NAG CL Interface
f01fqc (complex_gen_matrix_pow)
1
Purpose
f01fqc computes an abitrary real power ${A}^{p}$ of a complex $n$ by $n$ matrix $A$.
2
Specification
void 
f01fqc (Integer n,
Complex a[],
Integer pda,
double p,
NagError *fail) 

The function may be called by the names: f01fqc or nag_matop_complex_gen_matrix_pow.
3
Description
For a matrix
$A$ with no eigenvalues on the closed negative real line,
${A}^{p}$ (
$p\in \mathbb{R}$) can be defined as
where
$\mathrm{log}\left(A\right)$ is the principal logarithm of
$A$ (the unique logarithm whose spectrum lies in the strip
$\left\{z:\pi <\mathrm{Im}\left(z\right)<\pi \right\}$).
${A}^{p}$ is computed using 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 $q\in \left(1,1\right)$ and $r\in \mathbb{Z}$. 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 and a Padé approximant.
4
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 SIAM J. Matrix Anal. Appl. 34(3) 1341–1360
5
Arguments

1:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

2:
$\mathbf{a}\left[\mathit{dim}\right]$ – Complex
Input/Output

Note: the dimension,
dim, of the array
a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the
$n$ by
$n$ matrix
$p$th power,
${A}^{p}$. Alternatively, if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NEGATIVE_EIGVAL, contains an
$n$ by
$n$ nonprincipal power of
$A$.

3:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.

4:
$\mathbf{p}$ – double
Input

On entry: the required power of $A$.

5:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NEGATIVE_EIGVAL

$A$ has eigenvalues on the negative real line. The principal $p$th power is not defined so a nonprincipal power is returned.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_SINGULAR

$A$ is singular so the $p$th power cannot be computed.
 NW_SOME_PRECISION_LOSS

${A}^{p}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
7
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 ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$) and noninteger $p$, 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.
8
Parallelism and Performance
f01fqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fqc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The cost of the algorithm is $O\left({n}^{3}\right)$. The exact cost depends on the matrix $A$ but if $p\in \left(1,1\right)$ then the cost is independent of $p$.
$O\left(4\times {n}^{2}\right)$ complex allocatable memory is required by the function.
If estimates of the condition number of
${A}^{p}$ are required then
f01kec should be used.
10
Example
This example finds
${A}^{p}$ where
$p=0.2$ and
10.1
Program Text
10.2
Program Data
10.3
Program Results