DGBRFS(3)                  MathKeisan LAPACK routine                 DGBRFS(3)



NAME
       DGBRFS - the computed solution to a system of linear equations when the
       coefficient matrix is banded, and provides error  bounds  and  backward
       error estimates for the solution

SYNOPSIS
       SUBROUTINE DGBRFS( TRANS,  N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
                          B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )

           CHARACTER      TRANS

           INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

           INTEGER        IPIV( * ), IWORK( * )

           DOUBLE         PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,  *
                          ), BERR( * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE
       DGBRFS  improves  the computed solution to a system of linear equations
       when the coefficient matrix is banded, and provides  error  bounds  and
       backward error estimates for the solution.


ARGUMENTS
       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)

       N       (input) INTEGER
               The order of the matrix A.  N >= 0.

       KL      (input) INTEGER
               The number of subdiagonals within the band of A.  KL >= 0.

       KU      (input) INTEGER
               The number of superdiagonals within the band of A.  KU >= 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 original band matrix A, stored in rows 1 to  KL+KU+1.   The
               j-th  column  of A is stored in the j-th column of the array AB
               as   follows:   AB(ku+1+i-j,j)   =    A(i,j)    for    max(1,j-
               ku)<=i<=min(n,j+kl).

       LDAB    (input) INTEGER
               The leading dimension of the array AB.  LDAB >= KL+KU+1.

       AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
               Details  of  the LU factorization of the band matrix A, as com-
               puted by DGBTRF.  U is  stored  as  an  upper  triangular  band
               matrix  with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
               multipliers used during the factorization are  stored  in  rows
               KL+KU+2 to 2*KL+KU+1.

       LDAFB   (input) INTEGER
               The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

       IPIV    (input) INTEGER array, dimension (N)
               The pivot indices from DGBTRF; for 1<=i<=N, row i of the matrix
               was interchanged with row IPIV(i).

       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/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
               On entry, the solution matrix X, as  computed  by  DGBTRS.   On
               exit, the improved 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

PARAMETERS
       ITMAX is the maximum number of steps of iterative refinement.



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