powermeth - A . (If this is too analytic for your taste,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
The Power Method Assume that A C n × n has a n linearly independent eigenvectors v 1 , v 2 , . . . , v n . Then any x C n can be represented uniquely as x = n X i =1 c i v i . (1) Here we are interested in what (if any) direction A k x heads toward as k → ∞ . Specifically, we have a sequence { x k } of vectors defined by x 0 = x, x k +1 = Ax k = A k x 0 , k = 0 , 1 , 2 , . . . (2) and we would like to know in what direction it is ultimately pointing. Recall that if v i is an eigenvector of A , then there is a scalar λ i , called an eigenvalue, for which Av i = λ i v i . Then A k v i = λ k i v i (you do the induction). Using (1) (and linearity) we find that A k x = n X i =1 c i λ k i v i . (3) Now suppose that | λ 1 | > | λ i | , i = 2 , 3 , . . . , n. Then A k x λ k 1 = c 1 v 1 + n X i =2 c i ( λ i λ 1 ) k v i (4) Here it is clear (yes?) that unless c 1 = 0, ( cA ) k x v 1 . Thus we call v 1 the dominant eigenvector of A . This result is as simple as it is powerful: if v 1 is the dominant eigenvector of A , then for almost all x C n , x v 1 under repeated application of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: A . (If this is too analytic for your taste, then change to the basis { v 1 , v 2 , . . . , v n } . Under this basis A has coordinates = diag( 1 , 2 , . . . , n ), and k y e 1 as long as y (1) 6 = 0.) The power method consists of scaling iteration (2) to avoid underow or overow, and guring out when to stop. We solve both problems by approximating 1 at each step. The code below (if it terminates) gives a small backward error (i.e. gives an eigenpair for a matrix close to A ). i = argmax( | x | ) x = x/x ( i ) For k = 1 , 2 , . . . until done y = Ax i = argmax( | y | ) = y ( i ) r = y-x , if k r k is small enough, then stop x = y/...
View Full Document

This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

Ask a homework question - tutors are online