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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_matrix_cond_pow (f01ke)

Purpose

nag_matop_complex_gen_matrix_cond_pow (f01ke) computes an estimate of the relative condition number ${\kappa }_{{A}^{p}}$ of the $p$th power (where $p$ is real) of a complex $n$ by $n$ matrix $A$, in the $1$-norm. The principal matrix power ${A}^{p}$ is also returned.

Syntax

[a, condpa, ifail] = f01ke(a, p, 'n', n)
[a, condpa, ifail] = nag_matop_complex_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
 $Ap= expplogA$
where $\mathrm{log}\left(A\right)$ is the principal logarithm of $A$ (the unique logarithm whose spectrum lies in the strip $\left\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \right\}$).
The Fréchet derivative of the matrix $p$th power of $A$ is the unique linear mapping $E⟼L\left(A,E\right)$ such that for any matrix $E$
 $A+Ep - Ap - LA,E = oE .$
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
 $κAp = LA A Ap ,$
where $‖L\left(A\right)‖$ is the norm of the Fréchet derivative of the matrix power at $A$.
nag_matop_complex_gen_matrix_cond_pow (f01ke) uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute ${\kappa }_{{A}^{p}}$ and ${A}^{p}$. The real number $p$ is expressed as $p=q+r$ where $q\in \left(-1,1\right)$ 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 the estimate of ${\kappa }_{{A}^{p}}$, nag_matop_complex_gen_matrix_cond_pow (f01ke) first estimates $‖L\left(A\right)‖$ by computing an estimate $\gamma$ of a quantity $K\in \left[{n}^{-1}{‖L\left(A\right)‖}_{1},n{‖L\left(A\right)‖}_{1}\right]$, such that $\gamma \le K$. 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.
If $A$ is nonsingular but has negative real eigenvalues nag_matop_complex_gen_matrix_cond_pow (f01ke) will return a non-principal matrix $p$th power and its condition number.

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least ${\mathbf{n}}$.
The second dimension of the array a must be at least ${\mathbf{n}}$.
The $n$ by $n$ matrix $A$.
2:     $\mathrm{p}$ – double scalar
The required power of $A$.

Optional Input Parameters

1:     $\mathrm{n}$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: ${\mathbf{n}}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be ${\mathbf{n}}$.
The second dimension of the array a will be ${\mathbf{n}}$.
The $n$ by $n$ principal matrix $p$th power, ${A}^{p}$, unless ${\mathbf{ifail}}={\mathbf{1}}$, in which case a non-principal $p$th power is returned.
2:     $\mathrm{condpa}$ – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{3}}$, an estimate of the relative condition number of the matrix $p$th power, ${\kappa }_{{A}^{p}}$. Alternatively, if ${\mathbf{ifail}}={\mathbf{4}}$, the absolute condition number of the matrix $p$th power.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
$A$ has eigenvalues on the negative real line. The principal $p$th power is not defined in this case, so a non-principal power was returned.
${\mathbf{ifail}}=2$
$A$ is singular so the $p$th power cannot be computed.
${\mathbf{ifail}}=3$
${A}^{p}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
${\mathbf{ifail}}=4$
The relative condition number is infinite. The absolute condition number was returned instead.
${\mathbf{ifail}}=5$
An unexpected internal error occurred. This failure should not occur and suggests that the function has been called incorrectly.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
Constraint: $\mathit{lda}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

nag_matop_complex_gen_matrix_cond_pow (f01ke) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zd) to produce an estimate $\gamma$ of a quantity $K\in \left[{n}^{-1}{‖L\left(A\right)‖}_{1},n{‖L\left(A\right)‖}_{1}\right]$, such that $\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zd).
For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$), 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.

The amount of complex allocatable memory required by the algorithm is typically of the order $10×{n}^{2}$.
The cost of the algorithm is $O\left({n}^{3}\right)$ 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_complex_gen_matrix_pow (f01fq) should be used. If the Fréchet derivative of the matrix power is required then nag_matop_complex_gen_matrix_frcht_pow (f01kf) should be used. The real analogue of this function is nag_matop_real_gen_matrix_cond_pow (f01je).

Example

This example estimates the relative condition number of the matrix power ${A}^{p}$, where $p=0.4$ and
 $A = 1+2i 3 2 1+3i 1+i 1 1 2+i 1 2 1 2i 3 i 2+i 1 .$
```function f01ke_example

fprintf('f01ke example results\n\n');

% Principal power p of matrix A

a = [ 1+2i  3      2     1+3i;
1+ i  1      1     2+ i;
1     2      1       2i;
3        i   2+i   1];

p = 0.4;

[pa, condpa, ifail] = f01ke(a,p);

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

fprintf('Estimated condition number is: %6.2f\n', condpa)

```
```f01ke example results

A^p:
0.9742 + 0.5211i   0.8977 - 0.1170i   0.6389 - 0.3900i   0.0975 + 0.6205i
0.1586 + 0.2763i   1.0176 - 0.0250i   0.0623 - 0.3471i   0.6431 + 0.2560i
0.2589 - 0.5817i   0.5633 + 0.3969i   1.1470 + 0.4042i  -0.3771 + 0.3113i
0.8713 - 0.0270i  -0.5734 + 0.0868i   0.2816 + 0.3739i   1.3568 - 0.2709i

Estimated condition number is:   6.86
```