Takeshi Kiyono

Curve Fitting by a One-Pass Method With a Piecewise Cubic Polynomial

A method is described for fitting a piecewise cubic polynomial to a sequence of data by a onepass method. The polynomial pieces are calculated as the data is scanned only once from left to right. The algorithm is shown to be stable when a piecewise cubic polynomial is used which is continuous with its first derivative, while it becomes unstable if the polynomial is made continuous up to the second derivative. The knots of the approximating function are determined successively using a criterion by Powell.

Numerical solution of Plateau’s problem by a finite element method

This paper concerns the application of a finite element method to the numerical solution of a nonrestricted form of the Plateau problem, as well as to a free boundary prob- lem of Plateau type. The solutions obtained here are examined for several examples and are considered to be sufficiently accurate. It is also observed that the hysteresis effect, which is a feature of a nonlinear problem, appears in this problem. 1. Introduction. Methods for the numerical solution of the Plateau problem have so far been examined by D. Greenspan (3), (4), using the combination technique of difference and variational methods, and by P. Concus (5), using a finite difference method. These two methods can be applied only to the so-called restricted form of the Plateau problem described by Forsythe and Wasow (2, Section 18.9), that is, to the problems where the boundary condition is represented by a single-valued function. Thus, they cannot be applied to the problem where the boundary condition is repre- sented by a multi-valued function, such as Courants example described later. This paper shows that such multi-valued boundary-value problems can be solved numerically by a finite element method. In this case, two solution methods, one for a free boundary problem and the other in a cyclindrical coordinate system, are presented.

Monte Carlo path-integral calculations for two-point boundary-value problems

A computational technique based on the method of path integral is studied with a view to finding approximate solutions of a class of two-point boundary-value problems. These solutions are “rough” solutions by Monte Carlo sampling. From the computational point of view, however, once these rough solutions are obtained for any nonlinear cases, they serve as good starting approximations for improving the solutions to higher accuracy. Numerical results of a few examples are also shown.

curve fitting by a piecewise cubic polynomial

This paper deals with curve fitting by a piecewise cubic polynomial which is continuous with its first derivative. A knot is inserted successively until a certain criterion is satisfied. Then a suboptimal algorithm is applied to minimize the sum of squares of residuals.ZusammenfassungDiese Arbeit befaßt sich mit der Kurvenanpassung durch stückweise kubische Polynome, die zusammen mit ihrer ersten Ableitung stetig sind. Es werden sukzessive Knotenpunkte hinzugenommen, bis ein gewisses Kriterium erfüllt ist. Dann wird ein suboptimaler Algorithmus angewendet, um die Fehlerquadratsumme zu verkleinern.

Numerical solution of integral equations in Chebyshev series

In this paper we discuss the Chebyshev series method with Newton iterations for the numerical solution of nonlinear integral equations. An existence theorem for nonlinear integral equations is given using a functional analytic approach. A method to compute and error bound to an approximate solution is discussed on the basis of the theorem.

Application of the Monte Carlo method to systems of nonlinear algebraic equations

It is, in general, a difficult problem to find all the roots of a given system of nonlinear algebraic equations with several unknowns, even though we meet with such demands in many fields of applied sciences. We have several methods for such problems, which have been extended from some powerful method for algebraic equations with a single unknown. Iterative procedures thus developed, are often found unsuccessful, since they do not necessarily converge, given arbitrary starting values. The conventional methods of iteration are guaranteed against divergence, only if the starting value is well enough in the neighborhood of the solution 1. Thus, as long as we have no means to place all the roots even very roughly, we have no working method to find all the roots of a given system of nonlinear algebraic equations. In the present paper, a new scheme of the Monte Carlo technique is proposed in order to find approximate locations of these roots, which are believed to provide successful starting values for the current classical methods of iteration. The results of a test on a digital computer are also illustrated, together with a sketch of the actual computer procedures. Apart from the novelty of the underlying concept, the method has a mer~e~iff that it can treat cases with many variables with no particular increase in complexity. 2. Principle

On the Solution of Laplace’s Equation by Certain Dual Series Equations

Certain dual trigonometric series equations are discussed in connection with a mixed boundary value problem of the Laplace equation. This paper gives an accurate and fast method of solution of the mixed boundary value problem of the Laplace equation by the dual series equations approach. The problem can be applied to the determination of the line capacitance of microstrip transmission lines which have gained in importance in recent years. The dual series equations are reduced to a Fredholm integral equation of the second kind. The method of solution is suited for boundary value problems and numerical computation.

Boundary contraction solution of the Neumann and mixed boundary value problems of the Laplace equation

The boundary contraction method is generalized in such a way that it is applicable to the Neumann and mixed boundary value problems over regions of irregular shape. Various variants of mixed boundary value problems can be solved numerically in a unified way with mesh points fewer than those required of ADI and SOR. The method is not iterative and therefore does not require the positive definiteness of eigenvalues which is the necessary condition of the stability of ADI and SOR. The method is also applicable to exterior problems. Thus the applicability of the contraction method to problems of practical importance is substantially improved.

Interpolation of multivariable functions with respect to random points

The numerical interpolation of multivariable functions has been solved before by the Monte Carlo method, where data points are assumed to be given on discrete lattice points. When data points are randomly distributed, it is very difficult to develop interpolation formulas. This paper deals with least squares interpolation of multivariable functions with respect to random points.ZusammenfassungBei numerischer Interpolation von Funktionen mehrerer Veränderlicher mit Stützstellen in diskreten Gitterpunkten wurden früher Monte-Carlo-Methoden verwendet. Bei zufällig verteilten Stützstellen ist die Entwicklung von Interpolationsformeln besonders schwierig. Diese Arbeit benützt die Methode der kleinsten Quadrate zur Entwicklung eines Interpolationsverfahrens bei zufälligen Stützstellen.

Numerical integration in the irregular region

Numerical evaluation of integrals in two or more dimensions has been developed mainly for simple regions such as cube, sphere, simplex, etc. This paper deals with numerical integration in a closed irregular region. The region is transformed into parallelpiped and an adaptive type formula is used for the integration.ZusammenfassungNumerische Integration in zwei und mehr Dimensionen ist hauptsächlich für einfache Gebiete wie Würfel, Kugel, Simplex und andere entwickelt worden. Dieses Papier behandelt die numerische Integration in einem geschlossenen unregelmäßigen Gebiet. Das Gebiet wird in Quader transformiert und eine an die geforderte Genauigkeit anpassungsfähige Formel wird für die Integration benützt.