f01kec computes an estimate of the relative condition number ${\kappa}_{{A}^{p}}$ of the $p$th power (where $p$ is real) of a complex $n\times n$ matrix $A$, in the $1$-norm. 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 $\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \}$).
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 $E$
where $\Vert L\left(A\right)\Vert $ is the norm of the Fréchet derivative of the matrix power at $A$.
f01kec 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 $q\in (\mathrm{-1},1)$ 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.
To obtain the estimate of ${\kappa}_{{A}^{p}}$, f01kec first estimates $\Vert L\left(A\right)\Vert $ by computing an estimate $\gamma $ of a quantity $K\in [{n}^{-1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}]$, 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 f01kec will return a non-principal matrix $p$th power and its condition number.
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, ${A}^{p}$, unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_NEGATIVE_EIGVAL, in which case a non-principal $p$th power is returned.
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: $\mathbf{condpa}$ – double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix $p$th power, ${\kappa}_{{A}^{p}}$. Alternatively, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_RCOND, the absolute condition number of the matrix $p$th power.
6: $\mathbf{fail}$ – 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 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_RCOND
The relative condition number is infinite. The absolute condition number was returned instead.
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
f01kec uses the norm estimation function f04zdc to produce an estimate $\gamma $ of a quantity $K\in [{n}^{\mathrm{-1}}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}]$, such that $\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for f04zdc.
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.
8Parallelism and Performance
f01kec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kec 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 amount of complex allocatable memory required by the algorithm is typically of the order $10\times {n}^{2}$.
The cost of the algorithm is $O\left({n}^{3}\right)$ floating-point operations; see Higham and Lin (2013).
If the matrix $p$th power alone is required, without an estimate of the condition number, then f01fqc should be used. If the Fréchet derivative of the matrix power is required then f01kfc should be used. The real analogue of this function is f01jec.
10Example
This example estimates the relative condition number of the matrix power ${A}^{p}$, where $p=0.4$ and