CPBRFS - 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 CPBRFS( 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

           REAL           BERR( * ), FERR( * ), RWORK( * )

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


PURPOSE

       CPBRFS 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) COMPLEX 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 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 CPBTRF, 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 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 array, dimension (LDX,NRHS)
               On  entry,  the  solution matrix X, as computed by
               CPBTRS.  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) COMPLEX array, dimension (2*N)

       RWORK   (workspace) REAL 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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