ex 24.3, 26.2, 27.4, 27.5, due Friday Nov 12.

ex 24.3: Use the matlab expm function to compute the matrix exponential. You don't need to turn in ten plots of ||e^(tA)|| versus t, for ten different matrices, just a few that illustrate the main cases worth commenting about.

ex 26.2: How to do contour-plot a singularity in matlab, by example.

% create a grid in the complex plane
x = [-1:.02:1]; 
y = [-1:.02:1]; 
[X,Y] = meshgrid(x,y);
Z = X + 1i*Y;        
         
% assign to W the values of 1/|z| at the gridpoints     
W = zeros(length(x),length(y));
for i=1:length(x)
    for j=1:length(y)
       W(i,j) = 1/abs(Z(i,j));
    end
end

% Plot W directly and scale the contour levels exponentially
% The disadvantage is that the color scaling doesn't work well 
%[C,h] = contour(x,y, W, 10.^[-1:.1:2]);
%caxis([10^-1 10^2])

% Plot log10(W) and scale the contour levels and color linearly
% ('contourf' fills the space between contour lines with color,
%  'contour' just plots colored contour lines.)

[C,h] = contourf(x,y, log10(W), -1:.2:2);     
caxis([-1 2])
colorbar

title('log10(1/|z|)')
xlabel('Re z')
ylabel('Im z')
axis square
axis equal
axis tight

eps-pseudospectra and ||e^(tA)|| versus t for 32 x 32 matrix A with -1 on main diagonal, mu on 1st and 2nd superdiagonal, for a few values of mu. Note that mu = 1 gives the matrix asked for in exer 26.2, and alpha =0 gives a nice real symmetric matrix with eigenvalues -1 and orthogonal eigenvectors. The right-hand plots show the asymptotic behavior e^(alpha t) as well, where alpha = -1 is the spectral abscissa of A (i.e. max Re lambda).

mu = 1.0, ampl = 3e05, l.b. = 5e04

mu = 0.7, ampl = 178, l.b. = 41.3

mu = 0.6, ampl = 10.3, l.b. = 3.3

mu = 0.5, ampl = 1, l.b. = .98

mu = 0.3, ampl = 1, l.b. = .82

The thing to notice is that transient amplification occurs when the eps-pseudospectra of A extend into the positive-real part of the complex plane. A more precise relationship is given by the Kreiss matrix theorem

In the above bound, read ||(zI-A)^{-1}|| to be the value eps^{-1} for a given eps-pseudospectra. The bound (Re z) ||(zI-A)^{-1}|| will be then be large when some eps-pseudospectrum extends far into the right-hand half of the complex plane.

Label the left and right-hand sides of this inequality as ampl (amplification) and l.b. (lower bound). The labels in the above plots give these values for the given matrix.

This was a lot to ask for, given that we didn't even discuss pseudospectra in class, let alone the Kreiss matrix theorem! But comparing the amplification and pseudospectra graphs for matrices A smoothly varying between the given and well-behaved forms, as done above, is within everyone's grasp.