# NAG Library Function Document

## 1Purpose

nag_matop_real_gen_matrix_cond_pow (f01jec) computes an estimate of the relative condition number ${\kappa }_{{A}^{p}}$ of the $p$th power (where $p$ is real) of a real $n$ by $n$ matrix $A$, in the $1$-norm. The principal matrix power ${A}^{p}$ is also returned.

## 2Specification

 #include #include
 void nag_matop_real_gen_matrix_cond_pow (Integer n, double a[], Integer pda, double p, double *condpa, NagError *fail)

## 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 $\left\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \right\}$).
The Fréchet derivative of the matrix $p$th power of $A$ is the unique linear mapping $E⟼L\left(A,E\right)$ such that for any matrix $E$
 $A+Ep - Ap - LA,E = oE .$
The derivative describes the first-order effect of perturbations in $A$ on the matrix power ${A}^{p}$.
The relative condition number of the matrix $p$th power can be defined by
 $κAp = LA A Ap ,$
where $‖L\left(A\right)‖$ is the norm of the Fréchet derivative of the matrix power at $A$.
nag_matop_real_gen_matrix_cond_pow (f01jec) 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 $q\in \left(-1,1\right)$ 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, a Padé approximant and the scaling and squaring method.
To obtain an estimate of ${\kappa }_{{A}^{p}}$, nag_matop_real_gen_matrix_cond_pow (f01jec) first estimates $‖L\left(A\right)‖$ by computing an estimate $\gamma$ of a quantity $K\in \left[{n}^{-1}{‖L\left(A\right)‖}_{1},n{‖L\left(A\right)‖}_{1}\right]$, 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.

## 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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/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: the $n$ by $n$ principal 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:    $\mathbf{condpa}$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 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation 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; nag_matop_complex_gen_matrix_cond_pow (f01kec) 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 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_matop_real_gen_matrix_cond_pow (f01jec) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04ydc) to produce an estimate $\gamma$ of a quantity $K\in \left[{n}^{-1}{‖L\left(A\right)‖}_{1},n{‖L\left(A\right)‖}_{1}\right]$, such that $\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_real_gen_norm_rcomm (f04ydc).
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.

## 8Parallelism and Performance

nag_matop_real_gen_matrix_cond_pow (f01jec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_cond_pow (f01jec) 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 amount of real allocatable memory required by the algorithm is typically of the order $10×{n}^{2}$.
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 nag_matop_real_gen_matrix_pow (f01eqc) should be used. If the Fréchet derivative of the matrix power is required then nag_matop_real_gen_matrix_frcht_pow (f01jfc) should be used. If $A$ has negative real eigenvalues then nag_matop_complex_gen_matrix_cond_pow (f01kec) can be used to return a complex, non-principal $p$th power and its condition number.

## 10Example

This example estimates the relative condition number of the matrix power ${A}^{p}$, where $p=0.2$ and
 $A = 3 3 2 1 1 1 0 2 1 4 4 2 3 1 3 1 .$

### 10.1Program Text

Program Text (f01jece.c)

### 10.2Program Data

Program Data (f01jece.d)

### 10.3Program Results

Program Results (f01jece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017