1. 5. Topic Method of Powers Stable Populations Linear Recurrences

2. 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.

3. 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 |

4. 5. Topic.2. Stable Populations

5. 5. Topic.3. Linear Recurrences