ON CALCULATION OF MULTIPLE EIGENVALUES OF THE LINEAR OPERATOR-FUNCTION BY REDUCTION PSEUDO-PERTURBATION METHODS

This article examines the theory of bifurcations, which applies to the problem of retaining of the approximately given multiple eigenvalues and their generalized eigenvectors. This approach allows the reduction of algebraic multiplicity of eigenvalue for one and transfers the problem to the similar one but with simple eigenvalue. The method of the false perturbation is used to construct iterative processes. ABSTRAK:Dalam artikel ini, kaedah teori bifurkasi diaplikasikan terhadap masalah untuk mengekalkan penghampiran pelbagai nilai eigen dan vektor eigen yang teritlak. Kaedah ini membenarkan pengurangan kegandaan aljabar nilai eigen kepada satu dan memindahkan permasalahan kepada yang hampir serupa tetapi dengan nilai eigen yang lebih mudah. Pengkaedahan usikan palsu digunakan untuk proses pelelaran.


INTRODUCTION
Many of engineering tasks connected with free fluctuations are reduced to eigenvalue problems so that many of them aren't solved analytically but resort to the approached methods.The problem of retaining of the approximately given eigenvalue and its generalized eigenvectors has been considered in the articles [1][2][3].The authors used method of the false perturbations which was introduced by Gavurin [4].According to this method the operator of the false perturbation builds in a way that the known approximations of the eigenvalue and generalized eigenvectors become the exact ones but for the perturbed operator-function.
Using the method of the theory of bifurcation [5], the iteration processes are then built in order to find the exact values of the eigenvalues and generalized eigenvectors of initial operator-function.The most general operator of the false perturbations was built in [3].In its formula the generalized eigenvectors of the direct operator-function and their adjoints were used in a symmetrical way.This operator was applied to several problems of mathematical physics [6,7].
The method of the finite-dimensional regularization [8] allows the reduction of geometric multiple of the eigenvalue to a unit (in this case operator-function has many linearly independent eigenvectors without generalized Jordan chains' elements).Here, using the results and methods developed in [8], it will be shown that algebraic multiplicity of the eigenvalue also could be reduced to a unit.The terminology and notations of [5] are used.

PROBLEM SETTING
We consider eigenvalue problem

(
) Let unknown eigenvalue λ be the Fredholm point of the linear operator-function A Jordan chains [5] are defined by formulas: ( ) According to [9] elements A Jordan chains, could be chosen in such a way that they satisfy following biorthogonality conditions: We assume that , Λ , ) ( 0 ψ are sufficiently good approximations to an unknowns eigenvalue λ and elements of Jordan chains: We suppose that numbers 0 K and 0 L (which are approximations of K and L from the formula (2.2)) are close to 1.
Next lemma was proved in [3].
Lemma.Going over linear combinations, can be defined as the systems that satisfies the following biorthogonality conditions: Now we want to apply methods developed in [8] to the problem of retaining of the approximately given multiple eigenvalues and their generalized eigenvectors.
Because of ) ( The solution of the equation ( ) , is taken as the initial approximation for eigenvalue λ , i.e.
We define pseudoperturbed operator using formulas Then, Using the Schmidt's regularization [5] the equation ( ) 0 could be reduced to the system: where, If we substitute x in the second equation (3.5) with the value found from the first, we construct the equation: which is called "bifurcation equation".The exact eigenvalue λ is a simple root of bifurcation equation.
Let ( ) -be the ball of radius ρ centred at 0 λ .

Theorem 3.2
If initial approximations is the sufficiently good, then there is the ball ( ) , in which equation will converge toward this solution.

Proof:
Firstly we verify the conditions of the Theorem 3.2 [10, p.446 If initial approximations are sufficiently good, we can chooseε such that 4 Then, according to Theorem 3.2 [10, p.446]-where, there is unique solution of the equation (3.6) in the ball ( ) . The iterations calculated by formula (3.7) converge toward this solution.It should be noted that on every step of iterative process it is necessary to solve only one equation:

Theorem 3.3
The elements of GJS { } i p n are the solutions of the following recurrent systems: Proof.
We want to apply method of false perturbations to initial equations (2.1) and (2.2).In order to do this we are using pseudo perturbed operator 0 D suggested by Loginov [8]: , which, according to theorem 1 [3], has the following properties: Because the initial equation has exactly n linearly independent solutions, the last system also has n linearly independent solutions.It means that the rank of the matrix of the coefficients of this system is equal to zero and therefore it is possible to assume Similarly (2.2) could be reduced to the system x in the second part of the system with the value found from the first one we can simplify the system: Because it was mentioned earlier that the rank of the matrix of this system equals 0 for each s , we can assume that Similarly, applying the same arguments to equations Unknown eigenvalue λ is the simple eigenvalue of the operator-function (3.1).
one, we obtain the system of the linear algebraic equations to find the unknown coefficients be taken as initial approximations forϕ and ψ .

-
is the bounded linear operator which exists according to the E. Schmidt's lemma[5].Then the initial segments of their Taylor series with 25 terms.All calculation experiments were carried out using program Maple 11.