zhpsvx(3)

NAME

ZHPSVX - use the diagonal pivoting factorization A =

U*D*U**H or A = L*D*L**H to compute the solution to a complex

system of linear equations A * X = B, where A is an N-by-N Hermi

tian matrix stored in packed format and X and B are N-by-NRHS ma

trices

SYNOPSIS

SUBROUTINE ZHPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV,  B,
LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
    CHARACTER      FACT, UPLO
    INTEGER        INFO, LDB, LDX, N, NRHS
    DOUBLE         PRECISION RCOND
    INTEGER        IPIV( * )
    DOUBLE          PRECISION BERR( * ), FERR( * ), RWORK(
* )
    COMPLEX*16     AFP( * ), AP( * ), B( LDB, * ), WORK( *
), X( LDX, * )

PURPOSE

ZHPSVX uses the diagonal pivoting factorization A =

U*D*U**H or A = L*D*L**H to compute the solution to a complex

system of linear equations A * X = B, where A is an N-by-N Hermi

tian matrix stored in packed format and X and B are N-by-NRHS ma

trices. Error bounds on the solution and a condition estimate

are also provided.

DESCRIPTION

The following steps are performed:

1. If FACT = 'N', the diagonal pivoting method is used to

factor A as
A = U * D * U**H, if UPLO = 'U', or
A = L * D * L**H, if UPLO = 'L',

where U (or L) is a product of permutation and unit up

per (lower)
triangular matrices and D is Hermitian and block diago

nal with
1-by-1 and 2-by-2 diagonal blocks.

2. If some D(i,i)=0, so that D is exactly singular, then

the routine
returns with INFO = i. Otherwise, the factored form of

A is used
to estimate the condition number of the matrix A. If

the
reciprocal of the condition number is less than machine

precision,
INFO = N+1 is returned as a warning, but the routine

still goes on
to solve for X and compute error bounds as described

below.

3. The system of equations is solved for X using the fac

tored form
of A.

4. Iterative refinement is applied to improve the computed

solution
matrix and calculate error bounds and backward error

estimates
for it.

ARGUMENTS

FACT (input) CHARACTER*1

Specifies whether or not the factored form of A

has been supplied on entry. = 'F': On entry, AFP and IPIV con

tain the factored form of A. AFP and IPIV will not be modified.

= 'N': The matrix A will be copied to AFP and factored.

UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N (input) INTEGER
The number of linear equations, i.e., 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 ma

trix 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further de

tails.

AFP (input or output) COMPLEX*16 array, dimension

(N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and

on entry 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 in the same storage format as A.

If FACT = 'N', then AFP is an output argument and

on exit 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 in the same storage format as A.

IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and

on entry contains details of the interchanges and the block

structure of D, as determined by ZHPTRF. If IPIV(k) > 0, then

rows and columns k and IPIV(k) were interchanged and D(k,k) is a

1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) <

0, then rows and columns k-1 and -IPIV(k) were interchanged and

D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IP

IV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k)

were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = 'N', then IPIV is an output argument and

on exit contains details of the interchanges and the block struc

ture of D, as determined by ZHPTRF.

B (input) COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.

LDB (input) INTEGER
The leading dimension of the array B. LDB >=

max(1,N).

X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution

matrix X.

LDX (input) INTEGER
The leading dimension of the array X. LDX >=

max(1,N).

RCOND (output) DOUBLE PRECISION
The estimate of the reciprocal condition number of

the matrix A. If RCOND is less than the machine precision (in

particular, if RCOND = 0), the matrix is singular to working pre

cision. This condition is indicated by a return code of INFO >

0.

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 element 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
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization

has been completed but the factor D is exactly singular, so the

solution and error bounds could not be computed. RCOND = 0 is re

turned. = N+1: D is nonsingular, but RCOND is less than machine

precision, meaning that the matrix is singular to working preci

sion. Nevertheless, the solution and error bounds are computed

because there are a number of situations where the computed solu

tion can be more accurate than the value of RCOND would suggest.

FURTHER DETAILS

The packed storage scheme is illustrated by the following

example when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

LAPACK version 3.0 15 June 2000

ZHPSVX(3)