DGBRFS - improve the computed solution to a system of lin­
       ear 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 lin­
       ear 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-

               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  solu­
               tion  vector X(j) (the j-th column of the solution
               matrix X).  If XTRUE is the true  solution  corre­
               sponding  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 reli­
               able  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

               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 ille­
               gal value


PARAMETERS

       ITMAX  is the maximum number of steps of iterative refine­
       ment.


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