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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zhptri (f07pw)

▸▿ 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_zhptri (f07pw) computes the inverse of a complex Hermitian indefinite matrix

A

, where

A

has been factorized by nag_lapack_zhptrf (f07pr), using packed storage.

Syntax

[ap, info] = f07pw(uplo, ap, ipiv, 'n', n)

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

Description

nag_lapack_zhptri (f07pw) is used to compute the inverse of a complex Hermitian indefinite matrix

A

, the function must be preceded by a call to nag_lapack_zhptrf (f07pr), which computes the Bunch–Kaufman factorization of

A

, using packed storage.

uplo ='U'

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

and

A^{- 1}

is computed by solving

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

for

X

uplo ='L'

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

and

A^{- 1}

is computed by solving

L^{H} P^{T} X P L = D^{- 1}

for

X

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.

$uplo ='U'$: $A = P U D U^{H} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: $A = P L D L^{H} P^{T}$ , 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_zhptrf (f07pr).

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_zhptrf (f07pr).

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} - 1.36 + 0.00 i & 1.58 + 0.90 i & 2.21 - 0.21 i & 3.91 + 1.50 i \\ 1.58 - 0.90 i & - 8.87 + 0.00 i & - 1.84 - 0.03 i & - 1.78 + 1.18 i \\ 2.21 + 0.21 i & - 1.84 + 0.03 i & - 4.63 + 0.00 i & 0.11 + 0.11 i \\ 3.91 - 1.50 i & - 1.78 - 1.18 i & 0.11 - 0.11 i & - 1.84 + 0.00 i \end{array}) .

Here

A

is Hermitian indefinite, stored in packed form, and must first be factorized by nag_lapack_zhptrf (f07pr).

Open in the MATLAB editor: f07pw_example

function f07pw_example


fprintf('f07pw example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored in packed form)
uplo = 'L';
n = int64(4);
ap = [-1.36 + 0i;  1.58 - 0.9i;   2.21 + 0.21i;  3.91 - 1.5i;
                  -8.87 + 0i;    -1.84 + 0.03i; -1.78 - 1.18i;
                                 -4.63 + 0i;     0.11 - 0.11i;
                                                -1.84 + 0i];

% Factorize
[apf, ipiv, info] = f07pr( ...
                           uplo, n, ap);

% Invert
[ainv, info] = f07pw( ...
                      uplo, apf, ipiv);

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

f07pw example results

 Inverse
          1       2       3       4
 1   0.0826
     0.0000

 2  -0.0335 -0.1408
     0.0440  0.0000

 3   0.0603  0.0422 -0.2007
    -0.0105 -0.0222  0.0000

 4   0.2391  0.0304  0.0982  0.0073
    -0.0926  0.0203 -0.0635  0.0000

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Chapter Contents

Chapter Introduction

NAG Toolbox