SPBRFS - improve the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  symmetric
       positive  definite  and  banded, and provides error bounds
       and backward error estimates for the solution


SYNOPSIS

       SUBROUTINE SPBRFS( UPLO,  N,  KD,  NRHS,  AB,  LDAB,  AFB,
                          LDAFB,  B,  LDB,  X,  LDX,  FERR, BERR,
                          WORK, IWORK, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, KD, LDAB,  LDAFB,  LDB,  LDX,  N,
                          NRHS

           INTEGER        IWORK( * )

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


PURPOSE

       SPBRFS  improves the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  symmetric
       positive  definite  and  banded, and provides error bounds
       and backward error estimates for the solution.


ARGUMENTS

       UPLO    (input) CHARACTER*1
               = 'U':  Upper triangle of A is stored;
               = 'L':  Lower triangle of A is stored.

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

       KD      (input) INTEGER
               The number of superdiagonals of the  matrix  A  if
               UPLO  = 'U', or the number of subdiagonals if UPLO
               = 'L'.  KD >= 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  upper or lower triangle of the symmetric band
               matrix A, stored in the first  KD+1  rows  of  the
               array.  The j-th column of A is stored in the j-th
               column of the array AB as follows: if UPLO =  'U',
               AB(kd+1+i-j,j)  = A(i,j) for max(1,j-kd)<=i<=j; if
               UPLO  =   'L',   AB(1+i-j,j)      =   A(i,j)   for
               j<=i<=min(n,j+kd).

               The  leading  dimension  of the array AB.  LDAB >=
               KD+1.

       AFB     (input) REAL array, dimension (LDAFB,N)
               The triangular factor U or  L  from  the  Cholesky
               factorization A = U**T*U or A = L*L**T of the band
               matrix A as computed by SPBTRF, in the same  stor­
               age format as A (see AB).

       LDAFB   (input) INTEGER
               The  leading dimension of the array AFB.  LDAFB >=
               KD+1.

       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
               SPBTRS.  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).

       WORK    (workspace) REAL 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

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


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