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NAG Toolbox: nag_matop_complex_gen_matrix_cond_exp (f01kg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_matop_complex_gen_matrix_cond_exp (f01kg) computes an estimate of the relative condition number κexpA of the exponential of a complex n by n matrix A, in the 1-norm. The matrix exponential eA is also returned.

Syntax

[a, condea, ifail] = f01kg(a, 'n', n)
[a, condea, ifail] = nag_matop_complex_gen_matrix_cond_exp(a, 'n', n)

Description

The Fréchet derivative of the matrix exponential of A is the unique linear mapping ELA,E such that for any matrix E 
eA+E - e A - LA,E = oE .  
The derivative describes the first-order effect of perturbations in A on the exponential eA.
The relative condition number of the matrix exponential can be defined by
κexpA = LA A expA ,  
where LA is the norm of the Fréchet derivative of the matrix exponential at A.
To obtain the estimate of κexpA, nag_matop_complex_gen_matrix_cond_exp (f01kg) first estimates LA by computing an estimate γ of a quantity Kn-1LA1,nLA1, such that γK.
The algorithms used to compute κexpA are detailed in the Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).
The matrix exponential eA is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives LA,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:     alda: – complex 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: n0.

Output Parameters

1:     alda: – complex 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 eA.
2:     condea – double scalar
An estimate of the relative condition number of the matrix exponential κexpA.
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
eA 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: n0.
   ifail=-3
Constraint: ldan.
   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_complex_gen_matrix_cond_exp (f01kg) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zd) 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 (f04zd).
For a normal matrix A (for which AHA=AAH) the computed matrix, eA, 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_complex_gen_matrix_cond_std (f01ka) uses a similar algorithm to nag_matop_complex_gen_matrix_cond_exp (f01kg) to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of A/expA). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_complex_gen_matrix_cond_exp (f01kg) and so its use is recommended over nag_matop_complex_gen_matrix_cond_std (f01ka).
The cost of the algorithm is On3 and the complex allocatable memory required is approximately 15n2; 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_complex_gen_matrix_exp (f01fc) should be used. If the Fréchet derivative of the matrix exponential is required then nag_matop_complex_gen_matrix_frcht_exp (f01kh) 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 eA, where
A = 1+0i 2+0i 2+0i 2+i 3+2i 1i+0 1i+0 2+i 3+2i 2+0i 1i+0 2+i 3+2i 3+2i 3+2i 1+i .  
function f01kg_example


fprintf('f01kg example results\n\n');

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

% Compute exp(a)
[expa, condea, ifail] = f01kg(a);

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

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


f01kg example results

exp(A):
   1.0e+03 *

  -0.1579 - 0.7544i  -0.1947 - 0.5551i  -0.1866 - 0.4755i  -0.1558 - 0.5202i
  -0.2069 - 0.6947i  -0.2255 - 0.5054i  -0.2124 - 0.4311i  -0.1866 - 0.4755i
  -0.2087 - 0.8082i  -0.2385 - 0.5908i  -0.2255 - 0.5054i  -0.1947 - 0.5551i
  -0.1334 - 1.0855i  -0.2087 - 0.8082i  -0.2069 - 0.6947i  -0.1579 - 0.7544i

Estimated condition number is:  15.29

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