SGBRFS - 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 SGBRFS( 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( * )

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


PURPOSE

       SGBRFS  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) REAL 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) REAL array, dimension (LDAFB,N)
               Details of the LU factorization of the band matrix
               A, as computed by SGBTRF.  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 SGBTRF; for 1<=i<=N, row  i
               of the matrix was interchanged with row IPIV(i).

       B       (input) REAL 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) REAL array, dimension (LDX,NRHS)
               On entry, the solution matrix X,  as  computed  by
               SGBTRS.   On exit, the improved solution matrix X.

       LDX     (input) INTEGER
               The leading dimension of  the  array  X.   LDX  >=
               max(1,N).

       FERR    (output) REAL 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) REAL 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).


       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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