DTBRFS(3)                  MathKeisan LAPACK routine                 DTBRFS(3)



NAME
       DTBRFS  - 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 solu-
       tion 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 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) 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



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