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sgbrfs

 NAME
      SGBRFS - improve the computed solution to a system of linear
      equations when the coefficient matrix is banded, and pro-
      vides error bounds and backward error estimates for the
      solution

 SYNOPSIS
      SUBROUTINE SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
                         LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR,
                         WORK, IWORK, INFO )

          CHARACTER      TRANS

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

          INTEGER        IPIV( * ), IWORK( * )

          REAL           AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
                         ), BERR( * ), FERR( * ), WORK( * ), X(
                         LDX, * )

 PURPOSE
      SGBRFS improves the computed solution to a system of linear
      equations when the coefficient matrix is banded, and pro-
      vides error bounds and backward error estimates for the
      solution.

 ARGUMENTS
      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 = Tran-
              spose)

      N       (input) INTEGER
              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) REAL array, dimension (LDAB,N)
              The original band matrix A, stored 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).

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

      AFB     (input) REAL array, dimension (LDAFB,N)
              Details of the LU factorization of the band matrix
              A, as computed by SGBTRF.  U is stored as an upper
              triangular band matrix with KL+KU superdiagonals in
              rows 1 to KL+KU+1, and the multipliers used during
              the factorization are stored in rows KL+KU+2 to
              2*KL+KU+1.

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

      IPIV    (input) INTEGER array, dimension (N)
              The pivot indices from SGBTRF; for 1<=i<=N, row i of
              the matrix was interchanged with row IPIV(i).

      B       (input) REAL array, dimension (LDB,NRHS)
              The right hand side matrix B.

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

      X       (input/output) REAL array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by
              SGBTRS.  On exit, the improved solution matrix X.

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

      FERR    (output) REAL 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) REAL 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) REAL array, dimension (3*N)

      IWORK   (workspace) INTEGER array, dimension (N)

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
              value

 PARAMETERS
      ITMAX is the maximum number of steps of iterative refine-
      ment.