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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zpptri (f07gw)

▸▿ 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_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.

uplo ='U'

A = U^{H} U

and

A^{- 1}

is computed by first inverting

U

and then forming

(U^{- 1}) U^{- H}

uplo ='L'

A = L L^{H}

and

A^{- 1}

is computed by first inverting

L

and then forming

L^{- H} (L^{- 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.

$uplo ='U'$: $A = U^{H} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{H}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (:)$ – complex array

The dimension of the array ap must be at least

\max (1, n \times (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

\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 $info > 0$: 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

{‖X A - I‖}_{2} \leq c (n) ε κ_{2} (A) and {‖A X - I‖}_{2} \leq c (n) ε κ_{2} (A),

where

c (n)

is a modest function of

n

ε

is the machine precision and

κ_{2} (A)

is the condition number of

A

defined by

κ_{2} (A) = {‖A‖}_{2} {‖A^{- 1}‖}_{2} .

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_dpptri (f07gj).

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}) .

Here

A

is Hermitian positive definite, stored in packed form, and must first be factorized by nag_lapack_zpptrf (f07gr).

Open in the MATLAB editor: f07gw_example

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 Contents

Chapter Introduction

NAG Toolbox