DTBRFS (3lapack)

provide error bounds and backward error estimates for the solution to a system of linear equations with a triangular 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 triangular 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 = Transpose)

  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 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) 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    (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