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dtbrfs

 NAME
      DTBRFS - provide error bounds and backward error estimates
      for the solution to a system of linear equations with a tri-
      angular band coefficient matrix

 SYNOPSIS
      SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
                         B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
                         INFO )

          CHARACTER      DIAG, TRANS, UPLO

          INTEGER        INFO, KD, LDAB, LDB, LDX, N, NRHS

          INTEGER        IWORK( * )

          DOUBLE         PRECISION AB( LDAB, * ), B( LDB, * ),
                         BERR( * ), FERR( * ), WORK( * ), X( LDX,
                         * )

 PURPOSE
      DTBRFS provides error bounds and backward error estimates
      for the solution to a system of linear equations with a tri-
      angular band coefficient matrix.

      The solution matrix X must be computed by DTBTRS or some
      other means before entering this routine.  DTBRFS does not
      do iterative refinement because doing so cannot improve the
      backward error.

 ARGUMENTS
      UPLO    (input) CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

      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)

      DIAG    (input) CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      KD      (input) INTEGER
              The number of superdiagonals or subdiagonals of the

              triangular band matrix A.  KD >= 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) DOUBLE PRECISION array, dimension (LDAB,N)
              The upper or lower triangular band matrix A, stored
              in the first kd+1 rows of the array. The j-th column
              of A is stored in the j-th column of the array AB as
              follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for
              max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j)    =
              A(i,j) for j<=i<=min(n,j+kd).  If DIAG = 'U', the
              diagonal elements of A are not referenced and are
              assumed to be 1.

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

      B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
              The solution matrix X.

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

      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) DOUBLE PRECISION 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