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5. Topic Method of Powers Stable Populations Linear Recurrences

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5. Topic.1. Method of Powers Sparse matrices: matrices with most entries equal to 0. Method of powers: Fast method for getting eigenvalues of sparse matrices. Let T be n n matrix with n distinct eigenvalues λ 1, …, λ n. The assoicated eigenvectors form a basis ζ 1, …, ζ n for R n. → Let λ j be the eigenvalue with the largest absolute value.

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Example: λ 2.9999. Implementation issues: 1.The T k v s are calculated successively, i.e., T k v = T ( T k 1 v ). 2.To avoid large numbers, each T k v should be normalized, i.e., T k v = T ( T k 1 v / || T k 1 v || ) so that || T k v || → |λ j |

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5. Topic.2. Stable Populations

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5. Topic.3. Linear Recurrences

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