Journal of Sound and < ibration (2002) 254(3), 599 } 612 doi:10.1006/jsvi.2001.4102, available online at http://www.idealibrary.com on COMPUTING THE LOWEST EIGENVALUE WITH RAYLEIGH QUOTIENT ITERATION S. RAJENDRAN School of Mechanical and Production Engineering, Nanyang ¹ echnological ;niversity, 50 Nanyang Avenue, Singapore 639798. E-mail: msrajendran@ntu.edu.sg (Received 7 August 2000, and in ,nal form 2 August 2001) 1. INTRODUCTION Free vibration analysis of structures involves solution of generalized eigenproblem given by Kp"Mp, (1) where K and M are nn matrices, p is an eigenvector, and  is the corresponding eigenvalue. In this work, it is assumed that the matrices K and M are real, symmetric and positive de"nite. The eigenvalues of such eigenproblems are all real and positive. Several eigensolution methods exist in the literature [1}7]. Inverse iteration is a vector iterative method primarily used for the computation of the smallest eigenvalue and the corresponding eigenvector. The inverse iteration also forms an integral part of hybrid eigensolution methods (e.g., see reference [1]) such as Lanczos methods, simultaneous/subspace iteration method, determinant search method, Rayleigh quotient iteration and Householder QR inverse iteration (HQRI) method. The convergence of inverse iteration can be very slow if the eigenvalues are closely spaced or the starting iteration vector is di"cient in the "rst eigenvector. Rayleigh quotient iteration [1] is a shifted inverse iteration method where the iterations are performed as an alternate sequence between the following pair of equations: [K! (x  ) M] x   "Mx  , x  " x   x    Mx   , (2, 3) where Rayleigh's quotient is computed as  (x   )" x    Kx   x    Mx   , x  Kx  x  Mx  ,x  Kx  , (x  ). (4) The iteration is terminated whenever the relative change in Rayleigh's quotient between successive iterations is less than the allowable tolerance. The convergence of this method is reported to be cubic [1, 8]. In spite of its excellent convergence characteristics, the method may, in principle, converge arbitrarily to any eigenvalue depending on the magnitude of the shift value, and hence assuring convergence to any particular eigenvalue is di$cult. This is in contrast with the inverse iteration method, where the convergence is always towards the "rst eigenvalue, provided the starting vector is not de"cient in the "rst eigenvector. 0022-460X/02/$35.00  2002 Elsevier Science Ltd. All rights reserved.