Takeo Ojika

Deflation Algorithm for the Multiple Roots of a System of Nonlinear Equations

Abstract T. Ojika, S. Watanabe, and T. Mitsui (in preparation) have been developing a subroutine package NAES (Nonlinear Algebraic Equations Solver) for the numerical solutions of the system of nonlinear equations. An algorithm, in the package, termed the deflation algorithm, for determining multiple roots for a system of nonlinear equations, is presented and its effectiveness is shown by solving a numerical example.

Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations

Abstract The Behavior of the Newton-Raphson method at the singular roots has been studied by a number of authors and the convergence of the Newton-Raphson sequence has been shown to be linear. In this paper a new method with symbolic and numerical manipulations, termed the modified deflation algorithm, is proposed for the singular root of a system of nonlinear algebraic equations. The basic idea of the present method is to replace a part of the original equations by a set of new equations which pass through the singular root. According to the method, both convergency and accuracy can greatly be improved. In addition it is often possible to obtain analytically the singular root from the new equations. In order to show the effectiveness of the present method two illustrative examples are solved.

Modified deflation algorithm for the solution of singular problems. II: Nonlinear multipoint boundary value problems

Abstract Various kinds of iterative methods have been proposed for the solution of nonlinear multipoint boundary-value problems MPBVPs. However, it is necessary for these methods that the adjusting matrix, which corresponds to the Jacobian of nonlinear equations, is nonsingular at the solution. In this paper an algorithm for the singular solution of nonlinear MPBVPs, which is an extension of the modified deflation algorithm for the singular root of nonlinear algebraic equations developed by the author is presented. According to the present method, the singular solution can ultimately be reduced to the usual simple solution and both convergency and accuracy can greatly be improved. The effectiveness of the present method is shown by solving two illustrative examples.

A numerical method for multipoint boundary value problems with application to a restricted three body problem

A type of parallel shooting method is proposed for the solution of nonlinear multipoint boundary value problems. It extends the usual quasilinearization method and a previous shooting method developed for such problems, and reduces to usual multiple shooting techniques for two point boundary value problems. The effectiveness of the method for stiff problems is illustrated by an application to the problem of finding periodic solutions of a restricted three body problem with given Jacobian constant and unknown period.

Multipoint boundary value problems with discontinuities II. Convergence of the initial value adjusting method

Abstract In this paper we give a proof for convergence of the initial value adjusting method, described in detail in part I, based on Kantorovichs theorem. Under standard assumptions on the dynamics, boundary conditions, and initial approximations, a quadratic convergence rate is obtained.

Structure analyses for large scale nonlinear multipoint boundary value problems

where fi and g, are twice continuously differentiable with respect to their arguments, fi is continuous in t on [tl , t,,,] and ’ denotes transposition. The MPBVP given by (1.1) and (1.2) can almost never be solved in a closed form. Usually iterative algorithms of various kinds are employed to execute a numerical solution. These algorithms generally solve the MPBVP by reducing it to a corresponding initial value problem and starting with a set of initial conditions x(t,) = (a,, a, ,..., a,)‘, and employ different iterative schemes to modify the initial conditions so as to satisfy the given boundary conditions [ 1-6, 8, 14-17, 19-21, 241. As the number of the ODES and the BCs as well as the entire interval [tl, t,] increases, however, these methods run into difficulties because of numerical errors, large computer storage requirements, and the excessive amount of computer time needed to solve simultaneously the entire system of equations along the entire interval. In order to establish a basic existence theorem [ 3, 7,9, 12,231 guaranteeing the global solvability of the corresponding initial value problem and to mitigate the difficulties mentioned above, it is often necessary and effective 139 0022-247X/83

On the Quadratic Convergence Properties of the e-secant Method for the Solution of a System of Nonlinear Equations and Its Application to a Chemical Reaction Problem

3.00

Initial value adjusting method and graph theoretical analysis for the solution of nonlinear multipoint boundary value problems with varying system dimensions

Many applied and theoretical problems in the real world can be reduced to systems of simultaneous equations. In particular, the mathematical treatment of physical phenomena often produces large systems of nonlinear algebraic equations. In general it is impossible to get their exact solutions, so numerical computations are necessary for the analysis of the original problems. One of the most famous and widely used iterative methods for the numerical solutions of nonlinear systems of algebraic equations is the Newton-Raphson method which was proposed by I. Newton (1642-1729) ]6] and J. Raphson (1648-1715) [lo]. Though it has been widely applied to various problems, the method has some disadvantages: (i) in advance, the partial derivatives of the systems of equations (the Jacobian matrix) must be calculated analytically, (ii) the

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

Abstract The initial value adjusting method for the solution of nonlinear multipoint boundary value problems in which the system dimensions vary over subintervals is proposed. To reduce the computer storage requirements and the excessive amount of computer time, an algorithm based on a digraph and its associated Boolean matrices is also proposed. The effectiveness of these algorithms is illustrated by an application to a five compartment model from pharmacokinetics.

An interaction-coordination algorithm with modified performance index for nonlinear optimal control problems

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.