Monte Carlo Algorithms for Calculating Eigenvalues
A new Monte Carlo approach for evaluating eigenvalues of real symmetric matrices is proposed and studied. Two Monte Carlo Almost Optimal (MAO) algorithms are presented. The first one is called Resolvent Monte Carlo algorithm (RMC) and uses Monte Carlo iterations by the resolvent matrix. The second one is called Inverse Monte Carlo Iterative algorithm (IMCI) and uses the presentation of the smallest eigenvalue by inverse Monte Carlo iterations. Estimators for speedup and for parallel efficiency are introduced. Estimates are obtained for both algorithms under consideration. Various typical models of computer architectures are considered.
Numerical tests are performed for a number of test matrices - general symmetric dense matrices, sparse symmetric matrices (including band symmetric matrices) with different behaviors on the vector machine CRAY Y—MP C92A.
Some information about the vectorization efficiency of the algorithms is obtained, showing that the studied algorithms are well-vectorized.
KeywordsMarkov Chain Mathematical Expectation Symmetric Matrice Small Eigenvalue Monte Carlo Algorithm
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