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# zhprfs

```
NAME
ZHPRFS - improve the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefin-
ite and packed, and provides error bounds and backward error
estimates for the solution

SYNOPSIS
SUBROUTINE ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
LDX, FERR, BERR, WORK, RWORK, INFO )

CHARACTER      UPLO

INTEGER        INFO, LDB, LDX, N, NRHS

INTEGER        IPIV( * )

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

COMPLEX*16     AFP( * ), AP( * ), B( LDB, * ), WORK( *
), X( LDX, * )

PURPOSE
ZHPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefin-
ite and packed, 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.

NRHS    (input) INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X.  NRHS >= 0.

AP      (input) COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array.  The j-th
column of A is stored in the array AP as follows: if
UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for
j<=i<=n.

AFP     (input) COMPLEX*16 array, dimension (N*(N+1)/2)
The factored form of the matrix A.  AFP contains the
block diagonal matrix D and the multipliers used to

obtain the factor U or L from the factorization A =
U*D*U**H or A = L*D*L**H as computed by ZHPTRF,
stored as a packed triangular matrix.

IPIV    (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure
of D as determined by ZHPTRF.

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
ZHPTRS.  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 bounds for each solution
vector X(j) (the j-th column of the solution matrix
X).  If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j).  The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.

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 entry 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 illegal
value

PARAMETERS
ITMAX is the maximum number of steps of iterative refine-
ment.
```