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====== homework 4 ======
ex 24.3, 26.2, 27.4, 27.5, due Friday Nov 12.
===== tips =====

**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.

<code>

% 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
</code>
{{:unh2010:iam931:hw4:contoureg.png?400}}
==== exer 26.2 ====
 
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. In the right-hand plots, ''e^(alpha 

mu = 1.0

{{:unh2010:iam931:hw4:ex26_2a10.png?400}} {{:unh2010:iam931:hw4:ex26_2b10.png?400}} 

mu = 0.7 

{{:unh2010:iam931:hw4:ex26_2a7.png?400}} {{:unh2010:iam931:hw4:ex26_2b7.png?400}}    

mu = 0.6

{{:unh2010:iam931:hw4:ex26_2a6.png?400}} {{:unh2010:iam931:hw4:ex26_2b6.png?400}}  

mu = 0.5

{{:unh2010:iam931:hw4:ex26_2a5.png?400}} {{:unh2010:iam931:hw4:ex26_2b5.png?400}} 

mu = 0.3

{{:unh2010:iam931:hw4:ex26_2a3.png?400}} {{:unh2010:iam931:hw4:ex26_2b3.png?400}}


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