ctbrfs(3F)

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

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

         CHARACTER      DIAG, TRANS, UPLO

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

         REAL           BERR( * ), FERR( * ), RWORK( * )

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

PURPOSE
     CTBRFS 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 CTBTRS or some other means
     before entering this routine.  CTBRFS 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)

     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) COMPLEX 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) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS)
             The 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 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) REAL 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) COMPLEX array, dimension (2*N)

     RWORK   (workspace) REAL array, dimension (N)

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