Accelerated computational schemes for symbolic and numeric algorithms

本研究计划的目标是开发一套算法，软件和硬件工具，加速一些标准的符号和数值算法。我们的目标符号算法是在该地区的符号求和。目标数值算法集中在初等函数和特殊函数的高精度计算上。我们的加速方法是基于计算问题的符号预处理，包括选择合适的表示形式，如封闭形式公式、零化线性微分或差分算子、递归链、组合和的积分表示以及基于所选表示的代码生成;--使用替代算法进行高精度计算，例如具有特殊模数选择的任意精度有理数的模数表示，自适应精度浮点运算;--通过工作站网络或网络分发计算困难的问题; --用于加速特定计算任务的专用硬件的VHDL原型，包括基于递归和自适应精度浮点运算的多维链技术的专用浮点单元的原型。

The objective of this research program is to develop a set of algorithms, software and hardware tools for acceleration of some standard symbolic and numeric algorithms. Our target symbolic algorithms are in the area of symbolic summation. The target numeric algorithms focus on high precision evaluation of elementary and special functions. Our approach to the acceleration is based on -- symbolic preprocessing of computational problems, including the choice of an appropriate representation, for example, closed form formula, annihilating linear differential or difference operator, chains of recurrences, integral representation of combinatorial sums and code generation based on chosen representation; -- use of alternative arithmetics for high precision computations such as modular representation of arbitrary precision rational numbers with special choice of moduli, adaptive precision floating-point arithmetic; -- distributing computationally hard problems over a network of workstations or through the web; -- VHDL prototyping of specialized hardware for acceleration of a particular computational task, including prototyping of specialized floating point units based on the technique of multidimensional chains of recurrences and adaptive precision floating-point arithmetic.