In the synthesis problem of a plane equidistant antenna array according to the given requirements to a power directivity pattern, a nonlinear integral equation is obtained, which has a non-unique solution. One of them is trivial, from which at certain electrodynamic parameters other solutions can branch off. In the process of finding the branching curves of the solutions of the integral equation, two-parameter eigenvalue problems with spectral parameters (electrodynamic parameters) that are contained non-linearly in the kernel of the linearized integral operator are arisen. A numerical algorithm for finding eigenvalues is considered for such problems. The numerical examples are given.

Methods and algorithms that form a one-parameter family of methods of bilateral approximations to the eigenvalues of nonlinear spectral problems are constructed. Their convergence is proved. Numerical results are given.

In this article, the research proposed by the author, the approach to the construction of methods and algorithms of bilateral approximations to the eigenvalues of nonlinear spectral problems, is continued. On the basis of Newton's method, some new algorithms of the bilateral approximations to their eigenvalues are constructed and substantiated.

A nonlinear synthesis problem of antennas according to the prescribed power (squared amplitude) radiation pattern (RP) is considered in the variational statement that yields in the possibility to take into account an additional restriction to the synthesized power RP. The problem of synthesis consists of finding such currents in antenna, which generates the RP with the best approximation to the given one. The respective Euler’s equation is reduced on the basis of used functional. This is nonlinear integral equation of Hammerstein’s type. The effective numerical methods are elaborated and applied for its solving. The computational results verify the effectiveness of approach proposed.

An algorithm for finding the number of eigenvalues of two-parameter spectral problems in a given region is proposed. At the heart of the algorithm lies the principle of the argument of the analytic function of one variable. Numerical results for a nonlinear two-parameter eigenvalue problems are given.

It is investigated the inverse eigenvalue problem that includes classic additive and multiplicative spectral problems. It is presented the method of transformation of the inverse eigenvalue problem to the direct multi-parameter one. It is proposed the numerical method of calculating the approximate solution of the spectral problem by solving the equivalent variation problem. There are several numerical experiments presented in order to illustrate the behavior of the method.

We consider matrix inverse eigenvalue problems. For their numerical solution, we propose an algorithm based on Newton’s iterative process, where, for constructing the Jacobian, a numerical procedure of calculation of the exact derivatives of a matrix determinant is used.

In the real abstract Hilbert space the nonlinear multiparameter spectral problem is assigned to the variation problem on a minimum of some functional. The equivalence of spectral and variation problems is proved. On the base of modified Newton method a numerical algorithm of finding its eigenvalues and eigenvectors is sproposed.

We consider an iteration algorithm for determining the eigenvalues of an algebraic two-parameter spectral problem with the use of Newton’s method and a new efficient numerical procedure for calculation of the derivative of a determinant. Numerical examples are given.

The computational aspects of the use of efficient numerical procedure to calculate derivatives of matrix determinant in the algorithms of finding the eigenvalue curves and bifurcation points of nonlinear two-parameter eigenvalue problems are considered. Some numerical examples are given.

AbstractThe iteration algorithm of finding bifurcation points of simple eigenvalue curves of the linear algebraic two-parameter eigenvalue problem is considered. The algorithm is based on the efficient numerical procedure of calculation of the derivative of matrix determinant. Numerical results are given.

The nonlinear synthesis problem of a plane equidistant antenna array according to the given requirements to an amplitude directivity pattern is considered. When finding the branching lines of solutions of nonlinear integral equations obtained as a result of solution of the synthesis problem, the two-parameter eigenvalue problems with spectral parameters analytically including in a kernel of the linearized integral operator are arisen. For such problems the numerical algorithms of finding the eigenvalue curves and the bifurcation points are considered. The numerical examples are presented.

In the finite-dimensional real Euclidean space the nonlinear generalized spectral problem is put in accordance to the variation problem on the minimum of some functional. The equivalence of spectral and variation problems is proved. On the base of gradient procedure the numerical algorithm of finding of its eigenvalues and eigenvectors is offered. Under certain conditions over the operators the local convergence of method is proved.

