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 3.2.1.3 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.
uplo=Nag_Upper
A=UHU, where U is upper triangular.
uplo=Nag_Lower
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

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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.
NE_SINGULAR
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