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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.