is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative

L (A, E)

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: $a (lda :)$ – 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 $A$ .
2: $e (lde :)$ – complex array: The first dimension of the array e must be at least $n$ .
The second dimension of the array e must be at least $n$ .

The $n$ by $n$ matrix $E$

Optional Input Parameters

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

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – 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 $e^{A}$ .
2: $e (lde :)$ – complex array: The first dimension of the array e will be $n$ .
The second dimension of the array e will be $n$ .

The Fréchet derivative $L (A, E)$
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: $lda \geq n$ .

$ifail = - 5$: Constraint: $lde \geq n$ .

$ifail = - 99$: 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

For a normal matrix

A

(for which

A^{H} A = A A^{H}

) 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), Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) for details and further discussion.

Further Comments

The cost of the algorithm is

O (n^{3})

and the complex allocatable memory required is approximately

9 n^{2}

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

If the matrix exponential alone is required, without the Fréchet derivative, then nag_matop_complex_gen_matrix_exp (f01fc) should be used.

If the condition number of the matrix exponential is required then nag_matop_complex_gen_matrix_cond_exp (f01kg) 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 finds the matrix exponential

e^{A}

and the Fréchet derivative

L (A, E)

, where

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

Open in the MATLAB editor: f01kh_example

function f01kh_example


fprintf('f01kh example results\n\n');

% Exponential of matrix A and Frechet derivative of exp(A)E.

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];

e =  [1,    2+ i, 2,    4+i;
      3+2i, 0,    1,    0+i;
      0+2i, 0+ i, 1,    0;
      1+ i, 2+2i, 0+3i, 1];

[expa, lae, ifail] = f01kh(a,e);

[ifail] = x04da('General', ' ', expa, 'exp(A):');
disp(' ');
[ifail] = x04da('General', ' ', lae, 'L_exp(A,E):');

f01kh example results

 exp(A):
             1          2          3          4
 1   -157.9003  -194.6526  -186.5627  -155.7669
     -754.3717  -555.0507  -475.4533  -520.1876

 2   -206.8899  -225.4985  -212.4414  -186.5627
     -694.7443  -505.3938  -431.0611  -475.4533

 3   -208.7476  -238.4962  -225.4985  -194.6526
     -808.2090  -590.8045  -505.3938  -555.0507

 4   -133.3958  -208.7476  -206.8899  -157.9003
    -1085.5496  -808.2090  -694.7443  -754.3717
 
 L_exp(A,E):
             1          2          3          4
 1   1571.5852   778.4238   500.2085   740.7485
    -4640.2429 -3719.8308 -3246.0234 -3424.1963

 2   1472.7846   731.6608   473.2569   692.0895
    -4273.5048 -3432.5961 -2990.9285 -3148.4635

 3   1996.4848  1107.9174   782.1266  1031.5808
    -4568.8881 -3714.9923 -3249.1926 -3400.8557

 4   3327.1347  2015.2763  1514.3130  1873.9421
    -5829.0773 -4810.2591 -4234.6812 -4404.0163

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox