ZTBRFS(3)                  MathKeisan LAPACK routine                 ZTBRFS(3)



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

SYNOPSIS
       SUBROUTINE ZTBRFS( 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

           DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * )

           COMPLEX*16     AB( LDAB, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE
       ZTBRFS provides error bounds and backward error estimates for the solu-
       tion to a system of linear equations with a triangular band coefficient
       matrix.

       The  solution  matrix  X must be computed by ZTBTRS or some other means
       before entering this routine.  ZTBRFS 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*16 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*16 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*16 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 vec-
               tor 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*16 array, dimension (2*N)

       RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

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



 LAPACK routine (version 3.1)    November 2006                       ZTBRFS(3)