F07PWF (ZHPTRI) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07PWF (ZHPTRI) computes the inverse of a complex Hermitian indefinite matrix A, where A has been factorized by F07PRF (ZHPTRF), using packed storage.

2  Specification

COMPLEX (KIND=nag_wp)  AP(*), WORK(N)
The routine may be called by its LAPACK name zhptri.

3  Description

F07PWF (ZHPTRI) is used to compute the inverse of a complex Hermitian indefinite matrix A, the routine must be preceded by a call to F07PRF (ZHPTRF), which computes the Bunch–Kaufman factorization of A, using packed storage.
If UPLO='U', A=PUDUHPT and A-1 is computed by solving UHPTXPU=D-1 for X.
If UPLO='L', A=PLDLHPT and A-1 is computed by solving LHPTXPL=D-1 for X.

4  References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

5  Parameters

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 – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     AP(*) – COMPLEX (KIND=nag_wp) arrayInput/Output
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 (ZHPTRF).
On exit: the factorization is overwritten by the n by n matrix A-1.
More precisely,
  • if UPLO='U', the upper triangle of A-1 must be stored with element Aij in APi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of A-1 must be stored with element Aij in APi+2n-jj-1/2 for ij.
4:     IPIV(*) – INTEGER arrayInput
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 (ZHPTRF).
5:     WORK(N) – COMPLEX (KIND=nag_wp) arrayWorkspace
6:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, di,i is exactly zero; D is singular and the inverse of A cannot be computed.

7  Accuracy

The computed inverse X satisfies a bound of the form cn is a modest linear function of n, and ε is the machine precision.

8  Further Comments

The total number of real floating point operations is approximately 83n3.
The real analogue of this routine is F07PJF (DSPTRI).

9  Example

This example computes the inverse of the matrix A, 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 .
Here A is Hermitian indefinite, stored in packed form, and must first be factorized by F07PRF (ZHPTRF).

9.1  Program Text

Program Text (f07pwfe.f90)

9.2  Program Data

Program Data (f07pwfe.d)

9.3  Program Results

Program Results (f07pwfe.r)

F07PWF (ZHPTRI) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012