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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_real_gen_matrix_cond_log (f01jj)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_matop_real_gen_matrix_cond_log (f01jj) computes an estimate of the relative condition number

κ_{\log} (A)

of the logarithm of a real

n

n

matrix

A

, in the

1

-norm. The principal matrix logarithm

\log (A)

is also returned.

Syntax

[a, condla, ifail] = f01jj(a, 'n', n)

[a, condla, ifail] = nag_matop_real_gen_matrix_cond_log(a, 'n', n)

Description

For a matrix with no eigenvalues on the closed negative real line, the principal matrix logarithm

\log (A)

is the unique logarithm whose spectrum lies in the strip

\{z : - π < Im (z) < π\}

The Fréchet derivative of the matrix logarithm of

A

is the unique linear mapping

E ⟼ L (A, E)

such that for any matrix

E

\log (A + E) - \log (A) - L (A, E) = o (‖E‖) .

The derivative describes the first order effect of perturbations in

A

on the logarithm

\log (A)

The relative condition number of the matrix logarithm can be defined by

κ_{\log} (A) = \frac{‖L (A)‖ ‖A‖}{‖\log (A)‖},

where

‖L (A)‖

is the norm of the Fréchet derivative of the matrix logarithm at

A

To obtain the estimate of

κ_{\log} (A)

, nag_matop_real_gen_matrix_cond_log (f01jj) first estimates

‖L (A)‖

by computing an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

The algorithms used to compute

κ_{\log} (A)

and

\log (A)

are based on a Schur decomposition, the inverse scaling and squaring method and Padé approximants. Further details can be found in Al–Mohy and Higham (2011) and Al–Mohy et al. (2012).

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

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – double array: The first dimension of the array a must be at least $n$ .
The second dimension of the array a must be at least $n$ .

The $n$ by $n$ matrix $A$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $n$ .
The second dimension of the array a will be $n$ .

The $n$ by $n$ principal matrix logarithm, $\log (A)$ .
2: $condla$ – double scalar: With $ifail = 0$ or $3$ , an estimate of the relative condition number of the matrix logarithm, $κ_{\log} (A)$ . Alternatively, if $ifail = 4$ , contains the absolute condition number of the matrix logarithm.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: $A$ is singular so the logarithm cannot be computed.

$ifail = 2$: $A$ has eigenvalues on the negative real line. The principal logarithm is not defined in this case; nag_matop_complex_gen_matrix_cond_log (f01kj) can be used to return a complex, non-principal log.

$ifail = 3$: $\log (A)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

$ifail = 4$: The relative condition number is infinite. The absolute condition number was returned instead.

$ifail = 5$: An unexpected internal error occurred. This failure should not occur and suggests that the function has been called incorrectly.

$ifail = - 1$: Constraint: $n \geq 0$ .

$ifail = - 3$: Constraint: $lda \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_matop_real_gen_matrix_cond_log (f01jj) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04yd) to produce an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_real_gen_norm_rcomm (f04yd).

For a normal matrix

A

(for which

A^{T} A = A A^{T}

), 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

\log (A)

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

\log (A)

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) and Section 11.2 of Higham (2008) for details and further discussion.

Further Comments

nag_matop_real_gen_matrix_cond_std (f01ja) uses a similar algorithm to nag_matop_real_gen_matrix_cond_log (f01jj) to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of

‖A‖ / ‖\log (A)‖

). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_real_gen_matrix_cond_log (f01jj) and so its use is recommended over nag_matop_real_gen_matrix_cond_std (f01ja).

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 (n^{3})

floating-point operations; see Al–Mohy et al. (2012).

If the matrix logarithm alone is required, without an estimate of the condition number, then nag_matop_real_gen_matrix_log (f01ej) should be used. If the Fréchet derivative of the matrix logarithm is required then nag_matop_real_gen_matrix_frcht_log (f01jk) should be used. If

A

has negative real eigenvalues then nag_matop_complex_gen_matrix_cond_log (f01kj) can be used to return a complex, non-principal matrix logarithm and its condition number.

Example

This example estimates the relative condition number of the matrix logarithm

\log (A)

, where

A = (\begin{array}{r} 4 & - 1 & 0 & 1 \\ 2 & 5 & - 2 & 2 \\ 1 & 1 & 3 & - 1 \\ 2 & 0 & 2 & 8 \end{array}) .

Open in the MATLAB editor: f01jj_example

function f01jj_example


fprintf('f01jj example results\n\n');

% Logarithm and conditioning of matrix A 
a =  [4, -1,  0,  1;
      2,  5, -2,  2;
      1,  1,  3, -1;
      2,  0,  2,  8];

% Compute log(a)
[loga, condla, ifail] = f01jj(a);

% Display results
disp('Log(A):');
disp(loga);

fprintf('Estimated condition number is: %6.2f\n', condla);

f01jj example results

Log(A):
    1.4081   -0.2051   -0.1071    0.1904
    0.4396    1.7096   -0.5147    0.2226
    0.2560    0.2613    1.2485   -0.2413
    0.3030   -0.0107    0.3834    2.0891

Estimated condition number is:   5.50

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Chapter Contents

Chapter Introduction

NAG Toolbox