization A = P*L*U of the equili-
brated matrix A (see the description of A for the form of the
equilibrated matrix).
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
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 = P*L*U as computed by
ZGETRF; 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 = P*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 factorization A = P*L*U of the equili-
brated 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 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
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 multi-
plied on the left by diag(R); if EQUED = 'N' or 'C', R is not
accessed. R is an input variable if FACT = 'F'; otherwise, R is an
output variable. If FACT = 'F' and EQUED = 'R' or 'B', each ele-
ment 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 multi-
plied 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 ele-
ment of C must be positive.
B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N-by-NRHS 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*16 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 particular, 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 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 accurate, 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (2*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 com-
pleted, 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 is singular to working precision, and the solution and error
bounds have not been computed.
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