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====== Differences ====== This shows you the differences between two versions of the page.

--- unh2010:iam931:hw4 [2010/11/07 18:28]
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+++ unh2010:iam931:hw4 [2010/11/15 06:42] (current)
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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:** The
+**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. 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
+{{:unh2010:iam931:hw4:ex26_2a10.png?400}} {{:unh2010:iam931:hw4:ex26_2b10.png?400}}
+mu = 0.7, ampl = 178, l.b. = 41.3
+{{:unh2010:iam931:hw4:ex26_2a7.png?400}} {{:unh2010:iam931:hw4:ex26_2b7.png?400}}
+mu = 0.6, ampl = 10.3, l.b. = 3.3
+{{:unh2010:iam931:hw4:ex26_2a6.png?400}} {{:unh2010:iam931:hw4:ex26_2b6.png?400}}
+mu = 0.5, ampl = 1, l.b. = .98
+{{:unh2010:iam931:hw4:ex26_2a5.png?400}} {{:unh2010:iam931:hw4:ex26_2b5.png?400}}
+mu = 0.3, ampl = 1, l.b. = .82
+{{:unh2010:iam931:hw4:ex26_2a3.png?400}} {{:unh2010:iam931:hw4:ex26_2b3.png?400}}
+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
+<latex>
+\sup_{t\geq 0} ||e^{tA}|| \geq \sup_{Re\; z > 0} (Re\; z)||(zI-A)^{-1}||
+</latex>
+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.

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