Research on High Dimension Problems in Numerical Analysis.

研究了求解高维问题的快速数值算法及其在并行或向量结构的快速计算机上的高效实现。在代数计算领域，我们发现了多项式的数值分解的新算法和多项式余项序列的算法。在常微分方程组领域，我们提出了适合于并行计算的两步Runge-Kutta方法的新格式，并研究了特征曲线法作为并行常微分方程组的应用来求解对流和扩散问题；在随机微分方程上，我们通过根树分析得到了行型格式的阶条件。对于延迟-微分方程组，我们研究了线性系统的稳定性，提出了几种判定线性系统稳定性和数值解的方法；对于孤子方程，我们找到了求解二维孤子的KP方程的新算法；对于高维积分，我们提出了高精度的好格点方法。在高维线性方程组上，我们提出了GPBi-CG算法，这是一种基于BiCG算法的广义乘积型算法，用于求解非对称线性方程组。对于高维数学问题的可视化，我们研究了具有正性、单调性和凸性等约束条件的Sprint曲线的生成问题。

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.