ABSTRACT
The first example of the power method, Example 2.1. converged very slowly since the two largest eigenvalues of matrix A are close together (i.e., 13.870584 and 8.620434). Convergence can be accelerated by using the results of an early iteration, say iteration 5, to shift the eigenvalues by the approximate eigenvalue, and then using the inverse power method on the shifted matrix to accelerate convergence. From Example 2.1, after 5 iterations, A(5) = 13.694744 and X(5)T = [-0.192991 -0.188895 1.000000]. Thus, shift matrix A by 5 = 13.694744:
[ -5.694744 -2.000000
Ashifted = (A - 51) = -2.000000 -9.694744 -2.000000 -2.000000
The corresponding Land U matrices are
[
1.000000 0.000000 0.000000] L = 0.351201 1.000000 0.000000
0.351201 0.144300 1.000000
[
-5.694744 -2.000000 -2.000000] U = 0.000000 -8.992342 -1.297598
0.000000 0.000000 0.194902
(2.88)
(2.89)
Let X(O)T = x(5)T and continue scaling the third component of x to unity. Applying the inverse power method to matrix Ashifted yields the results presented in Table 2.5. These results were obtained on a 13-digit precision computer with an absolute convergence tolerance of 0.00000 I. The eigenvalue Ashifted of the shifted matrix Ashifted is
1 1 Ashifted = A = 5.686952 = 0.175841
(2.90)
