If $A$ is a matrix whose eigenvalues are in the open left half of the complex plane, then the corresponding dynamical system defined by the equation $\frac{du}{dt} = Au$ is asymptotically stable, with all solutions decaying to zero as $t \to \infty$. Since the solution is $u(t) = e^{tA} u(0)$, another way to say this is that the quantity $| e^{tA} |$ decays to zero as $t \to \infty$.

Along the way, however, there may be transient growth, and this is important, for example, in some problems in fluid mechanics. A recent paper by Whidborne and Amar [2] considers the following matrix taken from an earlier paper by Plitschke and Wirth:

tic
A = [-1 0 0 0 0 0 -625; 0 -1 -30 400 0 0 250; -2 0 -1 0 0 0 30;
     5 -1 5 -1 0 0 200; 11 1 25 -10 -1 1 -200;
     200 0 0 -150 -100 -1 -1000; 1 0 0 0 0 0 -1]
A =
  Columns 1 through 6
          -1           0           0           0           0           0
           0          -1         -30         400           0           0
          -2           0          -1           0           0           0
           5          -1           5          -1           0           0
          11           1          25         -10          -1           1
         200           0           0        -150        -100          -1
           1           0           0           0           0           0
  Column 7
        -625
         250
          30
         200
        -200
       -1000
          -1

Here (adapted from linalg/NonnormalQuiz) is a code to compute and plot $| e^{tA} |$ as a function of $t$:

e = chebfun(@(t) norm(expm(t*A)),[0 2.5],'vectorize','splitting','on');
LW = 'linewidth'; FS = 'fontsize'; plot(e,'b',LW,2)
xlabel('t',FS,14), ylabel('||e^{tA}||',FS,14)
title('amplitude',FS,16)
Warning: Function not resolved using 6050 pts.

Actually Whidborne and Amar plot the square of this function. The following figure matches their Figure 1.

e2 = e.^2;
plot(e2,'b',LW,2)
xlabel('t',FS,14), ylabel('||e^{tA}||^2',FS,14)
title('energy',FS,16)

They are interested in calculating the maximum energy:

fprintf('Maximum energy = %15.8f\n',max(e2))
Maximum energy = 358147.98785176

Here's the time for this Example:

toc
Elapsed time is 11.016822 seconds.

References

  1. L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton U. Press, 2005.

  2. J. F. Whidborne and N. Amar, Computing the maximum transient energy growth, BIT Numerical Mathematics, 51 (2011), 447-457.