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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zsptri (f07qw)

▸▿ 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_zsptri (f07qw) computes the inverse of a complex symmetric matrix

A

, where

A

has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.

Syntax

[ap, info] = f07qw(uplo, ap, ipiv, 'n', n)

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

Description

nag_lapack_zsptri (f07qw) is used to compute the inverse of a complex symmetric matrix

A

, the function must be preceded by a call to nag_lapack_zsptrf (f07qr), 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: $ap (:)$ – complex 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_zsptrf (f07qr).

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_zsptrf (f07qr).

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: $ap (:)$ – complex 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 real floating-point operations is approximately

\frac{8}{3} n^{3}

The real analogue of this function is nag_lapack_dsptri (f07pj).

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} - 0.39 - 0.71 i & 5.14 - 0.64 i & - 7.86 - 2.96 i & 3.80 + 0.92 i \\ 5.14 - 0.64 i & 8.86 + 1.81 i & - 3.52 + 0.58 i & 5.32 - 1.59 i \\ - 7.86 - 2.96 i & - 3.52 + 0.58 i & - 2.83 - 0.03 i & - 1.54 - 2.86 i \\ 3.80 + 0.92 i & 5.32 - 1.59 i & - 1.54 - 2.86 i & - 0.56 + 0.12 i \end{array}) .

Here

A

is symmetric, stored in packed form, and must first be factorized by nag_lapack_zsptrf (f07qr).

Open in the MATLAB editor: f07qw_example

function f07qw_example


fprintf('f07qw example results\n\n');

% Get Inverse of A, where A is complex symmetric matrix such that the
% lower triangular part is stored in packed format
uplo = 'L';
n = int64(4);
ap = [ -0.39 -  0.71i,   5.14 -  0.64i,  -7.86 - 2.96i,   3.80 + 0.92i, ...
        8.86 +  1.81i,  -3.52 +  0.58i,   5.32 - 1.59i,   ...
       -2.83 -  0.03i,  -1.54 -  2.86i,  ...
       -0.56 + 0.12i];

% Factorize
[apf, ipiv, info] = f07qr( ...
                           uplo, n, ap);

% Invert
[ainv, info] = f07qw(uplo, apf, ipiv);

% Display packed inverse: Integer labels, 80 colunms wide, no indent
rlabs = {'       '};
clabs = {'       '};
ncols  = int64(80);
indent = int64(0);

[ifail] = x04dd( ...
                 uplo, 'N', n, ainv, 'Brac', ' ', 'Inverse', 'Int', rlabs, ...
                 'Int', clabs, ncols, indent);

f07qw example results

 Inverse
                      1                   2                   3
 1  ( -0.1562, -0.1014)
 2  (  0.0400,  0.1527) (  0.0946, -0.1475)
 3  (  0.0550,  0.0845) ( -0.0326, -0.1370) ( -0.1320, -0.0102)
 4  (  0.2162, -0.0742) ( -0.0995, -0.0461) ( -0.1793,  0.1183)

                      4
 1
 2
 3
 4  ( -0.2269,  0.2383)

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Chapter Contents

Chapter Introduction

NAG Toolbox