Research on High Dimension Problems in Numerical Analysis.

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We investigated fast numerical algorithms for solving high dimensional problems and efficient implementation for fast computers with parallel or vector architecture.On the field of algebraic calculation, we found new algorithms for numerical factorization of polynomials and an algorithm for sequence of polynomial remainders. And we also investigate stable numerical algorithms for Grobner basis and basis of quotient rings of polynomials.On the field of ordinary differential equations, we proposed new scheme of two-step Runge-Kutta methods suitable for parallel computing and investigated characteristic curve methods solving convection and diffusion problems as an application of parallel ODE solvers.On stochastic differential equations, we found the order conditions of ROW-type scheme by rooted tree analysis. We also proposed new stability criteria for numerical scheme.On delay-differential equations, we investigate the stability of linear systems and proposed several methods for determine stability of linear systems and numerical solver.On soliton equations, we found new algorithm for KP equations which represents a two dimensional soliton.On high dimensional integration, we proposed high precision good lattice points methods. We also developed a new method named sparse grid method.On high dimensional linear equations, we proposed GPBi-CG,which is a generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems. We also proposed rotated alternative LU decomposition which is suitable for vector architecture.For visualization of high dimensional mathematical problems, we investigate the generation of sprine curves with several constraints, i.e., positivity, monotonisity and convexity.