NAG Library Routine Document
F01KFF
1 Purpose
F01KFF computes the Fréchet derivative $L\left(A,E\right)$ of the $p$th power (where $p$ is real) of the complex $n$ by $n$ matrix $A$ applied to the complex $n$ by $n$ matrix $E$. The principal matrix power ${A}^{p}$ is also returned.
2 Specification
INTEGER 
N, LDA, LDE, IFAIL 
REAL (KIND=nag_wp) 
P 
COMPLEX (KIND=nag_wp) 
A(LDA,*), E(LDE,*) 

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
$\left\{z:\pi <\mathrm{Im}\left(z\right)<\pi \right\}$). If
$A$ is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but a nonprincipal
$p$th power can be defined by using a nonprincipal logarithm.
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 ${A}^{p}$.
F01KFF uses the algorithms of
Higham and Lin (2011) and
Higham and Lin (2013) to compute
${A}^{p}$ and
$L\left(A,E\right)$. 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. The Padé approximant is differentiated in order to obtain the Fréchet derivative of
${A}^{q}$ and
$L\left(A,E\right)$ is then computed using a combination of the chain rule and the product rule for Fréchet derivatives.
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, ${A}^{p}$. Alternatively if ${\mathbf{IFAIL}}={\mathbf{1}}$, 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 F01KFF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{E}\left({\mathbf{LDE}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
E
must be at least
${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $E$.
On exit: the Fréchet derivative $L\left(A,E\right)$.
 5: $\mathrm{LDE}$ – INTEGERInput

On entry: the first dimension of the array
E as declared in the (sub)program from which F01KFF is called.
Constraint:
${\mathbf{LDE}}\ge {\mathbf{N}}$.
 6: $\mathrm{P}$ – REAL (KIND=nag_wp)Input

On entry: the required power of $A$.
 7: $\mathrm{IFAIL}$ – 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.
 ${\mathbf{IFAIL}}=4$

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: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=5$

On entry, ${\mathbf{LDE}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDE}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=99$
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 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.
If the condition number of the matrix power is required then
F01KEF should be used.
8 Parallelism and Performance
F01KFF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01KFF 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 complex allocatable memory required by the algorithm is approximately $6\times {n}^{2}$.
The cost of the algorithm is
$O\left({n}^{3}\right)$ floatingpoint operations; see
Higham and Lin (2011) and
Higham and Lin (2013).
If the matrix
$p$th power alone is required, without the Fréchet derivative, then
F01FQF should be used. If the condition number of the matrix power is required then
F01KEF should be used. The real analogue of this routine is
F01JFF.
10 Example
This example finds
${A}^{p}$ and the Fréchet derivative of the matrix power
$L\left(A,E\right)$, where
$p=0.2$,
10.1 Program Text
Program Text (f01kffe.f90)
10.2 Program Data
Program Data (f01kffe.d)
10.3 Program Results
Program Results (f01kffe.r)