Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.3

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.3

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10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an matrix A are obtained by solving its characteristic equation For large values of n , polynomial equations like this one are difficult and time-consuming to solve. Moreover, numerical techniques for approximating roots of polynomial equations of high degree are sensitive to rounding errors. In this section you will look at an alterna- tive method for approximating eigenvalues. As presented here, the method can be used only to find the eigenvalue of A that is largest in absolute value—this eigenvalue is called the dominant eigenvalue of A . Although this restriction may seem severe, dominant eigenval- ues are of primary interest in many physical applications. Not every matrix has a dominant eigenvalue. For instance, the matrix with eigenvalues of and has no dominant eigenvalue. Similarly, the matrix with eigenvalues of and has no dominant eigenvalue. E X A M P L E 1 Finding a Dominant Eigenvalue Find the dominant eigenvalue and corresponding eigenvectors of the matrix Solution From Example 4 of Section 7.1 you know that the characteristic polynomial of A is So the eigenvalues of A are and of which the dominant one is From the same example you know that the dominant eigenvectors of A those corresponding to are of the form t 0. x t 3 1 , 2 2 2 2. 2 2, 1 1 2 3 2 1 2 . A 2 1 12 5 . 3 1 1 2, 2 2, A 2 0 0 0 2 0 0 0 1 2 1 1 1 A 1 0 0 1 n c n 1 n 1 c n 2 n 2 . . . c 0 0. n n 586 CHAPTER 10 NUMERICAL METHODS Definition of Dominant Eigenvalue and Dominant Eigenvector Let and be the eigenvalues of an matrix A . is called the dominant eigenvalue of A if The eigenvectors corresponding to are called dominant eigenvectors of A . 1 i 2, . . . , n . 1 > i , 1 n n n 1 , 2 , . . . ,
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The Power Method Like the Jacobi and Gauss-Seidel methods, the power method for approximating eigenval- ues is iterative. First assume that the matrix A has a dominant eigenvalue with correspond- ing dominant eigenvectors. Then choose an initial approximation of one of the dominant eigen vectors of A . This initial approximation must be a nonzero vector in Finally, form the sequence given by For large powers of k , and by properly scaling this sequence, you will see that you obtain a good approximation of the dominant eigenvector of A . This procedure is illustrated in Example 2. E X A M P L E 2 Approximating a Dominant Eigenvector by the Power Method Complete six iterations of the power method to approximate a dominant eigenvector of Solution Begin with an initial nonzero approximation of Then obtain the following approximations. Iteration “Scaled” Approximation 190 2.99 1.00 x 6 A x 5 2 1 12 5 280 94 568 190 94 2.98 1.00 x 5 A x 4 2 1 12 5 136 46 280 94 46 2.96 1.00 x 4 A x 3 2 1 12 5 64 22 136 46 22 2.91 1.00 x 3 A x 2 2 1 12 5 28 10 64 22 10 2.80 1.00 x 2 A x 1 2 1 12 5 10 4 28 10 4 2.50 1.00 x 1 A x 0 2 1 12 5 1 1 10 4 x 0 1 1 .
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