nag_zpptri (f07gwc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zpptri (f07gwc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpptri (f07gwc) computes the inverse of a complex Hermitian positive definite matrix A, where A has been factorized by nag_zpptrf (f07grc), using packed storage.

2  Specification

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

3  Description

nag_zpptri (f07gwc) is used to compute the inverse of a complex Hermitian positive definite matrix A, the function must be preceded by a call to nag_zpptrf (f07grc), which computes the Cholesky factorization of A, using packed storage.
If uplo=Nag_Upper, A=UHU and A-1 is computed by first inverting U and then forming U-1U-H.
If uplo=Nag_Lower, A=LLH and A-1 is computed by first inverting L and then forming L-HL-1.

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=UHU, where U is upper triangular.
A=LLH, 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 Cholesky factor of A stored in packed form, as returned by nag_zpptrf (f07grc).
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
Diagonal element value of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of A cannot be computed.

7  Accuracy

The computed inverse X satisfies
XA-I2cnεκ2A   and   AX-I2cnεκ2A ,
where cn is a modest function of n, ε is the machine precision and κ2A is the condition number of A defined by
κ2A=A2A-12 .

8  Further Comments

The total number of real floating point operations is approximately 83n3.
The real analogue of this function is nag_dpptri (f07gjc).

9  Example

This example computes the inverse of the matrix A, where
A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .
Here A is Hermitian positive definite, stored in packed form, and must first be factorized by nag_zpptrf (f07grc).

9.1  Program Text

Program Text (f07gwce.c)

9.2  Program Data

Program Data (f07gwce.d)

9.3  Program Results

Program Results (f07gwce.r)

nag_zpptri (f07gwc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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