26 lines
590 B
Org Mode
26 lines
590 B
Org Mode
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* Power Method
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v_{k+1} = A v_k, k = 0,1,2
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** Properties
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1. \frac{A v_k}{||v_k||} \rightarrow v_1
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2. \frac{v_k^T A v_k}{v_k^T v_k} \rightarrow \lambda_1
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3. If \lambda is a n eigenvalue of A, then \frac{1}{\lambda} is an eigenvalue of A^-1
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4. Av = \lambda v
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Av - \mu v = (\lambda-\mu)v = (A - \mu I)v
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5. If \lambda is an eigenvalue of A, then \lambda - \mu is an eigenvalue of A \cdot \mu I
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** Shifting Eigenvalues
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1. Partition [\lambda_n, \lambda_1]
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* Lanczos Algorithm
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#+BEGIN_SRC c
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for (int i = 0; i < n; i++) {
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sum = a0;
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v_dot_v(a[i], x);
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b[i] = sum;
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}
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#+END_SRC
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