Lecture 9 - 9 Overview of Eigenvalue Algorithms As already...

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9 Overview of Eigenvalue Algorithms As already mentioned earlier, any eigenvalue algorithm needs to be an iterative one. Some not so good approaches are: 1. Compute the roots of the characteristic polynomial. This is usually a very ill- conditioned problem, and should be used only for small pencil and paper calcu- lations. 2. The power iteration listed in most textbooks yields only the largest eigenvalue, and usually converges very slowly. We will, however, consider this method as the basis for more efficient methods later. More promising approaches are those based on the linear algebra facts reviewed earlier such as diagonalization or triangularization. In particular, the Schur factoriza- tion reveals all eigenvalues for any square matrix. Another advantage of this sort of approach is that there will be some similarities to the matrix factorization algorithms studied earlier. While it is impractical to compute the exact Schur factorization, we can use the following iterative approach to get close: Q * j . . . Q * 2 Q * 1 AQ 1 Q 2 . . . Q j T for j → ∞ . Here the Q j should ideally be “simple” unitary matrices, and T upper triangular. As mentioned earlier, the eigenvalues of real matrices can be complex, so the algo- rithms we devise will have to be able to handle the complex case, also.
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