Solution of systems of nonlinear algebraic equations in three variables. Methods and algorithms. 3

An approach to constructing methods for solving systems of nonlinear algebraic equations in three variables (SNAEs-3) is suggested. This approach is based on the interrelationship between solutions of SNAEs-3, and solutions of spectral problems for two- and three-parameter polynomial matrices and for pencils of two-parameter matrices. Methods for computing all of the finite zero-dimensional roots of a SNAE-3 requiring no initial approximations of them are suggested. Some information on k-dimensional (k>0) roots of SNAEs-3 useful for a further analysis of them is obtained. Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 159–190. Translated by V. N. Kublanovskaya     

Solution of the Cauchy problem. Methods and algorithms

The Cauchy problem for systems of algebraic-differential equations with constant coefficients, i.e., for systems of ordinary differential equations that cannot be resolved for the highest derivative, both regular of index γ>0 and singular of arbitrary index γ, is considered, Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 70–123. Translated by V. N. Kublanovskaya.     

Solution of an infinite system of algebraic equations

A system of infinite algebraic equations is solved explicitly using a Carleman-type problem.     

Perturbation of systems of linear algebraic equations

A classical perturbation result for nonsingular systems of linear algebraic equations is extended to general consistent systems under any norm. An optimal perturbation result is also obtained for general linear least squares problems under a Euclidean norm.     

Part IV Parallel algorithms for partial differential equations

Part IV Parallel algorithms for partial differential equations     

Solution of polynomial equations in the field of algebraic numbers

A method of solving polynomial equations in a ring D[x] is described, where D is an arbitrary order of field ?(ω) and ω is an algebraic integer number.     

Numerical solution to systems of delay integrodifferential algebraic equations

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.     

Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations

In this paper, we propose structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations, respectively. Assume that the partial multiplicities of purely imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property (P). Under these assumptions, we prove that if these structured doubling algorithms do not break down, then they converge to the desired Hermitian solutions globally and linearly. Numerical experiments show that the structured doubling algorithms perform efficiently and reliably.     

Controllability and stabilization for linear systems of algebraic and differential equations

In this paper, we study several reachability and controllability definitions for linear, autonomous systems of degenerate ordinary differential equations. We try to identify the most interesting among them; we prove the connections with stabilizability and full stabilizability properties of the systems, which are also studied. In particular, we show the connection of the stabilizability property with the existence of special solutions of a suitable version of the Riccati equation.This paper was written under the auspices of GNAFA-CNR (Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche), Rome Italy.     

Parallel and superfast algorithms for Hankel systems of equations

Summary Utilizing kernel structure properties a unified construction of Hankel matrix inversion algorithms is presented. Three types of algorithms are obtained: 1)O(n ²) complexity Levinson type, 2)O (n) parallel complexity Schur-type, and 3)O(n log² n) complexity asymptotically fast ones. All algorithms work without additional assumption (like strong nonsingularity).     

Asymptotically normal estimates of solutions of systems of linear algebraic equations. I

Statistical estimates with respect to observations over the coefficient matrix of the system of equations are found for the regularized solutions of systems of linear algebraic equations. Under certain conditions, it is proved that this estimate is asymptotically normal.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 755–762, June, 1990.     

Asymptotically normal estimates of solutions of systems of linear algebraic equations. II

For the regularized solutions of systems of linear algebraic equations we find statistical estimates with respect to observations over the coefficient matrix of the system of equations. Under certain conditions it is proved that these estimates are asymptotically normal.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 900–907, July, 1990.     

Solution of general ordinary differential equations using the algebraic theory approach

The algebraic theory for numerical methods, as developed by Herrera [3–7], provides a broad theoretical framework for the development and analysis of numerical approximations. To this point, the technique has only been applied to ordinary differential equations with constant coefficients. The present work extends the theory by developing a methodology for equations with variable coefficients. Approximation of the coefficients by piecewise polynomials forms the foundation of the approach. Analysis of the method provides firm error estimates. Furthermore, the analysis points to particular procedures that produce optimal accuracy. Example calculations illustrate the computational procedure and verify the theoretical convergence rates.     

The logarithmic derivative for resultants of systems of algebraic equations

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 6, pp. 96–103, November–December, 1990.     

A result of systems of nonlinear complex algebraic differential equations

In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gackstatter and Laine's result concerning complex differential equations to the systems of algebraic differential equations.     

Regularization of singular systems of linear algebraic equations by shifts

Regularization of singular systems of linear algebraic equations by shifts is examined. New equivalent conditions for the shift regularizability of such systems are derived.