where $\mathrm{log}\left(A\right)$ is the principal logarithm of $A$ (the unique logarithm whose spectrum lies in the strip $\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \}$).
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, entirely in real arithmetic, using a real Schur decomposition and a Padé approximant.
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 $n\times n$ matrix $p$th power, ${A}^{p}$.
3: $\mathbf{pda}$ – IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.
4: $\mathbf{p}$ – doubleInput
On entry: the required power of $A$.
5: $\mathbf{fail}$ – 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: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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. f01fqc can be used to find a complex, non-principal $p$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.
7Accuracy
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 ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$) and non-integer $p$, 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.
8Parallelism and Performance
f01eqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01eqc 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 cost of the algorithm is $O\left({n}^{3}\right)$. The exact cost depends on the matrix $A$ but if $p\in (\mathrm{-1},1)$ then the cost is independent of $p$.
$O(4\times {n}^{2})$ of real allocatable memory is required by the function.
If estimates of the condition number of ${A}^{p}$ are required then f01jec should be used.