On a Family of Newton-Like Methods for Bilateral Approximation of the Eigenvalues of Nonlinear Spectral Problems

I. INTERODUCTION Nonlinear eigenvalue problems arise in various scientific and technical fields. Over the last years, significant achievements have been made in the development of numerical methods for addressing such problems. Additional information on possible applications of nonlinear eigenvalue problems and numerical methods for solving them can be obtained, in particular, from [1]. For the problems which only have real eigenvalues it would be desirable to have not only a simple approximation (for example, monotony at one side) and an asymptotic error of calculation, but also obtain the upper and lower bounds of eigenvalues in the calculation process. In many cases this allows to evaluate the reliability of the iterative approximation. It is meaning at each step of the iterative process a convenient aposteriori estimation of the calculation error can be obtained. For linear spectral problems, there are several general approaches to constructing the lower bounds of eigenvalues, but they cannot be generalized on nonlinear problems. The exception is spectral problems for polynomial operator bundles of self-adjoint operators, on which can be generalized the methods based on inclusion theorems . This work is a continuation of the study proposed by the author of the approach to construct the methods and algorithms of bilateral approximations to the eigenvalues of spectral problems, nonlinear with respect to the spectral parameter [2, 3, 4]. Here, to calculate a simple isolated eigenvalue, a one parameter family of bilateral methods of Newton 's type is constructed. Their convergence is substantiated.


I. INTERODUCTION
Nonlinear eigenvalue problems arise in various scientific and technical fields. Over the last years, significant achievements have been made in the development of numerical methods for addressing such problems. Additional information on possible applications of nonlinear eigenvalue problems and numerical methods for solving them can be obtained, in particular, from [1]. For the problems which only have real eigenvalues it would be desirable to have not only a simple approximation (for example, monotony at one side) and an asymptotic error of calculation, but also obtain the upper and lower bounds of eigenvalues in the calculation process. In many cases this allows to evaluate the reliability of the iterative approximation. It is meaning at each step of the iterative process a convenient aposteriori estimation of the calculation error can be obtained.
For linear spectral problems, there are several general approaches to constructing the lower bounds of eigenvalues, but they cannot be generalized on nonlinear problems. The exception is spectral problems for polynomial operator bundles of self-adjoint operators, on which can be generalized the methods based on inclusion theorems . This work is a continuation of the study proposed by the author of the approach to construct the methods and algorithms of bilateral approximations to the eigenvalues of spectral problems, nonlinear with respect to the spectral parameter [2,3,4]. Here, to calculate a simple isolated eigenvalue, a oneparameter family of bilateral methods of Newton 's type is constructed. Their convergence is substantiated. .

II. PRELIMINARIES
We consider the nonlinear eigenvalue problem where ()  D is a square matrix of order n , all elements of which are sufficiently smooth (at least three times continuously differentiable) functions of the parameter R  , n yR  . The eigenvalues is sought as solutions of determinant equation To determine the isolated eigenvalue of matrix ()  D we proposed and justify the Newton-type iterative processes which give the alternate approximations to the eigenvalue   of the equation (2) The matrix elements in the decompositions (5) can be calculated using the corresponding recurrence relations written in [3]. Next, by   we denote a precise simple root of equation (2) which obviously has the same zeros as the function () is satisfied and which has the following properties.

III. A FAMILY OF BILATERAL APPROXIMATION METHODS
Using the properties of the function () z   , we can construct a family of iterative processes of Newton type methods that give bilateral approximations in the form (3). For example, for cases (A) or (D), the iterative process can be written in the form Similarly, an iterative process can be constructed for other cases.
The following theorem serves to justify the bilateral convergence of the iterative process (7).
The theorem is proved by the method of mathematical induction, relying on Theorem 1. It should be noted that an iterative process (7), which provides the bilateral approximations to the eigenvalue, with respect to (6), will be presented in the form  The algorithm shows that in order to obtain the alternate approximations, on each step of the algorithm, one must refer to the algorithm of calculating the decomposition (5). In some cases, the algorithm constructed based on the iterative process of including approximations is more optimal with respect to the number of calls to the calculation of decomposition (5) [3]: With this algorithm we get the including approximations to the eigenvalue in the form using one initial approximation, for example, to the left hand of   .
If we now replace the values of the function and its derivatives at the desired points by the relations (A.4), then the iterative process (9) will have the form 1   Thus, we see that with the algorithm 2, unlike the algorithm 1, two approximations (from left and right hand sides of the root) are calculated performing only one call to the calculation of decomposition (5).

IV. NUMERICAL RESULTS
In the study of the stability of systems of ordinary differential equations with delay, the eigenvalue problems in which the spectral parameter enters nonlinearly are arisen. Thus, the process of propagation of weak perturbations in multi-connected systems (such as electric and acoustic fields) is described by a system of linear differential-algebraic equations of the form We consider the partial case of such system with one delay parameter and also assume that the coefficients of the equations and the parameter are independent of the time variable. Then the system of equations will take the form where () ut is the vector of perturbations, ,, A B C are square real matrices whose coefficients are independent of t . Then the solutions of system (12) can be found in the form where x is the desired vector,  is the number. Substituting (13) into system (12), we obtain the eigenvalue problem Since (14)  Therefore, we consider a nonlinear spectral problem ( ) 0 where E is a unit matrix, A is a tridiagonal matrix with nonzero elements: 1, For such matrices, all the eigenvalues of problem (15) were calculated by each of algorithms 1 -2. They completely coincided with the numbers obtained in [5] by another algorithm, which is only applicable if the matrices in problem (14)

V. CONCLUSIONS
Approbation of the constructed algorithms on model and physical problems shows their reliability and efficiency, as well as advantages over the usual Newton's method in the sense that at each step of the iterative process we obtain bilateral estimates of the desired solution, and therefore at each step we get a convenient a posteriori estimation of calculation error.
As for the choice of the  parameter, this is the subject of a separate study. Here we note only next. When 1/ 2  is selected, the convergence rate increases and becomes cubic. When 1  , we can obtain the rate of convergence not less than cubic, using algorithm 2, in which for each subsequent "On a Family of Newton-Like Methods for Bilateral Approximation of the Eigenvalues of Nonlinear Spectral Problems"