dgbrfs(3F)

NAME
     DGBRFS - improve 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 computed
             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 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)

     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.