ZGESVX(3)      LAPACK routine of NEC Numeric Library Collection      ZGESVX(3)



NAME
       ZGESVX

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)



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 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.




ARGUMENTS
           FACT      (input)
                     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     (input)
                     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         (input)
                     N is INTEGER
                     The number of linear equations, i.e., the order of the
                     matrix A.  N >= 0.

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

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

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

           IPIV      (input/output)
                     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 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 equilibrated matrix A.

           EQUED     (input/output)
                     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         (input/output)
                     R is 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 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/output)
                     C is 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 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)
                     B is 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)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           X         (output)
                     X is COMPLEX*16 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)
                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(1,N).

           RCOND     (output)
                     RCOND is 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.

           FERR      (output)
                     FERR is DOUBLE PRECISION 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)
                     BERR is DOUBLE PRECISION 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      (output)
                     WORK is COMPLEX*16 array, dimension (2*N)

           RWORK     (output)
                     RWORK is DOUBLE PRECISION 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      (output)
                     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.



LAPACK routine                  31 October 2017                      ZGESVX(3)