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NAG Toolbox

NAG Toolbox: nag_lapack_zpptri (f07gw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zpptri (f07gw) computes the inverse of a complex Hermitian positive definite matrix A, where A has been factorized by nag_lapack_zpptrf (f07gr), using packed storage.

Syntax

[ap, info] = f07gw(uplo, n, ap)
[ap, info] = nag_lapack_zpptri(uplo, n, ap)

Description

nag_lapack_zpptri (f07gw) is used to compute the inverse of a complex Hermitian positive definite matrix A, the function must be preceded by a call to nag_lapack_zpptrf (f07gr), 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.

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
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 int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
3:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The Cholesky factor of A stored in packed form, as returned by nag_lapack_zpptrf (f07gr).

Optional Input Parameters

None.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
The factorization stores 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.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Diagonal element _ of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of A cannot be computed.

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 .  

Further Comments

The total number of real floating-point operations is approximately 83n3.
The real analogue of this function is nag_lapack_dpptri (f07gj).

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_lapack_zpptrf (f07gr).
function f07gw_example


fprintf('f07gw example results\n\n');

% Symmetric A in lower packed form
uplo = 'L';
n = int64(4);
ap = [3.23 + 0i    1.51 + 1.92i    1.90 - 0.84i    0.42 - 2.50i ...
                   3.58 + 0i      -0.23 - 1.11i   -1.18 - 1.37i ...
                                   4.09 + 0.00i    2.33 + 0.14i ...
                                                   4.29 + 0.00i];

[L, info] = f07gr( ...
                     uplo, n, ap);

% Invert
[ainv, info] = f07gw( ...
                      uplo, n, L);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, ainv, 'Inverse');


f07gw example results

 Inverse
             1          2          3          4
 1      5.4691
        0.0000

 2     -1.2624     1.1024
       -1.5491     0.0000

 3     -2.9746     0.8989     2.1589
       -0.9616    -0.5672     0.0000

 4      1.1962    -0.9826    -1.3756     2.2934
        2.9772    -0.2566    -1.4550     0.0000

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Chapter Introduction
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