There is no eigenvalue estimate If you want to calculate an estimate use the

There is no eigenvalue estimate if you want to

This preview shows page 8 - 12 out of 31 pages.

*There is no eigenvalue estimate. If you want to calculate an estimate, use the Rayleigh Quotient If is an eigenvector of unit norm (which in method 2, is), thus,
Image of page 8
Both forms guarantee a true eigenvalue/eigenvector with an number of iterations provided, 1) There is a unique dominant eigenvalue , (dominant must be real) 2) The initial guess, , must have a nonzero component in the direction of the true eigenvector Solution: Always choose to be ,or if Convergence for the power method is linear Inverse Power Method finds the smallest magnitude eigenvalue/eigenvector therefore is an eigenvalue of **the maximum magnitude eigenvalue of is the minimum magnitude eigenvalue of Method: choose calculate Let the largest magnitude value of calculate Iterate... After convergence, *Note that the convergence is still linear The Power Method and Inverse Power Method may seem limiting, but the smallest and largest eigenvalues/eigenvectors are still the most important in most applications, yet we can still find the other eigenvalues/eigenvectors by using the “spectral shift” Shifted-Inverse Power Method (Spectral Shift) can be used with an initial eigenvalue guess, , to find a true eigenvalue of a given matrix, closest to the value If then This implies that the eigenvectors remain unchanged, but the eigenvalues are shifted by Method: choose calculate Let the largest magnitude value of calculate Iterate... After convergence, In practice, the power methods are not used, except when you are solely interested in the
Image of page 9
largest or smallest eigenvalue/eigenvector **When you have an estimate of an eigenvalue, you can use the Shifted Inverse Power Method to hone in on the true eigenvalues/eigenvectors Rayleigh Quotient Iteration exists because the power methods are restricted to linear convergence *Iff A is symmetric, we can get cubic convergence Method: choose such that choose such that calculate *must calculate inverse b/c changes each iteration calculate calculate ...Iterate until convergence *Convergence is cubic for almost all **Method converges to just an eigenvalue/eigenvector; this depends on and Stopping Condition for Power Methods Let , then the stopping condition becomes is defined as is, because when , the true eigenvalue/eigenvector is found (we converge) Where Z is different depending on which method is used For Power Method For Inverse Power Method For Spectral Shift Method For Quotient Method __________________________________________________________________________ Nonlinear Systems a system is nonlinear if it does not meet the additivity or homogeneity properties a nonlinear problem is compounded because a nonlinear equation can have multiple solutions (0,1,2,3,..., solutions, *you never know how many) Examples
Image of page 10
2 solutions 3 solutions n solutions solutions when no solutions Two Types of Root-Finding Methods 1) Bracket 2) Open Bracket Methods assume we have a continuous function f(x) since and , there exists such that since and , there exists such that * This is only true if f(x) is continuous If there exists points such that , then there exists AT LEAST one x such that and , thus and
Image of page 11
Image of page 12

You've reached the end of your free preview.

Want to read all 31 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes