nag_lapack_zpftri (f07ww) computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) stored in Rectangular Full Packed (RFP) format.

Syntax

[ar, info] = f07ww(transr, uplo, n, ar)

[ar, info] = nag_lapack_zpftri(transr, uplo, n, ar)

Description

nag_lapack_zpftri (f07ww) is used to compute the inverse of a complex Hermitian positive definite matrix

A

, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. The function must be preceded by a call to nag_lapack_zpftrf (f07wr), which computes the Cholesky factorization of

A

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

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

Parameters

Compulsory Input Parameters

1: $transr$ – string (length ≥ 1)

Specifies whether the normal RFP representation of

A

or its conjugate transpose is stored.

$transr ='N'$: The matrix $A$ is stored in normal RFP format.
$transr ='C'$: The conjugate transpose of the RFP representation of the matrix $A$ is stored.

Constraint:

transr ='N'

'C'

2: $uplo$ – 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'

3: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $ar (n \times (n + 1) / 2)$ – complex array

The Cholesky factorization of

A

stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).

Optional Input Parameters

None.

Output Parameters

1: $ar (n \times (n + 1) / 2)$ – complex array: The factorization stores the $n$ by $n$ matrix $A^{- 1}$ stored in RFP format.
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: The leading minor of order $_$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$ . There is no function specifically designed to invert a Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling nag_lapack_zhetri (f07mw).

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_dpftri (f07wj).

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 RFP format, and must first be factorized by nag_lapack_zpftrf (f07wr).

Open in the MATLAB editor: f07ww_example

function f07ww_example


fprintf('f07ww example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 4.09 + 0.00i  2.33 - 0.14i;
       3.23 + 0.00i  4.29 + 0.00i;
       1.51 + 1.92i  3.58 + 0.00i;
       1.90 - 0.84i -0.23 - 1.11i;
       0.42 - 2.50i -1.18 - 1.37i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% Factorize a
[ar, info] = f07wr(transr, uplo, n, ar);

if info == 0
  % Compute inverse of a
  [ar, info] = f07ww( ...
                      transr, uplo, n, ar);
  % Convert inverse to full array form for display
  [a, info] = f01vh( ...
                     transr, uplo, n, ar);
  fprintf('\n');
  ncols  = int64(80);
  indent = int64(0);
  form   = 'f7.4';
  title  = 'Inverse, lower triangle:';
  diag   = 'n';
  [ifail] = x04db( ...
                   uplo, diag, a, 'brackets', form, title, ...
                   'int', 'int', ncols, indent);
else
  fprintf('\na is not positive definite.\n');
end

f07ww example results


 Inverse, lower triangle:
                    1                 2                 3                 4
 1  ( 5.4691, 0.0000)
 2  (-1.2624,-1.5491) ( 1.1024, 0.0000)
 3  (-2.9746,-0.9616) ( 0.8989,-0.5672) ( 2.1589,-0.0000)
 4  ( 1.1962, 2.9772) (-0.9826,-0.2566) (-1.3756,-1.4550) ( 2.2934,-0.0000)

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox