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zgbsvx


 NAME
      ZGBSVX - use the LU factorization to compute the solution to
      a complex system of linear equations A * X = B, A**T * X =
      B, or A**H * X = B,

 SYNOPSIS
      SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB,
                         AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X,
                         LDX, RCOND, FERR, BERR, WORK, RWORK, INFO
                         )

          CHARACTER      EQUED, FACT, TRANS

          INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N,
                         NRHS

          DOUBLE         PRECISION RCOND

          INTEGER        IPIV( * )

          DOUBLE         PRECISION BERR( * ), C( * ), FERR( * ),
                         R( * ), RWORK( * )

          COMPLEX*16     AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
                         ), WORK( * ), X( LDX, * )

 PURPOSE
      ZGBSVX uses the LU factorization to compute the solution to
      a complex system of linear equations A * X = B, A**T * X =
      B, or A**H * X = B, where A is a band matrix of order N with
      KL subdiagonals and KU superdiagonals, 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 by this subroutine:

      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 = L * U,
         where L is a product of permutation and unit lower tri-
      angular
         matrices with KL subdiagonals, and U is upper triangular
      with
         KL+KU superdiagonals.

      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 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, AFB 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.  AB, AFB, and IPIV are not modi-
              fied.  = 'N':  The matrix A will be copied to AFB
              and factored.
              = 'E':  The matrix A will be equilibrated if neces-
              sary, then copied to AFB 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.

      KL      (input) INTEGER
              The number of subdiagonals within the band of A.  KL
              >= 0.

      KU      (input) INTEGER
              The number of superdiagonals within the band of A.
              KU >= 0.

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

      AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
              On entry, the matrix A in band storage, in rows 1 to
              KL+KU+1.  The j-th column of A is stored in the j-th
              column of the array AB as follows: AB(KU+1+i-j,j) =
              A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

              If FACT = 'F' and EQUED is not 'N', then A must have
              been equilibrated by the scaling factors in R and/or
              C.  AB 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).

      LDAB    (input) INTEGER
              The leading dimension of the array AB.  LDAB >=
              KL+KU+1.

      AFB     (input or output) COMPLEX*16 array, dimension (LDAFB,N)
              If FACT = 'F', then AFB is an input argument and on
              entry contains details of the LU factorization of
              the band matrix A, as computed by ZGBTRF.  U is
              stored as an upper triangular band matrix with KL+KU
              superdiagonals in rows 1 to KL+KU+1, and the multi-
              pliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then
              AFB is the factored form of the equilibrated matrix
              A.

              If FACT = 'N', then AFB is an output argument and on
              exit returns details of the LU factorization of A.

              If FACT = 'E', then AFB is an output argument and on
              exit returns details of the LU factorization of the
              equilibrated matrix A (see the description of AB for
              the form of the equilibrated matrix).

      LDAFB   (input) INTEGER
              The leading dimension of the array AFB.  LDAFB >=
              2*KL+KU+1.

      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 = L*U as computed by ZGBTRF; 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 factori-
              zation A = 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 = 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 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 (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 A is singular to working precision, and
              the solution and error bounds have not been com-
              puted.