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Academic Commons Search Resultsen-usEstimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start
https://academiccommons.columbia.edu/catalog/ac:143005
Kuczynski, Jacek; Wozniakowski, Henryk10.7916/D8B56SVVMon, 19 Jun 2017 14:40:38 +0000This paper addresses the problem of computing an approximation to the largest eigenvalue of an n x n large symmetric positive definite matrix with relative error at most ε. Only algorithms that use Krylov information [b, Ab, . .. , Akb] consisting of k matrix-vector multiplications for some unit vector b are considered. If the vector b is chosen deterministically, then the problem cannot be solved no matter how many matrix-vector multiplications are performed and what algorithm is used. If, however, the vector b is chosen randomly with respect to the uniform distribution over the unit sphere, then the problem can be solved on the average and probabilistically. More precisely, for a randomly chosen vector b, the power and Lanczos algorithms are studied. For the power algorithm (method), sharp bounds on the average relative error and on the probabilistic relative failure are proven. For the Lanczos algorithm only upper bounds are presented. In particular, ln ( n )/k characterizes the average relative error of the power algorithm, whereas O( ( ln ( n )/ k )2 ) is an upper bound on the average relative error of the Lanczos algorithm. In the probabilistic case, the algorithm is characterized by its probabilistic relative failure, which is defined as the measure of the set of vectors b for which the algorithm fails. It is shown that the probabilistic relative failure goes to zero roughly as sqrt n (1 - ε) ^k for the power algorithm and at most as sqrt n e^{ - (2k - 1)\sqrt ε } for the Lanczos algorithm. These bounds are for a worst case distribution of eigenvalues which may depend on k. The behavior in the average and probabilistic cases of the two algorithms for a fixed matrix A is also studied as the number of matrix-vector multiplications k increases. The bounds for the power algorithm depend then on the ratio of the two largest eigenvalues and their multiplicities. The bounds for the Lanczos algorithm depend on the ratio between the difference of the two largest eigenvalues and the difference of the largest and the smallest eigenvalues.Computer sciencehw13Computer ScienceReportsImplementation of the GMR Algorithm for Large Symmetric Eigenproblems
https://academiccommons.columbia.edu/catalog/ac:141241
Kuczynski, Jacek10.7916/D8CZ3G58Mon, 19 Jun 2017 13:36:02 +0000We present an implementation of the generalized minimal residual (gmr) algorithm for finding an eigenpair of a large symmetric matrix. We report some numerical results for this algorithm and compare them with the results obtained for the Lanczos algorithm. A Fortran implementation of the gmr algorithm is included. The input of this subroutine is a matrix which has been partially reduced to tridiagonal form. Such a form can be obtained by the Lanczos process. The Fortran subroutine is also available via anonymous FTP as "pub/gmrval" on Columbia.edu [128.59.16.1] the Arpanet.Computer scienceComputer ScienceReportsOn the Optimal Solution of Large Eigenpair Problems
https://academiccommons.columbia.edu/catalog/ac:141244
Kuczynski, Jacek10.7916/D8862QFVMon, 19 Jun 2017 13:36:01 +0000The problem of approximation of an eigenpair of a large n × n matrix A is considered. We study algorithms which approximate an eigenpair of A using the partial information on A given by b, Ab, …, Ajb, j << n, i.e., by Krylov subspaces. A new algorithm called the generalized minimal residual (gmr) algorithm is analyzed. Its optimality for some classes of matrices is proved. We compare the gmr algorithm with the widely used Lanczos algorithm for symmetric matrices. The gmr and Lanczos algorithms cost essentially the same per step and they have the same stability characteristics. Since the gmr algorithm never requires more steps than the Lanczos algorithm, and sometimes uses substantially fewer steps, the gmr algorithm seems preferable. We indicate how to modify the gmr algorithm in order to approximate p eigenpairs of A. We also show some other problems which can be nearly optimally solved by gmr-type algorithms. The gmr algorithm for symmetric matrices was implemented and some numerical results are described. The detailed implementation, more numerical results, and the Fortran subroutine can be found in KuczyÅ„ski (“Implementation of the gmr Algorithm for Large Symmetric Eigenproblems,“ Report, Columbia University, 1985). The Fortran subroutine is also available via anonymous FTP as “pub/gmrval“ on COLUMBIA-EDU [128.59.16.1] on the Arpanet.Computer scienceComputer ScienceReports