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NAG Toolbox: nag_lapack_dpptri (f07gj)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dpptri (f07gj) computes the inverse of a real symmetric positive definite matrix A, where A has been factorized by nag_lapack_dpptrf (f07gd), using packed storage.

Syntax

[ap, info] = f07gj(uplo, n, ap)
[ap, info] = nag_lapack_dpptri(uplo, n, ap)

Description

nag_lapack_dpptri (f07gj) is used to compute the inverse of a real symmetric positive definite matrix A, the function must be preceded by a call to nag_lapack_dpptrf (f07gd), which computes the Cholesky factorization of A, using packed storage.
If uplo='U', A=UTU and A-1 is computed by first inverting U and then forming U-1U-T.
If uplo='L', A=LLT and A-1 is computed by first inverting L and then forming L-TL-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=UTU, where U is upper triangular.
uplo='L'
A=LLT, 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: – double 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_dpptrf (f07gd).

Optional Input Parameters

None.

Output Parameters

1:     ap: – double 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 floating-point operations is approximately 23n3.
The complex analogue of this function is nag_lapack_zpptri (f07gw).

Example

This example computes the inverse of the matrix A, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .  
Here A is symmetric positive definite, stored in packed form, and must first be factorized by nag_lapack_dpptrf (f07gd).
function f07gj_example


fprintf('f07gj example results\n\n');

% Symmetric matrix A, lower triangular part packed in ap
uplo = 'L';
n = int64(4);
ap = [4.16 -3.12  0.56 -0.10 ...
            5.03 -0.83  1.18 ...
                  0.76  0.34 ...
                        1.18];

[L, info] = f07gd( ...
                   uplo, n, ap);

[ainv, info] = f07gj( ...
                      uplo, n, L);

[ifail] = x04cc( ...
                 uplo, 'N', n, ainv, 'Inverse');


f07gj example results

 Inverse
             1          2          3          4
 1      0.6995
 2      0.7769     1.4239
 3      0.7508     1.8255     4.0688
 4     -0.9340    -1.8841    -2.9342     3.4978

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Chapter Contents
Chapter Introduction
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