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# zgesvx

```
NAME
ZGESVX - use the LU factorization to compute the solution to
a complex system 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 system 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 tri-
angular
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 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 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 supplied on entry, and if not, whether
the matrix A should be equilibrated before it is
factored.  = 'F':  On entry, AF 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.  A, AF, and IPIV are not modified.  =
'N':  The matrix A will be copied to AF and fac-
tored.
= 'E':  The matrix A will be equilibrated if neces-
sary, 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 '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).

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 factori-
zation 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 factoriza-
tion A = P*L*U of the original matrix A.

If FACT = 'E', then AF is an output argument and on
exit returns the factors L and U from the factoriza-
tion A = P*L*U of the equilibrated 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 factoriza-
tion 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 factori-
zation A = P*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.

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 '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) 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 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 is singular to working precision, and the
solution and error bounds have not been computed.
```