nag_zhptri (f07pwc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zhptri (f07pwc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhptri (f07pwc) computes the inverse of a complex Hermitian indefinite matrix A, where A has been factorized by nag_zhptrf (f07prc), using packed storage.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zhptri (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex ap[], const Integer ipiv[], NagError *fail)

3  Description

nag_zhptri (f07pwc) is used to compute the inverse of a complex Hermitian indefinite matrix A, the function must be preceded by a call to nag_zhptrf (f07prc), which computes the Bunch–Kaufman factorization of A, using packed storage.
If uplo=Nag_Upper, A=PUDUHPT and A-1 is computed by solving UHPTXPU=D-1 for X.
If uplo=Nag_Lower, 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies how A has been factorized.
A=PUDUHPT, where U is upper triangular.
A=PLDLHPT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     ap[dim]ComplexInput/Output
Note: the dimension, dim, 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 nag_zhptrf (f07prc).
On exit: the factorization is overwritten by the n by n matrix A-1.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
5:     ipiv[dim]const IntegerInput
Note: the dimension, dim, 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 nag_zhptrf (f07prc).
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
dvalue,value 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  Parallelism and Performance

nag_zhptri (f07pwc) is not threaded by NAG in any implementation.
nag_zhptri (f07pwc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of real floating-point operations is approximately 83n3.
The real analogue of this function is nag_dsptri (f07pjc).

10  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 nag_zhptrf (f07prc).

10.1  Program Text

Program Text (f07pwce.c)

10.2  Program Data

Program Data (f07pwce.d)

10.3  Program Results

Program Results (f07pwce.r)

nag_zhptri (f07pwc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014