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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_real_gen_matrix_cond_pow (f01je)

▸▿ 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_cond_pow (f01je) computes an estimate of the relative condition number

κ_{A^{p}}

of the

p

th power (where

p

is real) of a real

n

n

matrix

A

, in the

1

-norm. The principal matrix power

A^{p}

is also returned.

Syntax

[a, condpa, ifail] = f01je(a, p, 'n', n)

[a, condpa, ifail] = nag_matop_real_gen_matrix_cond_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\}

The Fréchet derivative of the matrix

p

th power of

A

is the unique linear mapping

E ⟼ L (A, E)

such that for any matrix

E

{(A + E)}^{p} - A^{p} - L (A, E) = o (‖E‖) .

The derivative describes the first-order effect of perturbations in

A

on the matrix power

A^{p}

The relative condition number of the matrix

p

th power can be defined by

κ_{A^{p}} = \frac{‖L (A)‖ ‖A‖}{‖A^{p}‖},

where

‖L (A)‖

is the norm of the Fréchet derivative of the matrix power at

A

nag_matop_real_gen_matrix_cond_pow (f01je) uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute

κ_{A^{p}}

and

A^{p}

. 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 using a Schur decomposition, a Padé approximant and the scaling and squaring method.

To obtain an estimate of

κ_{A^{p}}

, nag_matop_real_gen_matrix_cond_pow (f01je) first estimates

‖L (A)‖

by computing an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

. This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of

A^{q}

are obtained by differentiating the Padé approximant. Fréchet derivatives of

A^{p}

are then computed using a combination of the chain rule and the product rule for Fréchet derivatives.

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 and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
$n$ , 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 $n$ principal matrix $p$ th power, $A^{p}$ .
2: $condpa$ – double scalar: If $ifail = 0$ or $3$ , an estimate of the relative condition number of the matrix $p$ th power, $κ_{A^{p}}$ . Alternatively, if $ifail = 4$ , the absolute condition number of the matrix $p$ th power.
3: $ifail$ – 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 in this case; nag_matop_complex_gen_matrix_cond_pow (f01ke) 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.

$ifail = 4$: The relative condition number is infinite. The absolute condition number was returned instead.

$ifail = 5$: 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

nag_matop_real_gen_matrix_cond_pow (f01je) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04yd) to produce an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_real_gen_norm_rcomm (f04yd).

For a normal matrix

A

(for which

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

), the Schur decomposition is diagonal and the computation of the fractional part of the matrix power 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. See Higham and Lin (2011) and Higham and Lin (2013) for details and further discussion.

Further Comments

The amount of real allocatable memory required by the algorithm is typically of the order

10 \times n^{2}

The cost of the algorithm is

O (n^{3})

floating-point operations; see Higham and Lin (2013).

If the matrix

p

th power alone is required, without an estimate of the condition number, then nag_matop_real_gen_matrix_pow (f01eq) should be used. If the Fréchet derivative of the matrix power is required then nag_matop_real_gen_matrix_frcht_pow (f01jf) should be used. If

A

has negative real eigenvalues then nag_matop_complex_gen_matrix_cond_pow (f01ke) can be used to return a complex, non-principal

p

th power and its condition number.

Example

This example estimates the relative condition number of the matrix power

A^{p}

, where

p = 0.2

and

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

Open in the MATLAB editor: f01je_example

function f01je_example


fprintf('f01je example results\n\n');

% Principal power p of matrix A and relative condition number.

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

p = 0.2;

[pa, condpa, ifail] = f01je(a,p);

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

fprintf('\nEstimated relative condition number is : %6.2f\n', condpa);

f01je example results

A^p:
    1.2368    0.1977    0.1749   -0.0314
   -0.0543    1.1643   -0.0947    0.3145
    0.0537    0.3514    1.3254    0.0214
    0.3339   -0.2125    0.1880    1.0581


Estimated relative condition number is :   2.75

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Chapter Contents

Chapter Introduction

NAG Toolbox