f01kfc computes the Fréchet derivative $L(A,E)$ of the $p$th power (where $p$ is real) of the complex $n\times n$ matrix $A$ applied to the complex $n\times n$ matrix $E$. The principal matrix power ${A}^{p}$ is also returned.
where $\mathrm{log}\left(A\right)$ is the principal logarithm of $A$ (the unique logarithm whose spectrum lies in the strip $\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \}$). If $A$ is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but a non-principal $p$th power can be defined by using a non-principal logarithm.
The Fréchet derivative of the matrix $p$th power of $A$ is the unique linear mapping $E\u27fcL(A,E)$ such that for any matrix $E$
The derivative describes the first-order effect of perturbations in $A$ on the matrix power ${A}^{p}$.
f01kfc uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute ${A}^{p}$ and $L(A,E)$. The real number $p$ is expressed as $p=q+r$ where $q\in (\mathrm{-1},1)$ 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 $L(A,E)$ is then computed using a combination of the chain rule and the product rule for Fréchet derivatives.
4References
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
Note: the dimension, dim, of the array a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $(i,j)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[(j-1)\times {\mathbf{pda}}+i-1\right]$.
On entry: the $n\times n$ matrix $A$.
On exit: the $n\times n$ principal matrix $p$th power, ${A}^{p}$. Alternatively if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_NEGATIVE_EIGVAL, a non-principal $p$th power is returned.
3: $\mathbf{pda}$ – IntegerInput
On entry: the stride separating matrix row elements in the array a.
Note: the dimension, dim, of the array e
must be at least
${\mathbf{pde}}\times {\mathbf{n}}$.
The $(i,j)$th element of the matrix $E$ is stored in ${\mathbf{e}}\left[(j-1)\times {\mathbf{pde}}+i-1\right]$.
On entry: the $n\times n$ matrix $E$.
On exit: the Fréchet derivative $L(A,E)$.
5: $\mathbf{pde}$ – IntegerInput
On entry: the stride separating matrix row elements in the array e.
Constraint:
${\mathbf{pde}}\ge {\mathbf{n}}$.
6: $\mathbf{p}$ – doubleInput
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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pde}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. 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, so a non-principal power was 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.
7Accuracy
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 f01kec should be used.
8Parallelism and Performance
f01kfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kfc 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 implementation-specific information.
9Further Comments
The complex allocatable memory required by the algorithm is approximately $6\times {n}^{2}$.
If the matrix $p$th power alone is required, without the Fréchet derivative, then f01fqc should be used. If the condition number of the matrix power is required then f01kec should be used. The real analogue of this function is f01jfc.
10Example
This example finds ${A}^{p}$ and the Fréchet derivative of the matrix power $L(A,E)$, where $p=0.2$,