nag_matop_complex_gen_matrix_frcht_log (f01kkc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_matop_complex_gen_matrix_frcht_log (f01kkc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_matop_complex_gen_matrix_frcht_log (f01kkc) computes the Fréchet derivative LA,E of the matrix logarithm of the complex n by n matrix A applied to the complex n by n matrix E. The principal matrix logarithm logA is also returned.

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_matop_complex_gen_matrix_frcht_log (Integer n, Complex a[], Integer pda, Complex e[], Integer pde, NagError *fail)

3  Description

For a matrix with no eigenvalues on the closed negative real line, the principal matrix logarithm logA is the unique logarithm whose spectrum lies in the strip z:-π<Imz<π.
The Fréchet derivative of the matrix logarithm of A is the unique linear mapping ELA,E such that for any matrix E 
logA+E - logA - LA,E = oE .
The derivative describes the first order effect of perturbations in A on the logarithm logA.
nag_matop_complex_gen_matrix_frcht_log (f01kkc) uses the algorithm of Al–Mohy et al. (2012) to compute logA and LA,E. The principal matrix logarithm logA is computed using a Schur decomposition, a Padé approximant and the inverse scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative LA,E. If A is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but nag_matop_complex_gen_matrix_frcht_log (f01kkc) will return a non-principal logarithm and Fréchet derivative.

4  References

Al–Mohy A H and Higham N J (2011) Improved inverse scaling and squaring algorithms for the matrix logarithm SIAM J. Sci. Comput. 34(4) C152–C169
Al–Mohy A H, Higham N J and Relton S D (2012) Computing the Fréchet derivative of the matrix logarithm and estimating the condition number MIMS EPrint 2012.72
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

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 logarithm, logA. Alterntively, if fail.code= NE_NEGATIVE_EIGVAL, a non-principal logarithm is returned.
3:     pdaIntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4:     e[dim]ComplexInput/Output
Note: the dimension, dim, of the array e must be at least pde×n.
The i,jth element of the matrix E is stored in e[j-1×pde+i-1].
On entry: the n by n matrix E
On exit: with fail.code= NE_NOERROR, NE_NEGATIVE_EIGVAL or NW_SOME_PRECISION_LOSS, the Fréchet derivative LA,E
5:     pdeIntegerInput
On entry: the stride separating matrix row elements in the array e.
Constraint: pden.
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.
On entry, pde=value and n=value.
Constraint: pden.
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 logarithm is not defined in this case, so a non-principal logarithm was returned.
A is singular so the logarithm cannot be computed.
logA has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7  Accuracy

For a normal matrix A (for which AHA=AAH), the Schur decomposition is diagonal and the computation of the matrix logarithm reduces to evaluating the logarithm of the eigenvalues of A and then constructing logA using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. The sensitivity of the computation of logA and LA,E is worst when A has an eigenvalue of very small modulus or has a complex conjugate pair of eigenvalues lying close to the negative real axis. See Al–Mohy and Higham (2011), Al–Mohy et al. (2012) and Section 11.2 of Higham (2008) for details and further discussion.

8  Parallelism and Performance

nag_matop_complex_gen_matrix_frcht_log (f01kkc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_gen_matrix_frcht_log (f01kkc) 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 floating-point operations. The complex allocatable memory required is approximately 5n2; see Al–Mohy et al. (2012) for further details.
If the matrix logarithm alone is required, without the Fréchet derivative, then nag_matop_complex_gen_matrix_log (f01fjc) should be used. If the condition number of the matrix logarithm is required then nag_matop_complex_gen_matrix_cond_log (f01kjc) should be used. The real analogue of this function is nag_matop_real_gen_matrix_frcht_log (f01jkc).

10  Example

This example finds the principal matrix logarithm logA and the Fréchet derivative LA,E, where
A = 1+4i 3i 0i 2i+ 2i 3i+0 1i+0 1+i 0i 2+0i 2i+0 i 1+2i 3+2i 1+2i 3+i   and   E = 1i+0 1+2i 2i+0 2+i 1+3i 0i 1i+0 0i+ 2i 4+0i 1i+0 1i+ 1i+0 2+2i 3i 1i+ .

10.1  Program Text

Program Text (f01kkce.c)

10.2  Program Data

Program Data (f01kkce.d)

10.3  Program Results

Program Results (f01kkce.r)

nag_matop_complex_gen_matrix_frcht_log (f01kkc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

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