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.

               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  solu­
               tion  vector X(j) (the j-th column of the solution
               matrix X).  If XTRUE is the true  solution  corre­
               sponding  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 reli­
               able  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 ille­
               gal value


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