nag_matop_real_gen_matrix_pow (f01eqc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_matop_real_gen_matrix_pow (f01eqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_matop_real_gen_matrix_pow (f01eqc) computes the principal real power Ap, for arbitrary p, of a real n by n matrix A.

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_matop_real_gen_matrix_pow (Integer n, double a[], Integer pda, double p, NagError *fail)

3  Description

For a matrix A with no eigenvalues on the closed negative real line, Ap (p) can be defined as
Ap= expplogA
where logA is the principal logarithm of A (the unique logarithm whose spectrum lies in the strip z:-π<Imz<π).
Ap is computed using the real version of 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-1,1 and r. Then Ap=AqAr. The integer power Ar is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power Aq is computed, entirely in real arithmetic, using a real Schur decomposition and a Padé approximant.

4  References

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 MIMS Eprint 2013.1 Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester

5  Arguments

1:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The i,jth element of the matrix A is stored in a[j-1×pda+i-1].
On entry: the n by n matrix A.
On exit: the n by n matrix pth power, Ap.
3:     pdaIntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4:     pdoubleInput
On entry: the required power of A.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, pda=value and n=value.
Constraint: pdan.
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.
A has eigenvalues on the negative real line. The principal pth power is not defined. nag_matop_complex_gen_matrix_pow (f01fqc) can be used to find a complex, non-principal pth power.
A is singular so the pth power cannot be computed.
Ap has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7  Accuracy

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 ATA=AAT) and non-integer p, the Schur decomposition is diagonal and the algorithm reduces to evaluating powers of the eigenvalues of A and then constructing Ap using the Schur vectors. This should give a very accurate result. In general however, no error bounds are available for the algorithm.

8  Parallelism and Performance

nag_matop_real_gen_matrix_pow (f01eqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_pow (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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The cost of the algorithm is On3. The exact cost depends on the matrix A but if p-1,1 then the cost is independent of p. O4×n2 of real allocatable memory is required by the function.
If estimates of the condition number of Ap are required then nag_matop_real_gen_matrix_cond_pow (f01jec) should be used.

10  Example

This example finds Ap where p=0.2 and
A = 3 3 2 1 3 1 0 2 1 1 4 3 3 0 3 1 .

10.1  Program Text

Program Text (f01eqce.c)

10.2  Program Data

Program Data (f01eqce.d)

10.3  Program Results

Program Results (f01eqce.r)

nag_matop_real_gen_matrix_pow (f01eqc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014