The problem of construction of numerical bilateral approximation for determination of branching points of one nonlinear integral operator, arising in the theory of antennas synthesis according to the given amplitude directivity pattern, is considered. The basic difficulty consists in that the kernel of integral operator nonlinearly depends on the parameter, which play role of the spectral one. Thus the problem is reduced to a nonlinear eigenvalue problem with application the technique of the alternating approximations of eigenvalues. The technique is based on a generalization of the known Rayleigh ratio for iinear problem onto nonlinear (initial and some auxiliary) eigenvalue problems. These generalized Rayleigh ratioes are used for constructing an iterative process of alternating eigenvalue approximations.

In this paper a new approach to construction of iterative methods of bilateral approximations is proposed and investigated. The conditions on initial approximation, which ensures the convergence of iterative processes, are obtained.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 83, Issue 4 p. 282-286 Short Communication On bilateral convergence of Halley's method B.M. Podlevskyi, B.M. Podlevskyi [email protected] Institute of Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, 3''b'' Naukova str., 79000, Lviv, UkraineSearch for more papers by this author B.M. Podlevskyi, B.M. Podlevskyi [email protected] Institute of Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, 3''b'' Naukova str., 79000, Lviv, UkraineSearch for more papers by this author First published: 24 March 2003 https://doi.org/10.1002/zamm.200310035Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL References [1]G.H. Brown, On Halley's variation of Newton's method, Amer. Math. Monthly 84 (1977) 726– 727 [2]G. Alefeld, On the convergence of Halley's method, Amer. Math. Monthly 88 (1981) 530– 536 [3]W. Gander, Halley's iteration method, Amer. Math. Monthly 92 (1985) 131– 134 [4]J.F. Traub, Iterative Methods for the Solution of Equation (Chelsea Publishing Company, New York, 1982) [5]W. Gautschi, Numerical Analysis: An Introduction (Birkhäuser, Boston 1997) [6]B. Doring, Einige Sätze über das Verfahren der tangierenden Hyperbeln in Banach-Räumen, Apl. Mat. 15 (1970) 418– 464 [7]J.M. Orteda and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, London, 1970) [8]B.M. Podlevskyi, The bilateral approximation methods for the solution of nonlinear equations (Preprint Publ. Inst. Appl. Probl. Mech. Math. NASU, Lviv, 2001) (Ukrainian) [9]B.M. Podlevskyi, On the one approach to design the bilateral iterative methods for the solution of nonlinear equations (Ukrainian), Dop. Nat. Acad. Nauk Ukrainy 5 (1998) 37– 41 [10]B. M Podlevskyi, On new properties of Halley's method (Ukrainian), Dop. Nat. Acad. Nauk Ukrainy 12 (1999) 21– 26 Citing Literature Volume83, Issue4April 2003Pages 282-286 ReferencesRelatedInformation

The problem of building a numerical algorithm for determination of branching points of one nonlinear integral operator, which arises in the theory of antennas synthesis according to the given amplitude radiation pattern, is considered. The basic difficulty consists in that the kernel of an integral operator nonlinearly depends on two parameters, which play role of the spectral ones. For such problems, except a special ease, the existing numerical algorithms are not applicable. For building an algorithm for solving such problems, the equivalent variational statement is used, the problem is reduced to a sequence of linear two-parameter eigenvalue problems with application of one gradient procedure for simultaneous evaluation of two spectral parameters being the branching points of an initial nonlinear integral operator.

One iterative method for the solution of the generalized spectral problem for a matrix with elements analytically depending on the spectral parameter is developed and justified

The case of the eigenvalue problem of the quadratic matrix beam is proposed. The method permits one to calculate all eigenpairs of the matrix beam. With this aim, the generalized eigenvalue problem is reduced to a linear problem. Then, the modified successive approximations method is used for calculation of the eigenvalues and eigenvectors. This permits the use of the information obtained from the iterations, to obtain the first eigenpair and all the following ones more quickly. Some numerical examples are considered