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NAME
     zpbrfs - improve the computed solution to a system of linear
     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, * )



     #include <sunperf.h>

     void zpbrfs(char uplo, int n, int kd, int  nrhs,  doublecom-
               plex  *zab,  int  ldab,  doublecomplex  *afb,  int
               ldafb, doublecomplex *zb, int  ldb,  doublecomplex
               *zx,  int  ldx,  double  *ferr,  double *berr, int
               *info) ;

PURPOSE
     ZPBRFS improves the computed solution to a system of  linear
     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).

     LDAB      (input) INTEGER
               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
               storage 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
               corresponding  to  X(j),  FERR(j)  is an estimated
               upper bound for the magnitude of the largest  ele-
               ment in (X(j) - XTRUE) divided by the magnitude of
               the largest element in X(j).  The estimate  is  as
               reliable  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 ille-
               gal value

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