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Unformatted text preview: AIMSLectureNotes 2006 Peter J. Olver 8. Numerical Computation of Eigenvalues In this part, we discuss some practical methods for computing eigenvalues and eigen- vectors of matrices. Needless to say, we completely avoid trying to solve (or even write down) the characteristic polynomial equation. The very basic power method and its vari- ants, which is based on linear iteration, is used to effectively approximate selected eigenval- ues. To determine the complete system of eigenvalues and eigenvectors, the remarkable QR algorithm, which relies on the GramSchmidt orthogonalization procedure, is the method of choice, and we shall close with a new proof of its convergence. 8.1. The Power Method. We have already noted the role played by the eigenvalues and eigenvectors in the solution to linear iterative systems. Now we are going to turn the tables, and use the iterative system as a mechanism for approximating the eigenvalues, or, more correctly, selected eigenvalues of the coefficient matrix. The simplest of the resulting computational procedures is known as the power method . We assume, for simplicity, that A is a complete n n matrix. Let v 1 , . . ., v n denote its eigenvector basis, and 1 , . . ., n the corresponding eigenvalues. As we have learned, the solution to the linear iterative system v ( k +1) = A v ( k ) , v (0) = v , (8 . 1) is obtained by multiplying the initial vector v by the successive powers of the coefficient matrix: v ( k ) = A k v . If we write the initial vector in terms of the eigenvector basis v = c 1 v 1 + + c n v n , (8 . 2) then the solution takes the explicit form given in Theorem 7.2, namely v ( k ) = A k v = c 1 k 1 v 1 + + c n k n v n . (8 . 3) This is not a very severe restriction. Most matrices are complete. Moreover, perturbations caused by round off and/or numerical inaccuracies will almost inevitably make an incomplete matrix complete. 9/27/07 131 c circlecopyrt 2006 Peter J. Olver Suppose further that A has a single dominant real eigenvalue, 1 , that is larger than all others in magnitude, so | 1 | > | j | for all j > 1 . (8 . 4) As its name implies, this eigenvalue will eventually dominate the iteration (8.3). Indeed, since | 1 | k | j | k for all j > 1 and all k 0, the first term in the iterative formula (8.3) will eventually be much larger than the rest, and so, provided c 1 negationslash = 0, v ( k ) c 1 k 1 v 1 for k . Therefore, the solution to the iterative system (8.1) will, almost always, end up being a multiple of the dominant eigenvector of the coefficient matrix. To compute the corresponding eigenvalue, we note that the i th entry of the iterate v ( k ) is approximated by v ( k ) i c 1 k 1 v 1 ,i , where v 1 ,i is the i th entry of the eigenvector v 1 ....
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