unh2010:iam931:hw4
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unh2010:iam931:hw4 [2010/11/13 20:46] gibson |
unh2010:iam931:hw4 [2010/11/15 06:42] (current) gibson |
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| 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). | 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 | + | mu = 1.0, ampl = 3e05, l.b. = 5e04 |
| {{:unh2010:iam931:hw4:ex26_2a10.png?400}} {{:unh2010:iam931:hw4:ex26_2b10.png?400}} | {{:unh2010:iam931:hw4:ex26_2a10.png?400}} {{:unh2010:iam931:hw4:ex26_2b10.png?400}} | ||
| - | mu = 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}} | {{:unh2010:iam931:hw4:ex26_2a7.png?400}} {{:unh2010:iam931:hw4:ex26_2b7.png?400}} | ||
| - | mu = 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}} | {{:unh2010:iam931:hw4:ex26_2a6.png?400}} {{:unh2010:iam931:hw4:ex26_2b6.png?400}} | ||
| - | mu = 0.5 | + | mu = 0.5, ampl = 1, l.b. = .98 |
| {{:unh2010:iam931:hw4:ex26_2a5.png?400}} {{:unh2010:iam931:hw4:ex26_2b5.png?400}} | {{:unh2010:iam931:hw4:ex26_2a5.png?400}} {{:unh2010:iam931:hw4:ex26_2b5.png?400}} | ||
| - | mu = 0.3 | + | mu = 0.3, ampl = 1, l.b. = .82 |
| {{:unh2010:iam931:hw4:ex26_2a3.png?400}} {{:unh2010:iam931:hw4:ex26_2b3.png?400}} | {{:unh2010:iam931:hw4:ex26_2a3.png?400}} {{:unh2010:iam931:hw4:ex26_2b3.png?400}} | ||
| Line 74: | Line 74: | ||
| <latex> | <latex> | ||
| - | \sup_{t\geq 0} ||e^{tA}|| \geq \sup_{Re\; z > 0} (Re\; z)(zI-A)^{-1} | + | \sup_{t\geq 0} ||e^{tA}|| \geq \sup_{Re\; z > 0} (Re\; z)||(zI-A)^{-1}|| |
| </latex> | </latex> | ||
| - | In the above bound, read ''(zI-A)^{-1}'' to be the value eps^{-1} for a given eps-pseudospectra. The bound | + | 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 | + | ''(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. | + | 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. | 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/hw4.1289709961.txt.gz · Last modified: 2010/11/13 20:46 by gibson
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