NAG Library Routine Document
F01KEF
1 Purpose
F01KEF 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.
2 Specification
INTEGER 
N, LDA, IFAIL 
REAL (KIND=nag_wp) 
P, CONDPA 
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
$\left\{z:\pi <\mathrm{Im}\left(z\right)<\pi \right\}$).
The Fréchet derivative of the matrix
$p$th power of
$A$ is the unique linear mapping
$E\u27fcL\left(A,E\right)$ such that for any matrix
$E$
The derivative describes the firstorder 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
where
$\Vert L\left(A\right)\Vert $ is the norm of the Fréchet derivative of the matrix power at
$A$.
F01KEF 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
$r\in \mathbb{Z}$. 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, a Padé approximant and the scaling and squaring method.
To obtain the estimate of ${\kappa}_{{A}^{p}}$, F01KEF first estimates $\Vert L\left(A\right)\Vert $ by computing an estimate $\gamma $ of a quantity $K\in \left[{n}^{1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}\right]$, 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 F01KEF will return a nonprincipal matrix $p$th power and its condition number.
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: the $n$ by $n$ principal matrix $p$th power, ${A}^{p}$, unless ${\mathbf{IFAIL}}={\mathbf{1}}$, in which case a nonprincipal $p$th power is returned.
 3: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01KEF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{P}$ – REAL (KIND=nag_wp)Input

On entry: the required power of $A$.
 5: $\mathrm{CONDPA}$ – REAL (KIND=nag_wp)Output

On exit: 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.
 6: $\mathrm{IFAIL}$ – 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 in this case, so a nonprincipal 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 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
F01KEF uses the norm estimation routine
F04ZDF to produce an estimate
$\gamma $ of a quantity
$K\in \left[{n}^{1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}\right]$, such that
$\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for
F04ZDF.
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.
8 Parallelism and Performance
F01KEF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01KEF 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 amount of complex allocatable memory required by the algorithm is typically of the order $10\times {n}^{2}$.
The cost of the algorithm is
$O\left({n}^{3}\right)$ floatingpoint operations; see
Higham and Lin (2013).
If the matrix
$p$th power alone is required, without an estimate of the condition number, then
F01FQF should be used. If the Fréchet derivative of the matrix power is required then
F01KFF should be used. The real analogue of this routine is
F01JEF.
10 Example
This example estimates the relative condition number of the matrix power
${A}^{p}$, where
$p=0.4$ and
10.1 Program Text
Program Text (f01kefe.f90)
10.2 Program Data
Program Data (f01kefe.d)
10.3 Program Results
Program Results (f01kefe.r)