DPBRFS - 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 DPBRFS( 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( * )

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


PURPOSE

       DPBRFS  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DPBTRF, 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) 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
               DPBTRS.   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
               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)

               = 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.


Man(1) output converted with man2html