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Unformatted text preview: . Furthermore, the rate of convern
gence depends on how fast [(λ − α)/(λi − α)] → 0, and this can be slow
when there is another eigenvalue λi close to the desired λ. If λi is too
close to λ, roundoﬀ error can divert inverse iteration toward an eigenvector
associated with λi instead of λ in spite of a theoretically correct α. D
C Note: In the standard version of inverse iteration a constant value of α is used at
each step to approximate an eigenvalue λ, but there is variation called Rayleigh
quotient iteration that uses the current iterate xn to improve the value of α
at each step by setting α = xT Axn /xT xn . The function R(x) = xT Ax/xT x is
called the Rayleigh quotient. It can be shown that if x is a good approximation to
an eigenvector, then R(x) is a good approximation of the associated eigenvalue.
More is said about this in Example 7.5.1 (p. 549). Example 7.3.9
The QR Iteration algorithm for computing the eigenvalues of a general matrix came from an elegantly simple idea that was proposed by Heinz Rutishauser
in 1958 and reﬁned by J. F. G. Francis in 1961-1962. The underlying concept is
to alternate between computing QR factors (Rutishauser used LU factors) and Copyright c 2000 SIAM Buy online from SIAM
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Eigenvalues and Eigenvectors
reversing their order as shown below. Starting with A1 = A ∈ n×n , Factor: A1 = Q1 R1 ,
Set: A2 = R1 Q1 , Factor: A2 = Q2 R2 , It is illegal to print, duplicate, or distribute this material
Please report violations to email@example.com Set: A3 = R2 Q2 ,
. In general, Ak+1 = Rk Qk , where Qk and Rk are the QR factors of Ak .
Notice that if Pk = Q1 Q2 · · · Qk , then each Pk is an orthogonal matrix such
PT AP1 = QT Q1 R1 Q1 = A2 ,
PT AP2 = QT QT AQ1 Q2 = QT A2 Q2 = A3 ,
H PT APk = Ak+1 .
k In other words, A2 , A3 , A4 , . . . are each orthogonally similar to A, and...
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