Estimating the Largest Eigenvalue of a Positive Definite Matrix

The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the dominant eigenvalue after a very few iterations-even when the order of the matrix is large. Let A be a positive definite matrix of order n with eigenvalues Xj > X2 > • • • > X„ > 0 corresponding to the orthonormal system of eigenvectors xx,x2, . . . ,xn. In some applications, one must obtain an estimate of Xx without going to the expense of computing the complete eigensystem of A. A simple technique that is applicable to a variety of problems is the power method. Starting with a vector u0 of Euclidean norm unity (||«0||2 = 1), one iterates as follows: 1 1.1 1.2 1.3 1 loop for k := 0, 1, 2, vk := Auk; Pk :=uIvk' "fc + i := vk¡Hh> end loopThe theory of the method is well understood (e.g., see [4]). If Xx > X2 and x\u0 ^ 0, then the vectors uk converge linearly to Xj at a rate proportional to (X2/Xx)fc. The Rayleigh quotients pk converge to \x at a rate proportional to (X2/Xj)2fc. Convergence of the method can be hindered in two ways. First, if x^u0 is pathologically small compared to some of the numbers xfu0 (i > 1), then it will take many iterations for uk to become a good approximation to xr Second, if X2 is very near Xp the final rate of convergence will be slow. We can do very little about the first problem, except to note that it is unlikely to occur with a randomly chosen starting vector u0. Moreover, if our object is to compute the eigenvector xx, the only way to accelerate slow convergence due to an unfavorable ratio X2/Xj is to use more elaborate methods, such as simultaneous iteration [2], [3] or Lanczos tridiagonalization [1]. However, if we are only interested in a rough approximation to Xj, it will Received November 29, 1978. AMS (MOS) subject classifications (1970). Primary 65F15. ♦This work was supported in part by the office of Naval Research under Contract No. N0014-76-C-0391. © 1979 American Mathematical Society 0025-5718/79/0000-0162/$02.0