Implementation of Matrix Decomposition for Solving Linear Systems Arising in Orthogonal Spline Collocation for Separable Elliptic Boundary Value Problems

本项目涉及开发实现新的矩阵分解算法的软件，这些算法是为求解正交样条配置（即高斯点的样条配置）应用于矩形区域上某些可分离的二阶线性问题时产生的线性代数方程组而制定的。我们将研究两类算法。在第一种方法中，近似解是在均匀网格上定义的分段Hermite双立方，而第二种方法提供了在非均匀分区上任意高阶的近似。该算法是模块化的，具有很强的自然并行性，在解不可分边值问题、一般区域问题和时间相关问题中也有应用。新软件将用Fortran语言编写，并开发用于各种共享内存机器。

This project is concerned with the development of software implementing new matrix decomposition algorithms which have been formulated for solving the systems of linear algebraic equations arising when orthogonal spline collocation (that is, spline collocation at Gauss points) is applied to certain separable, second order, linear problems on rectangular regions. Two classes of algorithms will be examined. In the first, the approximate solution is a piecewise Hermite bicubic defined on a uniform mesh, while the second provides approximations of arbitrarily high order on nonuniform partitions. The new algorithms, which are modular and possess a great deal of natural parallelism, also have application in the solution of nonseparable boundary value problems, problems on general regions, and time dependent problems. The new software will be written in Fortran and developed for use on various shared memory machines.