lecture16 - Lecture 16 Numerical Methods for Eigenvalues As...

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Unformatted text preview: Lecture 16 Numerical Methods for Eigenvalues As mentioned above, the ew ’s and ev ’s of an n × n matrix where n ≥ 4 must be found numerically instead of by hand. The numerical methods that are used in practice depend on the geometric meaning of ew ’s and ev ’s which is equation (14.1). The essence of all these methods is captured in the Power method, which we now introduce. The Power Method In the command window of Matlab enter the following: > A = hilb(5) > x = ones(5,1) > x = A*x > el = max(x) > x = x/el Compare the new value of x with the original. Repeat the last three lines (you can use the scroll up button). Compare the newest value of x with the previous one and the original. Notice that there is less change between the second two. Repeat the last three commands over and over until the values stop changing. You have completed what is known as the Power Method . Now try the command: > [v e] = eig(A) The last entry in e should be the final el we computed. The last column in v is x/norm(x) . Below we explain why our commands gave this eigenvalue and eigenvector. For illustration consider a 2 × 2 matrix whose ew ’s are 1 / 3 and 2 and whose corresponding ev ’s are v 1 and v 2 . Let x be any vector which is a combination of v 1 and v 2 , e.g., x = v 1 + v 2 . Now let x 1 be A times x . It follows from (14.1) that x 1 = A v 1 + A v 2 = 1 3 v 1 + 2 v 2 ....
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This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.

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lecture16 - Lecture 16 Numerical Methods for Eigenvalues As...

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