nag_matop_complex_gen_matrix_cond_pow (f01kec) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_matop_complex_gen_matrix_cond_pow (f01kec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_matop_complex_gen_matrix_cond_pow (f01kec) computes an estimate of the relative condition number κAp of the pth power (where p is real) of a complex n by n matrix A, in the 1-norm. The principal matrix power Ap is also returned.

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_matop_complex_gen_matrix_cond_pow (Integer n, Complex a[], Integer pda, double p, double *condpa, 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).
The Fréchet derivative of the matrix pth power of A is the unique linear mapping ELA,E 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 Ap.
The relative condition number of the matrix pth power can be defined by
κAp = LA A Ap ,
where LA is the norm of the Fréchet derivative of the matrix power at A.
nag_matop_complex_gen_matrix_cond_pow (f01kec) uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute κAp and Ap. 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 using a Schur decomposition, a Padé approximant and the scaling and squaring method.
To obtain the estimate of κAp, nag_matop_complex_gen_matrix_cond_pow (f01kec) first estimates LA by computing an estimate γ of a quantity Kn-1LA1,nLA1, such that γK. This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of Aq are obtained by differentiating the Padé approximant. Fréchet derivatives of Ap 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 nag_matop_complex_gen_matrix_cond_pow (f01kec) will return a non-principal matrix pth power and its condition number.

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]ComplexInput/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 principal matrix pth power, Ap, unless fail.code= NE_NEGATIVE_EIGVAL, in which case a non-principal pth power is returned.
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:     condpadouble *Output
On exit: if fail.code= NE_NOERROR or NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix pth power, κAp. Alternatively, if fail.code= NE_RCOND, the absolute condition number of the matrix pth power.
6:     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 in this case, so a non-principal power was returned.
The relative condition number is infinite. The absolute condition number was returned instead.
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

nag_matop_complex_gen_matrix_cond_pow (f01kec) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zdc) to produce an estimate γ of a quantity Kn-1LA1,nLA1, such that γK. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zdc).
For a normal matrix A (for which AHA=AAH), 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 Ap 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.

8  Parallelism and Performance

nag_matop_complex_gen_matrix_cond_pow (f01kec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_gen_matrix_cond_pow (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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The amount of complex allocatable memory required by the algorithm is typically of the order 10×n2.
The cost of the algorithm is On3 floating-point operations; see Higham and Lin (2013).
If the matrix pth power alone is required, without an estimate of the condition number, then nag_matop_complex_gen_matrix_pow (f01fqc) should be used. If the Fréchet derivative of the matrix power is required then nag_matop_complex_gen_matrix_frcht_pow (f01kfc) should be used. The real analogue of this function is nag_matop_real_gen_matrix_cond_pow (f01jec).

10  Example

This example estimates the relative condition number of the matrix power Ap, where p=0.4 and
A = 1+2i 3 2 1+3i 1+i 1 1 2+i 1 2 1 2i 3 i 2+i 1 .

10.1  Program Text

Program Text (f01kece.c)

10.2  Program Data

Program Data (f01kece.d)

10.3  Program Results

Program Results (f01kece.r)

nag_matop_complex_gen_matrix_cond_pow (f01kec) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

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