Michio Sakakihara

Acceleration of the alternating least squares algorithm for principal components analysis

Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of Young et al. (1978) and PRINCALS of Gifi (1989) in which the alternating least squares algorithm is utilized. These algorithms based on the least squares estimation may require many iterations in their application to very large data sets and variable selection problems and may take a long time to converge. In this paper, we derive a new iterative algorithm for accelerating the convergence of PRINCIPALS and PRINCALS by using the vector @e algorithm of Wynn (1962). The proposed acceleration algorithm speeds up the convergence of the sequence of the parameter estimates obtained from PRINCIPALS or PRINCALS. Numerical experiments illustrate the potential of the proposed acceleration algorithm.

The Gauss-Seidel iterative method with the preconditioning matrix (I +S +Sm)

We propose a preconditioned iterative method with the preconditioning matrixPsm =I +S +Sm forAx =b, whereA is an irreducibly diagonal dominantZ-matrix with unit diagonal. The convergence property and the comparison theorem of the proposed method are discussed. Moreover, some numerical examples are reported to confirm the theoretical analysis.

Boundary elements in steady convective diffusion problems

A boundary element method is developed to solve the steady convective diffusion equation in n dimensions. For the formation a transformation into the selfadjoint or symmetric operator is used under a certain assumption, and a boundary integral equation is derived from the Greens second identity. For the discretization of the boundary integral equation, constant or linear boundary elements are employed. A simple model problem is treated in numerical experiments, and a comparison with the finite element methods is given. It is shown that the present boundary element solution is stable with respect to large Peclet numbers and is with the second-order accuracy.

Preconditioned Iterative Methods for Linear Equations Arising from the Boundary Element Method

A precondition for the Gauss–Seidel iterative method to solve a linear system of equations arising from the boundary element method for the Laplace and convective diffusion with first-order reaction problems is presented in this paper. The present precondition is based on the elementary matrix operation. We discuss the effect of the precondition in comparison with the Gauss elimination (GE) method in some numerical experiments.

BOUNDARY METHOD FOR TRANSIENT DIFFUSION PROBLEM BY USE OF FRACTIONAL STEP PROCEDURES

ABSTRACT The boundary element technique combined the method of fractional steps for the heat equation is presented in this article. An error eatimate of the present method for a two-dimensional problem imposed the Dirichlet boundary condition is proposed. In order to discuss the feature of the method numerical results are shown.

Iterative blow-up time approximation for initial value problems

where u(t) = (ul( t) , . . . ,un(t)) T and f is a nonlinear vector valued function of u and t. We study that a solution of the equation blows up at a finite time. In general: it is difficult to calculate the blow-up time by the use of a numerical method since the solution takes the infinite at the time. The problem to detect the blow-up time of the solution for a given initial problem is appeared in some important engineering problems as the blow-up of a concrete pavement adjoining a rigid structure. Although some a prior estimates of the blow-up time have been presented in the study of the semilinear equation which is a mathematical model of a diffusion-reaction phenomena and a nonlinear boundary value problem of the parabolic partial differential equat ion. If we apply a difference method in space variable to the nonlinear parabolic problems, we obtain an initial value problem as (1). The aim of this presentation is to propose a method to obtain an accurate approximate blow-up time by the use of some methods which generate truncated formal power series for the solution of the given initial value problem. The symbolic Picard , Newton and Gauss-Seidel Newton iterations are applied to generate a truncated formal power series of the solution for the problem (1). In order to detect the blow-up time we propose the method with the Pade approximation theory. The blow-up time depends on some given parameters as the initial value and parameter in the nonlinear right hand side of the equation (1). By the use of the present symbolic approach we can also obtain an approximate function ,which gives us an approximate blow-up time, of those parameters. We show some superior of the present method by illustrating calculating results for some problems. Moreover we will discuses some mathematical background of the present method. MICHIO SAKAKIHARA* AND SHIGEKAZU NAKAGAWA**

Galerkin Boundary Element Method with Single Layer Potential

Galerkin method for an integral equation on a boundary δΩ of a bounded domain in R2, arising from a Dirichlet boundary value problem for an elliptic partial differential equation is considered in this paper. By using a single layer potential corresponding to the problem we obtain an integral equation on the boundary. The main result of the paper is that the integral equation has a unique solution in the Sobolev space H-1/2 (δΩ). We also give its H1 (Ω)-error estimate.