NAG Library Routine Document

f07gwf  (zpptri)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f07gwf (zpptri) computes the inverse of a complex Hermitian positive definite matrix A, where A has been factorized by f07grf (zpptrf), using packed storage.

2
Specification

Fortran Interface
Subroutine f07gwf ( uplo, n, ap, info)
Integer, Intent (In):: n
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (Inout):: ap(*)
Character (1), Intent (In):: uplo
C Header Interface
#include nagmk26.h
void  f07gwf_ ( const char *uplo, const Integer *n, Complex ap[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zpptri.

3
Description

f07gwf (zpptri) is used to compute the inverse of a complex Hermitian positive definite matrix A, the routine must be preceded by a call to f07grf (zpptrf), which computes the Cholesky factorization of A, using packed storage.
If uplo='U', A=UHU and A-1 is computed by first inverting U and then forming U-1U-H.
If uplo='L', 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:     uplo – Character(1)Input
On entry: specifies how A has been factorized.
uplo='U'
A=UHU, where U is upper triangular.
uplo='L'
A=LLH, 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 Cholesky factor of A stored in packed form, as returned by f07grf (zpptrf).
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:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
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
Parallelism and Performance

f07gwf (zpptri) 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 routine. 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 routine is f07gjf (dpptri).

10
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 f07grf (zpptrf).

10.1
Program Text

Program Text (f07gwfe.f90)

10.2
Program Data

Program Data (f07gwfe.d)

10.3
Program Results

Program Results (f07gwfe.r)

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