Applied Numerical Mathematics 59 (2009) 1537–1548 www.elsevier.com/locate/apnum Optimal iterate of the power and inverse iteration methods Davod Khojasteh Salkuyeh a,∗ , Faezeh Toutounian b a Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran b Department of Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran Received 10 February 2008; received in revised form 19 October 2008; accepted 20 October 2008 Available online 1 November 2008 Abstract The power method is an algorithm for computing the largest eigenvalue of matrix A in absolute value. To find the other eigen- values one can apply the power method to the matrix (A − σI) −1 for some shift σ . This scheme is called the inverse iteration method. Both of these two methods produce a convergence sequence and the limit is approximated by one of the iterates. In the chosen iterate, it may be difficult to estimate the global error, consisting of the truncation error and the round-off error. In this paper, by using the CESTAC method and the CADNA library, we propose a method for computing the optimal iterate, the iterate for which the global error is minimal. In the proposed method the accuracy of the computed eigenvalue may also be estimated. Some numerical examples are given to show the efficiency of the method.  2008 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 65F15; 65G50 Keywords: Power method; Inverse iteration method; Round-off error; Common significant digits; CESTAC method; CADNA library 1. Introduction Many of the problems in scientific computing involve the computation of convergence sequences and the limit of the sequence is approximated by one of the iterates. Since the finite precision computations affect the stability of iterative algorithms and the accuracy of computed solutions, it is difficult to estimate the global error, consisting of the truncation error and the round off error, in the chosen iterate. In [11,16], the authors showed that under some assumptions on the speed of convergence of a sequence, by using the CESTAC method [17,19] and the CADNA library [1,4,10], the optimal iterate, i.e., the approximation for which the global error is minimized, can be dynamically computed. In this paper we show how, by using this library, the optimal iterate of the power method and inverse iteration method can be dynamically computed and its accuracy can be estimated. The power method is an iterative method for computing the largest eigenvalue of a matrix in absolute value. Let A be an n × n diagonalizable matrix and its eigenvalues satisfy |λ 1 | > |λ 2 |  ···  |λ n |. Let also x i be the eigenvector of A corresponding to the eigenvalue λ i . Then the power method is an iterative method which computes an approximation of the eigenpair (λ 1 ,x 1 ) of A. This algorithm may run as Algorithm 1 [8,9,22]. * Corresponding author. E-mail addresses: khojaste@uma.ac.ir (D. Khojasteh Salkuyeh), toutouni@math.um.ac.ir (F. Toutounian). 0168-9274/$30.00  2008 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2008.10.004