Kozo Ichida

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.

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.

Nonlinear Interpolation of Multivariable Functions by the Monte Carlo Method

The nonlinear interpolation of functions of very many variables is discussed. Deterministic termwise assessment of a prohibitively large number of terms naturally leads to a choice of random sampling from these numerous terms. After introduction of an appropriate higher order interpolation formula, a working algorithm is established by the Monte Carlo method. Numerical examples are also given.

Solution of partial differential equations by a modified random walk

A new Monte Carlo technique is applied to solve difference equations of elliptic and parabolic partial differential equations with given boundary values. Fixed random walk is extended to modified random walk, whereby a random walk is made on a maximum square. The average number of steps and the computational time in a modified random walk is much less than in a fixed random walk. Numerical examples support the utility of this method.

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.

Estimation of a Probability Density Function of Very Many Variables

The problem of estimating an unknown probability density function from a sequence of samples is well known in pattern classification and many other problems. We approximate the unknown density function by a multivariable spline that is constructed from the histogram of samples. This spline function is expressed as a sum of combinatorially many terms. To assess these numerous terms, the technique of Monte Carlo sampling is exploited and a combined sampling is devised to reduce the standard error.

Numerical interpolation and differentiation of multivariable functions

The problem of numerical interpolation of multivariable functions when their values are assumed to be given on discrete lattice points has been solved by the Monte Carlo method [5, 6,7]. This technique is used because the number of lattice points increases exponentially with the number of dimensions. This paper deals with numerical interpolation and differentiation of multivariable functions by a piecewise cubic polynomial in each variable. A sampling method which combines positive and negative coefficients is considered to prevent the degeneration of accuracy.