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


SYNOPSIS

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

           CHARACTER      UPLO

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

           DOUBLE         PRECISION  BERR( * ), FERR( * ), RWORK(
                          * )

           COMPLEX*16     AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,
                          * ), WORK( * ), X( LDX, * )


PURPOSE

       ZPBRFS  improves the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  Hermitian
       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 Hermitian  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) COMPLEX*16 array, dimension (LDAFB,N)
               The  triangular  factor  U  or L from the Cholesky
               factorization A = U**H*U or A = L*L**H of the band
               matrix  A as computed by ZPBTRF, 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) 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
               ZPBTRS.  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) 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



PARAMETERS

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


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