NAG CL Interface
f01jfc (real_gen_matrix_frcht_pow)
1
Purpose
f01jfc computes the Fréchet derivative $L\left(A,E\right)$ of the $p$th power (where $p$ is real) of the real $n$ by $n$ matrix $A$ applied to the real $n$ by $n$ matrix $E$. The principal matrix power ${A}^{p}$ is also returned.
2
Specification
void 
f01jfc (Integer n,
double a[],
Integer pda,
double e[],
Integer pde,
double p,
NagError *fail) 

The function may be called by the names: f01jfc or nag_matop_real_gen_matrix_frcht_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
$\left\{z:\pi <\mathrm{Im}\left(z\right)<\pi \right\}$).
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 ${A}^{p}$.
f01jfc 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 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:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
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: the $n$ by $n$ principal matrix $p$th power, ${A}^{p}$.

3:
$\mathbf{pda}$ – Integer
Input

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

4:
$\mathbf{e}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
e
must be at least
${\mathbf{pde}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $E$ is stored in ${\mathbf{e}}\left[\left(j1\right)\times {\mathbf{pde}}+i1\right]$.
On entry: the $n$ by $n$ matrix $E$.
On exit: the Fréchet derivative $L\left(A,E\right)$.

5:
$\mathbf{pde}$ – Integer
Input

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

6:
$\mathbf{p}$ – double
Input

On entry: the required power of $A$.

7:
$\mathbf{fail}$ – 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: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pde}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pde}}\ge {\mathbf{n}}$.
 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 in this case;
f01kfc can be used to find a complex, nonprincipal
$p$th power.
 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 a normal matrix
$A$ (for which
${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), 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
f01jec should be used.
8
Parallelism and Performance
f01jfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jfc 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 real 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
f01eqc should be used. If the condition number of the matrix power is required then
f01jec should be used. If
$A$ has negative real eigenvalues then
f01kfc can be used to return a complex, nonprincipal
$p$th power and its Fréchet derivative
$L\left(A,E\right)$.
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
10.2
Program Data
10.3
Program Results