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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_real_gen_matrix_cond_exp (f01jg)

▸▿ 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_exp (f01jg) computes an estimate of the relative condition number

κ_{\exp} (A)

of the exponential of a real

n

n

matrix

A

, in the

1

-norm. The matrix exponential

e^{A}

is also returned.

Syntax

[a, condea, ifail] = f01jg(a, 'n', n)

[a, condea, ifail] = nag_matop_real_gen_matrix_cond_exp(a, 'n', n)

Description

The Fréchet derivative of the matrix exponential of

A

is the unique linear mapping

E ⟼ L (A, E)

such that for any matrix

E

e^{A + E} - e^{A} - L (A, E) = o (‖E‖) .

The derivative describes the first-order effect of perturbations in

A

on the exponential

e^{A}

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

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

where

‖L (A)‖

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

A

To obtain the estimate of

κ_{\exp} (A)

, nag_matop_real_gen_matrix_cond_exp (f01jg) 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_{\exp} (A)

are detailed in the Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).

The matrix exponential

e^{A}

is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives

L (A, E)

which are used to estimate the condition number.

References

Al–Mohy A H and Higham N J (2009a) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989

Al–Mohy A H and Higham N J (2009b) Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation SIAM J. Matrix Anal. Appl. 30(4) 1639–1657

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Rev. 45 3–49

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.
$n$ , 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 $n$ matrix exponential $e^{A}$ .
2: $condea$ – double scalar: An estimate of the relative condition number of the matrix exponential $κ_{\exp} (A)$ .
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:

$ifail = 1$: The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.

$ifail = 2$: $e^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

$ifail = 3$: An unexpected internal error has occurred. Please contact NAG.

$ifail = - 1$: 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_exp (f01jg) 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 computed matrix,

e^{A}

, is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-normal matrices. See Section 10.3 of Higham (2008) for details and further discussion.

For further discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008).

Further Comments

nag_matop_real_gen_matrix_cond_std (f01ja) uses a similar algorithm to nag_matop_real_gen_matrix_cond_exp (f01jg) to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of

‖A‖ / ‖\exp (A)‖

). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_real_gen_matrix_cond_exp (f01jg) and so its use is recommended over nag_matop_real_gen_matrix_cond_std (f01ja).

The cost of the algorithm is

O (n^{3})

and the real allocatable memory required is approximately

15 n^{2}

; see Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) for further details.

If the matrix exponential alone is required, without an estimate of the condition number, then nag_matop_real_gen_matrix_exp (f01ec) should be used. If the Fréchet derivative of the matrix exponential is required then nag_matop_real_gen_matrix_frcht_exp (f01jh) should be used.

As well as the excellent book Higham (2008), the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).

Example

This example estimates the relative condition number of the matrix exponential

e^{A}

, where

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

Open in the MATLAB editor: f01jg_example

function f01jg_example


fprintf('f01jg example results\n\n');

% Exponential and conditioning of matrix A 
a =  [2, 2, 1, 2;
      3, 1, 4, 0;
      2, 3, 1, 2;
      0, 1, 3, 3];

% Compute exp(a)
[expa, condea, ifail] = f01jg(a);

% Display results
disp('exp(A):');
disp(expa);

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

f01jg example results

exp(A):
  404.4441  412.6036  496.7221  398.3043
  474.4388  482.8457  579.1310  460.6474
  466.9764  477.2769  574.3994  458.3804
  407.7005  420.8935  510.1939  410.4808

Estimated condition number is:   9.40

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Chapter Contents

Chapter Introduction

NAG Toolbox