Partitioning Inverse Monte Carlo Iterative Algorithm for Finding the Three Smallest Eigenpairs of Generalized Eigenvalue Problem

A new Monte Carlo approach for evaluating the generalized eigenpair of real symmetric matrices will be proposed. Algorithm for the three smallest eigenpairs based on the partitioning inverse Monte Carlo iterative (cid:2) IMCI (cid:3) method will be considered.


Introduction
It is well known that the problem of calculating the largest or smallest generalized eigenvalue problem is one of the most important problems in science and engineering 1, 2 . This problem arises naturally in many applications. Mathematically, it is a generalization of the symmetric eigenvalue problem, and it can be reduced to an equivalent symmetric eigenvalue problem. Let A, B ∈ Ê n×n be real symmetric matrices and the matrix B a positive definite matrix. Consider the problem of evaluating the eigenvalues of the pencil A, B , that is, the values for which Ax λBx. 1.1 A generalized eigenvalue problem 1.1 is said to be symmetric positive definite S/PD if A is symmetric and B is positive definite.

Input:
Initial vector x 0 begin Output: x j , λ j 1 end end Algorithm 1

Inverse Vector Iteration Method
Another procedure for eigenvalue prediction is to use the Rayleigh quotient given by 3 since B is positive definite, then 2.1 is well defined.

Monte Carlo Method for Matrix Computations
Suppose that the matrix A ∈ Ê n×n and two vectors f, h ∈ Ê n are given. Consider the following Markov chain T i with length i: where for j 1, . . . , i, k j ∈ {1, 2, . . . , n}. The statistical nature of constructing the chain 3.1 follows as where p α and p αβ show the probability of starting chain at α and transition probability from state α to β, respectively. In fact n α 1 Define the random variable W j using the following recursion for Now, define the following random variable: Theorem 3.1. Consider the following system: Let the nonsingular matrix M ∈ Ê n , such that MA I − L, then the system 3.6 can be presented in the following form:

Advances in Numerical Analysis
Suppose that x i is the ith iterative solution of the following recursion relation with x 0 f. If we set the random variable By simulating N random paths with length i The Monte Carlo estimation can be evaluated by which is an approximation of h, x i 1 . From all possible permissible densities, we apply the following:

3.14
The choice of the initial density vector and the transition probability matrix leads to an almost Optimal Monte Carlo MAO algorithm.
Theorem 3.2. Using the above choice p {p α } n α 1 and P {p αβ } n α,β 1 the variance of the unbiased estimator for obtaining the inverse matrix is minimized [4].
There is a global algorithm that evaluates the solution of system 3.6 for every matrix A. The complexity of algorithm is O n 2 lN , where l and N are the average length of Markov chian and the number of simulated paths, respectively 2 .

input:
A ∈ R n×n , f 0 ∈ R n begin Starting from initial vector f 0 For j 1, 2

Inverse Monte Carlo Iterative Algorithm (IMCI)
Inverse Monte Carlo iterative algorithm can be applied when A is a nonsingular matrix. In this method, we calculate the following functional in each steps: It is more efficient that we first evaluate the inverse matrix using the Monte Carlo algorithm 1, 2, 4 . The algorithm can be realized as in Algorithm 2.

Partitioning IMCI
Let the matrix A be partitioned into four blocks A 1 , A 2 , A 3 , and A 4 , where A 1 and A 4 are square matrices of order p and q such that p q n: By assumption that all the indicated matrix inversions are realized, it is easy to verify that where

5.3
Thus inverting a matrix of order n comes down to inverting four matrices, of which two have order p and two have order q, and several matrix multiplications. Therefore the basic Monte Carlo for solving Af j Bf j−1 will be called as the dimension of matrix Aand equals to threshold. This action causes the convergence acceleration. Now, we can use the following recursion algorithm to obtain the inverse of matrix A see Algorithm 3 .

Finding More Than One Generalized Eigenvalues
Assume that an eigenvalue λ 1 and its corresponding eigenvector v 1 have been computed using the partitioning IMCI algorithm. In the first step of the above algorithm, we deflate the matrix A to the matrix B. Then, we repeat again the first step of the algorithm to obtain the dominant eigenvalue of B which is the second dominant eigenvalue of A. Let p values of eigenvalues of pencil A, B be computed. Suppose that V p is a matrix such that the columns of V p are p vector of eigenvector of pencil A, B , that is, where ν i is eigenvector corresponding eigevalue λ i , i 1, 2, . . . , p.

Now, let
Hence, if we find the pth smallest eigenpair of pencil A, B , then we can evalute λ p 1 , that is, p 1 th smallest eigenvalue of pencil A, B .

Numerical Results
In this section, the experimental results for obtaining the three smallest eigenpairs outlined in Tables 1, 2, and 3. The numerical tests are performed on Intel R Core TM 2 CPU, 1.83 GHz personal machine.

Conclusion and Future Study
We have seen that Monte Carlo algorithms can be used for finding more than one eigenpair of generalized eigenvalue problems. We analyze the computational complexity, speedup, and efficiency of the algorithm in the case of dealing with sparse matrices. Finally, a new method