NAG Library Routine Document
F01FQF
1 Purpose
F01FQF computes an abitrary real power ${A}^{p}$ of a complex $n$ by $n$ matrix $A$.
2 Specification
INTEGER 
N, LDA, IFAIL 
REAL (KIND=nag_wp) 
P 
COMPLEX (KIND=nag_wp) 
A(LDA,*) 

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
MIMS Eprint 2013.1 Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester
http://eprints.ma.man.ac.uk/
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
A
must be at least
${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: if ${\mathbf{IFAIL}}={\mathbf{0}}$, the $n$ by $n$ matrix $p$th power, ${A}^{p}$. Alternatively, if ${\mathbf{IFAIL}}={\mathbf{1}}$, contains an $n$ by $n$ nonprincipal power of $A$.
 3: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01FQF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{P}$ – REAL (KIND=nag_wp)Input

On entry: the required power of $A$.
 5: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

$A$ has eigenvalues on the negative real line. The principal $p$th power is not defined so a nonprincipal power is 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.
 ${\mathbf{IFAIL}}=4$

An unexpected internal error occurred. This failure should not occur and suggests that the routine has been called incorrectly.
 ${\mathbf{IFAIL}}=1$

On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}\ge 0$.
 ${\mathbf{IFAIL}}=3$

On entry, ${\mathbf{LDA}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
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
F01FQF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01FQF 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 routine. 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 routine.
If estimates of the condition number of
${A}^{p}$ are required then
F01KEF should be used.
10 Example
This example finds
${A}^{p}$ where
$p=0.2$ and
10.1 Program Text
Program Text (f01fqfe.f90)
10.2 Program Data
Program Data (f01fqfe.d)
10.3 Program Results
Program Results (f01fqfe.r)