cgesvx.f man page

cgesvx.f —

Synopsis

Functions/Subroutines

subroutine cgesvx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CGESVX computes the solution to system of linear equations A * X = B for GE matrices

Function/Subroutine Documentation

subroutine cgesvx (characterFACT, characterTRANS, integerN, integerNRHS, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, characterEQUED, real, dimension( * )R, real, dimension( * )C, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, complex, dimension( * )WORK, real, dimension( * )RWORK, integerINFO)

CGESVX computes the solution to system of linear equations A * X = B for GE matrices

Purpose:

CGESVX 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 triangular
   matrix, and U is upper triangular.

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.

Parameters:

FACT

FACT is 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 factored.
= 'E':  The matrix A will be equilibrated if necessary, then
        copied to AF and factored.

TRANS

TRANS is 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

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A.  N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X.  NRHS >= 0.

A

A is COMPLEX 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

LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).

AF

AF is COMPLEX 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 CGETRF.  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 original 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 equilibrated matrix A (see the description of A for
the form of the equilibrated matrix).

LDAF

LDAF is INTEGER
The leading dimension of the array AF.  LDAF >= max(1,N).

IPIV

IPIV is 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 CGETRF; 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 equilibrated matrix A.

EQUED

EQUED is 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 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

R is 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

C is 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

B is COMPLEX 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

LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,N).

X

X is 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

LDX is INTEGER
The leading dimension of the array X.  LDX >= max(1,N).

RCOND

RCOND is 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

FERR is 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

BERR is 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

WORK is COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL array, dimension (2*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

INFO is 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

Definition at line 349 of file cgesvx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

cgesvx(3) is an alias of cgesvx.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK