unh2010:iam931:hw4
====== Differences ====== This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
unh2010:iam931:hw4 [2010/11/06 09:59] gibson |
unh2010:iam931:hw4 [2010/11/15 06:42] (current) gibson |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== homework 4 ====== | ====== homework 4 ====== | ||
| ex 24.3, 26.2, 27.4, 27.5, due Friday Nov 12. | 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. 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. | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
unh2010/iam931/hw4.1289062795.txt.gz · Last modified: 2010/11/06 09:59 by gibson
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
