610 p 573 that states ln3 n3 in in an in an in

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Unformatted text preview: ank computed with floating-point arithmetic can vary with the algorithm used and is often different than rank computed with exact arithmetic (recall Exercise 2.2.4). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 592 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Furthermore, computing higher-index eigenvalues with floating-point arithmetic is fraught with peril. To see why, consider the matrix ⎛ ⎞ 01 .. .. ⎜ ⎟ . . ⎜ ⎟ L( ) = ⎜ whose characteristic equation is λn − = 0. .. 1 ⎟ ⎝ ⎠ . 0 n×n For = 0, zero is the only eigenvalue (and it has index n ), but for all > 0, there are n distinct eigenvalues given by 1/n e2kπi/n for k = 0, 1, . . . , n − 1. For example, if n = 32, and if changes from 0 to 10−16 , then the eigenvalues of L( ) change in magnitude from 0 to 10−1/2 ≈ .316, which is substantial for such a small perturbation. Sensitivities of this kind present significant problems for floating-point algorithms. In addition to showing that high-index eigenvalues are sensitive to small perturbations, this example also shows that the Jordan structure is highly discontinuous. L(0) is in Jordan form, and there is just one Jordan block of size n, but for all = 0, the Jordan form of L( ) is a diagonal matrix—i.e., there are n Jordan blocks of size 1 × 1. Lest you think that this example somehow is an isolated case, recall from Example 7.3.6 (p. 532) that every matrix in C n×n is arbitrarily close to a diagonalizable matrix. All of the above observations make it clear that it’s hard to have faith in a Jordan form that has been computed with floating-point arithmetic. Consequently, numerical computation of Jordan forms is generally avoided. D E T H IG R Example 7.8.2 Y P The Jordan form of A conveys complete information about the eigenva...
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