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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dsptri (f07pj)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dsptri (f07pj) computes the inverse of a real symmetric indefinite matrix

A

, where

A

has been factorized by nag_lapack_dsptrf (f07pd), using packed storage.

Syntax

[ap, info] = f07pj(uplo, ap, ipiv, 'n', n)

[ap, info] = nag_lapack_dsptri(uplo, ap, ipiv, 'n', n)

Description

nag_lapack_dsptri (f07pj) is used to compute the inverse of a real symmetric indefinite matrix

A

, the function must be preceded by a call to nag_lapack_dsptrf (f07pd), which computes the Bunch–Kaufman factorization of

A

, using packed storage.

uplo ='U'

A = P U D U^{T} P^{T}

and

A^{- 1}

is computed by solving

U^{T} P^{T} X P U = D^{- 1}

uplo ='L'

A = P L D L^{T} P^{T}

and

A^{- 1}

is computed by solving

L^{T} P^{T} X P L = D^{- 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.

$uplo ='U'$: $A = P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: $A = P L D L^{T} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $ap (:)$ – double array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The factorization of

A

stored in packed form, as returned by nag_lapack_dsptrf (f07pd).

3: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

Details of the interchanges and the block structure of

D

, as returned by nag_lapack_dsptrf (f07pd).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $ap (:)$ – double array

The dimension of the array ap will be

\max (1, n \times (n + 1) / 2)

The factorization stores the

n

n

matrix

A^{- 1}

More precisely,

if $uplo ='U'$ , the upper triangle of $A^{- 1}$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A^{- 1}$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

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 $info > 0$: Element $_$ of the diagonal is exactly zero. $D$ is singular and the inverse of $A$ cannot be computed.

Accuracy

The computed inverse

X

satisfies a bound of the form

if $uplo ='U'$ , $|D U^{T} P^{T} X P U - I| \leq c (n) ε (|D| |U^{T}| P^{T} |X| P |U| + |D| |D^{- 1}|)$ ;
if $uplo ='L'$ , $|D L^{T} P^{T} X P L - I| \leq c (n) ε (|D| |L^{T}| P^{T} |X| P |L| + |D| |D^{- 1}|)$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision

Further Comments

The total number of floating-point operations is approximately

\frac{2}{3} n^{3}

The complex analogues of this function are nag_lapack_zhptri (f07pw) for Hermitian matrices and nag_lapack_zsptri (f07qw) for symmetric matrices.

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{matrix} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{matrix}) .

Here

A

is symmetric indefinite, stored in packed form, and must first be factorized by nag_lapack_dsptrf (f07pd).

Open in the MATLAB editor: f07pj_example

function f07pj_example


fprintf('f07pj example results\n\n');

% Indefinite matrix A (lower triangular part stored in packed format)
uplo = 'L';
n = int64(4);
ap = [2.07;   3.87;   4.20;   -1.15;
             -0.21;   1.87;    0.63;
                      1.15;    2.06;
                              -1.81];

% Factorize
[apf, ipiv, info] = f07pd( ...
                           uplo, n, ap);

% Invert
[ainv, info] = f07pj( ...
                      uplo, apf, ipiv);

[ifail] = x04cc( ...
                 uplo, 'Non-unit', n, ainv, 'Inverse');

f07pj example results

 Inverse
             1          2          3          4
 1      0.7485
 2      0.5221    -0.1605
 3     -1.0058    -0.3131     1.3501
 4     -1.4386    -0.7440     2.0667     2.4547

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Chapter Contents

Chapter Introduction

NAG Toolbox