NAG FL Interface
f07psf (zhptrs)

1 Purpose

f07psf solves a complex Hermitian indefinite system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by f07prf, using packed storage.

2 Specification

Fortran Interface
Subroutine f07psf ( uplo, n, nrhs, ap, ipiv, b, ldb, info)
Integer, Intent (In) :: n, nrhs, ipiv(*), ldb
Integer, Intent (Out) :: info
Complex (Kind=nag_wp), Intent (In) :: ap(*)
Complex (Kind=nag_wp), Intent (Inout) :: b(ldb,*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07psf_ (const char *uplo, const Integer *n, const Integer *nrhs, const Complex ap[], const Integer ipiv[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07psf, nagf_lapacklin_zhptrs or its LAPACK name zhptrs.

3 Description

f07psf is used to solve a complex Hermitian indefinite system of linear equations AX=B, the routine must be preceded by a call to f07prf which computes the Bunch–Kaufman factorization of A, using packed storage.
If uplo='U', A=PUDUHPT, where P is a permutation matrix, U is an upper triangular matrix and D is an Hermitian block diagonal matrix with 1 by 1 and 2 by 2 blocks; the solution X is computed by solving PUDY=B and then UHPTX=Y.
If uplo='L', A=PLDLHPT, where L is a lower triangular matrix; the solution X is computed by solving PLDY=B and then LHPTX=Y.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: uplo Character(1) Input
On entry: specifies how A has been factorized.
A=PUDUHPT, where U is upper triangular.
A=PLDLHPT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
4: ap* Complex (Kind=nag_wp) array Input
Note: the dimension of the array ap must be at least max1,n×n+1/2.
On entry: the factorization of A stored in packed form, as returned by f07prf.
5: ipiv* Integer array Input
Note: the dimension of the array ipiv must be at least max1,n.
On entry: details of the interchanges and the block structure of D, as returned by f07prf.
6: bldb* Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
7: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07psf is called.
Constraint: ldbmax1,n.
8: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where cn is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling f07pvf, and an estimate for κA (=κ1A) can be obtained by calling f07puf.

8 Parallelism and Performance

f07psf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately 8n2r.
This routine may be followed by a call to f07pvf to refine the solution and return an error estimate.
The real analogue of this routine is f07pef.

10 Example

This example solves the system of equations AX=B, where
A= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i  
B= 7.79+05.48i -35.39+18.01i -0.77-16.05i 4.23-70.02i -9.58+03.88i -24.79-08.40i 2.98-10.18i 28.68-39.89i .  
Here A is Hermitian indefinite, stored in packed form, and must first be factorized by f07prf.

10.1 Program Text

Program Text (f07psfe.f90)

10.2 Program Data

Program Data (f07psfe.d)

10.3 Program Results

Program Results (f07psfe.r)