# NAG CL Interfacef01fqc (complex_​gen_​matrix_​pow)

## 1Purpose

f01fqc computes an abitrary real power ${A}^{p}$ of a complex $n$ by $n$ matrix $A$.

## 2Specification

 #include
 void f01fqc (Integer n, Complex a[], Integer pda, double p, NagError *fail)
The function may be called by the names: f01fqc or nag_matop_complex_gen_matrix_pow.

## 3Description

For a matrix $A$ with no eigenvalues on the closed negative real line, ${A}^{p}$ ($p\in ℝ$) can be defined as
 $Ap= expplogA$
where $\mathrm{log}\left(A\right)$ is the principal logarithm of $A$ (the unique logarithm whose spectrum lies in the strip $\left\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \right\}$).
${A}^{p}$ is computed using the Schur–Padé algorithm described in Higham and Lin (2011) and Higham and Lin (2013).
The real number $p$ is expressed as $p=q+r$ where $q\in \left(-1,1\right)$ and $r\in ℤ$. 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 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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the $n$ by $n$ matrix $p$th power, ${A}^{p}$. Alternatively, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NEGATIVE_EIGVAL, contains an $n$ by $n$ non-principal power of $A$.
3: $\mathbf{pda}$Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
4: $\mathbf{p}$double Input
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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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 so a non-principal power is 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 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 ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$) 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

f01fqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fqc 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.

The cost of the algorithm is $O\left({n}^{3}\right)$. The exact cost depends on the matrix $A$ but if $p\in \left(-1,1\right)$ then the cost is independent of $p$. $O\left(4×{n}^{2}\right)$ complex allocatable memory is required by the function.
If estimates of the condition number of ${A}^{p}$ are required then f01kec should be used.

## 10Example

This example finds ${A}^{p}$ where $p=0.2$ and
 $A = 2i+ 3i+0 2i+0 1+3i 2+i 1i+0 1i+0 2+2i 2+i 2+2i 2i 2+4i 3i+ 2+2i 3i+0 1i+0 .$

### 10.1Program Text

Program Text (f01fqce.c)

### 10.2Program Data

Program Data (f01fqce.d)

### 10.3Program Results

Program Results (f01fqce.r)