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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_real_gen_matrix_pow (f01eq)

▸▿ 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_matop_real_gen_matrix_pow (f01eq) computes the principal real power

A^{p}

, for arbitrary

p

, of a real

n

n

matrix

A

Syntax

[a, ifail] = f01eq(a, p, 'n', n)

[a, ifail] = nag_matop_real_gen_matrix_pow(a, p, 'n', n)

Description

For a matrix

A

with no eigenvalues on the closed negative real line,

A^{p}

(

p \in ℝ

) can be defined as

A^{p} = \exp (p \log (A))

where

\log (A)

is the principal logarithm of

A

(the unique logarithm whose spectrum lies in the strip

\{z : - π < Im (z) < π\}

A^{p}

is computed using the real version of the Schur–Padé algorithm described in Higham and Lin (2011) and Higham and Lin (2013).

The real number

p

is expressed as

p = q + r

where

q \in (- 1, 1)

and

r \in ℤ

. Then

A^{p} = A^{q} A^{r}

. The integer power

A^{r}

is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power

A^{q}

is computed, entirely in real arithmetic, using a real Schur decomposition and a Padé approximant.

References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Higham N J and Lin L (2011) A Schur–Padé algorithm for fractional powers of a matrix SIAM J. Matrix Anal. Appl. 32(3) 1056–1078

Higham N J and Lin L (2013) An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives MIMS Eprint 2013.1 Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester http://eprints.ma.man.ac.uk/

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – double array: The first dimension of the array a must be at least $n$ .
The second dimension of the array a must be at least $n$ .

The $n$ by $n$ matrix $A$ .
2: $p$ – double scalar: The required power of $A$ .

Optional Input Parameters

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

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $n$ .
The second dimension of the array a will be $n$ .

The $n$ by $n$ matrix $p$ th power, $A^{p}$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: $A$ has eigenvalues on the negative real line. The principal $p$ th power is not defined. nag_matop_complex_gen_matrix_pow (f01fq) can be used to find a complex, non-principal $p$ th power.

$ifail = 2$: $A$ is singular so the $p$ th power cannot be computed.

$ifail = 3$: $A^{p}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

$ifail = 4$: An unexpected internal error occurred. This failure should not occur and suggests that the function has been called incorrectly.

$ifail = - 1$: Constraint: $n \geq 0$ .

$ifail = - 3$: Constraint: $lda \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

For positive integer

p

, the algorithm reduces to a sequence of matrix multiplications. For negative integer

p

, the algorithm consists of a combination of matrix inversion and matrix multiplications.

For a normal matrix

A

(for which

A^{T} A = A A^{T}

) and non-integer

p

, the Schur decomposition is diagonal and the algorithm reduces to evaluating powers of the eigenvalues of

A

and then constructing

A^{p}

using the Schur vectors. This should give a very accurate result. In general however, no error bounds are available for the algorithm.

Further Comments

The cost of the algorithm is

O (n^{3})

. The exact cost depends on the matrix

A

but if

p \in (- 1, 1)

then the cost is independent of

p

O (4 \times n^{2})

of real allocatable memory is required by the function.

If estimates of the condition number of

A^{p}

are required then nag_matop_real_gen_matrix_cond_pow (f01je) should be used.

Example

This example finds

A^{p}

where

p = 0.2

and

A = (\begin{array}{r} 3 & 3 & 2 & 1 \\ 3 & 1 & 0 & 2 \\ 1 & 1 & 4 & 3 \\ 3 & 0 & 3 & 1 \end{array}) .

Open in the MATLAB editor: f01eq_example

function f01eq_example


fprintf('f01eq example results\n\n');

% Principal power p of matrix A

a = [ 3 3 2 1;
      3 1 0 2;
      1 1 4 3;
      3 0 3 1];

p = 0.2;

[pa, ifail] = f01eq(a,p);

disp('A^p:');
disp(pa);

f01eq example results

A^p:
    1.2446    0.2375    0.2172   -0.1359
    0.0925    1.1239   -0.1453    0.3731
   -0.0769    0.1972    1.3131    0.1837
    0.3985   -0.2902    0.1085    1.1560

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Chapter Contents

Chapter Introduction

NAG Toolbox