DTBRFS - 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.

               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 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) 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)


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


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