NAG Library Function Document

nag_zhetri (f07mwc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zhetri (f07mwc) computes the inverse of a complex Hermitian indefinite matrix A, where A has been factorized by nag_zhetrf (f07mrc).

2
Specification

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

3
Description

nag_zhetri (f07mwc) is used to compute the inverse of a complex Hermitian indefinite matrix A, the function must be preceded by a call to nag_zhetrf (f07mrc), which computes the Bunch–Kaufman factorization of A.
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:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=PUDUHPT, where U is upper triangular.
uplo=Nag_Lower
A=PLDLHPT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     a[dim] ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: details of the factorization of A, as returned by nag_zhetrf (f07mrc).
On exit: the factorization is overwritten by the n by n Hermitian matrix A-1.
If uplo=Nag_Upper, the upper triangle of A-1 is stored in the upper triangular part of the array.
If uplo=Nag_Lower, the lower triangle of A-1 is stored in the lower triangular part of the array.
5:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array a.
Constraint: pdamax1,n.
6:     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_zhetrf (f07mrc).
7:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INTERNAL_ERROR
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
Element value of the diagonal 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_zhetri (f07mwc) 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 function. 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 83n3.
The real analogue of this function is nag_dsytri (f07mjc).

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 and must first be factorized by nag_zhetrf (f07mrc).

10.1
Program Text

Program Text (f07mwce.c)

10.2
Program Data

Program Data (f07mwce.d)

10.3
Program Results

Program Results (f07mwce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017