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

--- unh2010:iam931:hw4 [2010/11/13 20:04]
gibson
+++ unh2010:iam931:hw4 [2010/11/15 06:42] (current)
gibson
@@ Line 46: / Line 46: @@
 ==== exer 26.2 ====
-eps-pseudospectra and ''||e^(tA)||'' versus t for 32 x 32 matrix A with -1 on main diagonal, alpha on 1st and 2nd superdiagonal, for a few values of alpha. Note that alpha = 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.
+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).
-alpha = 1.0
+mu = 1.0, ampl = 3e05, l.b. = 5e04
 {{:unh2010:iam931:hw4:ex26_2a10.png?400}} {{:unh2010:iam931:hw4:ex26_2b10.png?400}}
-alpha = 0.7
+mu = 0.7, ampl = 178, l.b. = 41.3
 {{:unh2010:iam931:hw4:ex26_2a7.png?400}} {{:unh2010:iam931:hw4:ex26_2b7.png?400}}
-alpha = 0.6
+mu = 0.6, ampl = 10.3, l.b. = 3.3
 {{:unh2010:iam931:hw4:ex26_2a6.png?400}} {{:unh2010:iam931:hw4:ex26_2b6.png?400}}
-alpha = 0.5
+mu = 0.5, ampl = 1, l.b. = .98
 {{:unh2010:iam931:hw4:ex26_2a5.png?400}} {{:unh2010:iam931:hw4:ex26_2b5.png?400}}
-alpha = 0.4
+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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