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NAME

        ZGBRFS  - 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 ZGBRFS( TRANS,  N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
                           B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
 
            CHARACTER      TRANS
 
            INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
 
            INTEGER        IPIV( * )
 
            DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * )
 
            COMPLEX*16     AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
                           ), X( LDX, * )
 

PURPOSE

        ZGBRFS  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)
 
        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) COMPLEX*16 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) COMPLEX*16 array, dimension (LDAFB,N)
                Details  of  the LU factorization of the band matrix A, as com‐
                puted by ZGBTRF.  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 ZGBTRF; for 1<=i<=N, row i of the matrix
                was interchanged with row IPIV(i).
 
        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/output) COMPLEX*16 array, dimension (LDX,NRHS)
                On entry, the solution matrix X, as  computed  by  ZGBTRS.   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) 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
 

PARAMETERS

        ITMAX is the maximum number of steps of iterative refinement.
 

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