Section 4.1 The Power Method

Transcription

1 Section 4.1 The Power Method Key terms Dominant eigenvalue Eigenpair Infinity norm and 2-norm Power Method Scaled Power Method Power Method for symmetric matrices The notation used varies a bit from that in the text.

2 Terminology and Properties Dominant eigenvalue: eigenvalue of largest absolute value Eigenpair: an eigenvalue λ and an associated eigenvector p; (λ, p). Property 1. If (λ, p) is an eigenpair of A, then for any positive integer r, (λ r, p) is an eigenpair of A r. The power method is a way to approximate the dominant eigenvalue. We do this indirectly by first approximating an eigenvector associated with the dominant eigenvalue. Our development uses a number of properties of eigenvalues and information about bases and subspaces.

3 Property 1. If (λ, p) is an eigenpair of A, then for any positive integer r, (λ r, p) is an eigenpair of A r.

4 assuming the denominator is not zero. If we assume that then we have assuming that p 1, j 0. Hence, the ratio converges toward the dominant eigenvalue, and the convergence is linear with asymptotic rate constant λ 2 /λ 1.

5 We demonstrate this graphically in Example 1 and then from a numerical point of view in Example 2. Figure 1.

6 Next we compute vectors u 1 = Au 0, u 2 = Au 1, u 3 = Au 2,.... Then we form the ratio of corresponding components. As shown in the table below, the size of the entries in the vectors can grow quite rapidly and we may need the number k of vectors to be large to get an accurate approximation. (We have only included enough steps to indicate the behavior of the method.) It appears that the ratios of terms is approaching 6.

7 % Unscaled Power Method Code A=[7 4 5;-2 2-2;1 0 3]; %Example u=[1 1 1]'; %sets the initial guess for k=1:15 %<== number of iterations v=a*u %generates the next approximate eigenvector v./u %computes the ratio of corresponding entries %of the two most recent approximate eigenvectors & prints u=v; %updates the eigenvector for the next time through the loop end (The preceding code prints output at each iteration.) It is recommended that you use the following modification that constructs a table for the eigenvectors and eigenvalue approximations. pvec=[ ];pval=[ ]; for k=1:15 v=a*u; pvec=[pvec;v' ]; w=v./u; pval=[pval;w' ]; u=v; end pvec,pval

8 This can be entered as follows: pvec=[ ];pval=[ ]; for k=1:15, v=a*u; pvec=[pvec;v' ]; w=v./u; pval=[pval;w' ]; u=v; end, pvec, pval Remember to enter the matrix A and initial guess u prior to executing the code. Using this code with input data A=[7 4 5;-2 2 2;1 0 3]; u=[1 1 1]'; we obtained the information in the table. (We haven t shown all the output.) It appears that the ratios of corresponding entries are converging to 6. Hence we conjecture that the dominant eigenvalue is 6. There is a modification of the power method, called the scaled power method, which inhibits the growth of the size of the entries of the approximate eigenvectors u k and is recommended for use in computational work.

9 The SCALED POWER METHOD The scaled power method can be described as follows. Let initial guess u 0 be chosen as in Equation (1). We define the sequences of vectors

10 Scaled Power Method code Here we will use the -norm for the scaling and determine the approximation to the dominant eigenvalue at each step. pvec=[ ];pval=[ ]; [mv,kk]=max(abs(u));u=u/u(kk); pvec=u';pval=u(kk); format long e for k=1:15, v=a*u; pval=[pval;v(kk)]; [mv,kk]=max(abs(v)); w=v./v(kk); u=w; pvec=[pvec;w' ]; end, [pvec,pval] In this case the display contains the scaled eigenvectors w and the maximum value (with appropriate sign) of the unscaled eigenvector v as an approximate dominant eigenvalue. In order to get the sign of the approximations to the dominant eigenvalue correct we use the indicated code to find the entry in the approximate eigenvector that has largest magnitude. This code also appears in the for loop. A=[7 4 5;-2 2-2;1 0 3]; u =[1 1 1]'; %sets the initial guess

11 Next use code for the scaled power method. pvec=[ ];pval=[ ]; [mv,kk]=max(abs(u));u=u/u(kk); pvec=u';pval=u(kk); format long e for k=1:15, v=a*u; pval=[pval;v(kk)]; [mv,kk]=max(abs(v)); w=v./v(kk); u=w; pvec=[pvec;w' ]; end, [pvec,pval] This displayed in format short

12 The power method is an iterative scheme, so a convergence tolerance must be specified and a stopping condition implemented. Three possibilities for the stopping condition immediately come to mind. The iteration could be terminated when any of the following is true: checking for convergence of the eigenvalue checking for convergence of the eigenvector checking for convergence of the residual where TOL denotes the specified convergence tolerance and λ (k) is used to denote the approximation to the eigenvalue during the k-th iteration. The author argues that using the check for convergence of the eigenvector is preferred. (See page 267.)

13 power_method approximate the dominant eigenvalue and an associated eigenvector for an arbitrary matrix using the power method calling sequences: [lambda, v] = power_method ( A, x, TOL, Nmax ) lambda = power_method ( A, x, TOL, Nmax ) power_method ( A, x, TOL, Nmax ) inputs: A square matrix whose dominant eigenvalue is to be approximated x initial approximation to eigenvector corresponding to the dominant eigenvalue TOL absolute error convergence tolerance (convergence is measured in terms of the infinity norm of the difference between successive terms in the eigenvector seqeunce) Nmax maximum number of iterations to be performed outputs: lambda approximation to dominant eigenvalue of A v an eigenvector of A corresponding to the eigenvalue lambda - vector will be normalized to unit length in the maximum infinity

14 Example: Use m-file power_method to approximate the dominant eigenvalue of whose eigenvalues are λ 1 = -12, λ 2 = -3 and λ 3 = 3. Let's start with the vector x (0) = [1 0 0] T. A=[-2-2 3; ; ];x=[1 0 0]';TOL = 5E-6; Nmax=20; power_method ( A, x, TOL, Nmax ) Called the convergence column. See next slide.

15 The last column is an estimate for the asymptotic rate of linear convergence of the sequence {λ (j) } toward the value λ 1 = -12. This column was computed according to the formula Note the values in this column approach the value predicted by theory: λ 2 /λ 1 = 3/12 = If the matrix is symmetric there is a variation of the power method that uses the 2-norm instead of the infinity norm. Details are in the text. For a discussion about special cases and their effect on the available m- file for the power method see pp ; Some Final Comments Regarding the Power Method.