eu DOUBLE POWER METHOD ITERATION FOR PARALLEL EIGENVALUE PROBLEM

Abstract: In this paper, we introduce the double power iteration method witch can be seen as an extension of the classical power iteration in the sense that we calculate the two dominants eigenvalues at each stage. This work aims to propose a solution of slow convergence problem to the power iteration method and the calculation of the second dominant eigenvalue. We develop a parallel iterative procedure for the calculation of eigenvalues of a given matrix and we can expressed this method as a Quadrant Interlocking Factorization (QIF) which introduced by [7] and studied by [8]-[9] in other works.


Introduction
Although the power iteration method approximates only one eigenvalue of a matrix, it remains useful for certain computational problems.For instance, Google uses it to calculate the PageRank of documents in their search engine [1], and Twitter uses it to show users recommendations of who to follow [2].
For matrices that are well-conditioned and as sparse as the Web matrix, the power iteration method can be more efficient than other methods of finding the dominant eigenvector.But when the two dominants eigenvalues are of approximately the same magnitude this method may converge slowly or fail.
The central to the field of matrix computations is the problem of solving a system of linear equations Ax = b and the numerical solutions of the eigenvalues and the corresponding eigenvectors of a large and dense matrix plays an important role in numerous scientific applications.Rutishauser [5] proposed the LR algorithm for the calculation of the eigenvalues of a A. In this procedure, we obtain a sequence of matrices A (1) , A (2) , . . .which in general reduces to an upper triangular matrix.The most popular methods developed to solve this problem after [5] by factorization such as the QR algorithm, the Givens method, the Housholder transformation [10] [11] and by projection techniques on appropriate subspaces such as Lanczos and Davidson methods [13] [12].
In this paper, we present a double power iteration as iterative procedure to calculate the two dominants eigenvalues at each stage.This procedure still a variants of LR algorithm and the eigenvalues are obtained from simple 2 × 2 matrices derived from the main and cross diagonals of the limit matrix.So we can compute the eigenvalues of this limit matrix in parallel.We can express our approach as a Quadrant Interlocking Factorization and then we can give the solution of the linear equation Ax = b.
The paper is organized as follows.We start by giving mathematical framework for double power iteration algorithm for computing the two dominant eigenelements of a matrix A. Next, we prove the convergence of the algorithm.Finally, we factorize A in the form like those in [7] to evaluate all eigenvalues and we give an easy way to the solution of linear equation.

Double Power Iteration Method
Let A ∈ L(R n ) be a real matrix, I the identity matrix and {e i } 1≤i≤n the standard basis of R n .We denote by u t the transpose of vector u , B T the transpose of matrix B and by δ ij for i, j ∈ {1, ..., n}, the Kronecker delta The proposed double power iteration method is the solution of the following problem: Find x, y, z and t the real elements such that: for u and v two vectors on R n , we have This problem is equivalent to transform . . .
Let define a matrix Q applied for this transformation by v.e t n .So we write A (1) in the form Then for 1 ≤ i, j ≤ n, we have: i,1 = a i,n = 0 then, for j = 1 and j = n, we get the following system: Then ∀i, 1 ≤ i ≤ n, we have: Assume that the matrix A is diagonalizable, then there exist a basis {z i } 1≤i≤n formed by the eigenvectors of A such that: Under suitable conditions on A, the solution of this system can be obtained using iterative method like Jacobi or Gauss-Seidel method as follow: Algorithm for solving the system: For p = 0, 1, ... until convergence solve the following i set of 2 × 2 linear systems Convergence analysis: (E p ) is a 2x2 system equation who have a simple expression as follow: ∆ p =< Aū p , e 1 >< Av p , e n > − < Aū p , e n >< Av p , e 1 > is independent of the index i and can be computed only once.Then b.

Error Criteria
We denote by ε 1 p and ε 2 p the error calculation defined by

Let us give the expression of
and by the Lemma 2, we get ∀i, j > 2 A i → 0 where p → ∞ then for the large p, we get for i < j ≤ 2: with the same reasoning, we get : Then, the sequence {ū p } p and {v p } p converge respectively to ū = α 1 z 1 + α 2 z 2 and v = β 1 z 1 + β 2 z 2 .