Melvin R. Scott

Basin attractors for various methods

Abstract There are many methods for the solution of a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several methods of various orders and present the basin of attraction for several examples. It can be seen that not all higher order methods were created equal. Newton’s, Halley’s, Murakami’s and Neta–Johnson’s methods are consistently better than the others. In two of the examples Neta’s 16th order scheme was also as good.

Basins of attraction for several methods to find simple roots of nonlinear equations

There are many methods for solving a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed.

Basin attractors for various methods for multiple roots

Abstract There are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss all known methods of orders two to four and present the basin of attraction for several examples.

Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homotopy method

The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed-point problems, certain classes of zero finding and nonlinear programming problems, and two-point boundary-value approximations based on shooting and finite differences. The method is numerically stable and has been successfully applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally convergent for a class of spline-collocation approximations to nonlinear two-point boundary-value problems. Several numerical implementations of the algorithm are briefly described, and computational results are presented for a fairly difficult fluid-dynamics boundary-value problem.

ON THE CONVERSION OF BOUNDARY-VALUE PROBLEMS INTO STABLE INITIAL-VALUE PROBLEMS VIA SEVERAL INVARIANT IMBEDDING ALGORITHMS

Publisher Summary This chapter focuses on the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms. When the solution of the boundary-value problem is smooth and only a few output points are desired, all the algorithms operate with approximately the same degree of efficiency and return approximately the same accuracy. As a result of the stability of the equations, all of the codes normally produce greater accuracy than requested. The addition formulas, in particular, are especially susceptible to the number and placement of output points and to the strategy of choosing the break points to avoid the singularities. The Scott algorithm must be implemented with the restart process to avoid the loss of significance in subtraction. Since the imbedding algorithms are very stable and employ very sophisticated initial-value codes, which vary the step size, the integration is extremely efficient. The multiple shooting codes can be implemented to use the same initial-value codes. However, if the problem requires many subdivisions of the interval, then the number of equations becomes large and slows down the process. The imbedding will almost always be faster if there is much structure to the solution.

An initial value method for the eigenvalue problem for systems of ordinary differential equations

Abstract An initial value method is presented for the calculation of characteristic lengths and characteristic functions of systems of ordinary differential equations. Very general nonseparated boundary conditions can be handled and the equations tend to be quite stable numerically. Numerical examples are presented which demonstrate the efficacy of the method.

Choosing weight functions in iterative methods for simple roots

Weight functions with a parameter are introduced into an iteration process to increase the order of the convergence and enhance the behavior of the iteration process. The parameter can be chosen to restrict extraneous fixed points to the imaginary axis and provide the best basin of attraction. The process is demonstrated on several examples.

A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems

A comparison of several invariant imbedding algorithms for the numerical solution of two-point boundary-value problems is presented. These include the Scott algorithm, the Kagiwada-Kalaba algorithm, the addition formulas, and the sweep method. Advantages and disadvantages of each algorithm are discussed, and numerical examples are presented.

A PROPOSAL FOR THE CALCULATION OF CHARACTERISTIC FUNCTIONS FOR CERTAIN DIFFERENTIAL AND INTEGRAL OPERATORS VIA INITIAL-VALUE PROCEDURES,

Abstract : The Memorandum presents an intital-value method for the solution of the inhomogeneous Fredholm integral equation u(t) = g(t) + (lambda) the integral from 0 to c of (k(the absolute value of (t-y))u(y)dy). These equations and an expansion formula of classical analysis allow a proposal to be made for calculating the solutions to the homogeneous equation phi(t) = (lambda) the integral from 0 to c of (k(the absolute value of (t-y))phi(y)dy). The Memorandum discusses possible numerical difficulties and presents extensions to other types of linear operators. (Author)

Numerical solution of unstable initial value problems by invariant imbedding

This paper shows how a generalised Ricatti transformation, which grew out of the study of invariant imbedding, may be used to convert certain unstable linear second order initial value problems into equivalent initial value problems which are often quite stable.