Iterative Estimates for Eigenvalues
Power Method
Lay Example 5.8.2
We illustrate the use of the Power Method to approximate a specific eigenvalue and eigenvector of a given matrix.
Initial data.
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Use the Power Method to compute the μs and xs.
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Assemble Lay's Table 2, p366.
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x[k] | ![]() |
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a.x[k] | ![]() |
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μ[k] | 5.` | 8.` | 7.125` | 7.017543859649122` | 7.0025` | 7.0003570153516606` |
Check the results obtained from the Power Method.
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Check using Mathematica's Eigensystem command.
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Inverse Power Method
Lay Example 5.8.3
We illustrate the use of the Inverse Power Method to approximate a specific eigenvalue and eigenvector of a given matrix.
Initial data.
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Use the Inverse Power Method to compute the μs, νs, and xs.
This naive implementation of the Inverse Power Method uses the inverse of the matrix (a-α id) to compute y[k].
A more efficient version would solve a matrix equation for y[k].
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Assemble Lay's Table 3, p368.
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k=0 | 1 | 2 | 3 | 4 | 5 | 6 | |
x[k] | ![]() |
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y[k] | ![]() |
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μ[k] | 7.758804695837776` | 9.91970041059954` | 9.994930860390895` | 9.999646313779577` | 9.999975129049835` | 9.999998250104962` | 9.999999876874135` |
ν[k] | 2.02888583218707` | 2.0008094961145666` | 2.0000507171053` | 2.0000035369873026` | 2.00000024871012` | 2.0000000174989534` | 2.0000000012312587` |
Check the results obtained from the Inverse Power Method.
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Check using Mathematica's Eigensystem command.
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Created by Mathematica (April 1, 2005) | ![]() |