Alternative algorithms for accelerated symbolic and numeric summation

尽管CPU设计取得了无可争议的进步，但不同应用领域的一些计算任务仍然可以被归类为重量级或耗时的任务。对于此类问题，任何形式的加速都受到最终用户的欢迎。本文提出的研究是针对符号、符号-数字和数字求和中长期存在的与性能相关的问题。符号求和的目标是开发一套新的算法，以克服长期以来被计算机代数系统广泛采用的标准算法的缺点。这种方法的新颖之处在于（直接或间接地）使用积分表示和不同类型的积分变换的思想，以算法的方式解决求和问题。所得结果也将推广到求解多项式系数线性差分方程的算法中。对于符号-数值求和，目的是研究一种基于特殊选择模（坎宁安数）的模计算的快速收敛超几何级数高精度求值的替代方法。该方法的新颖之处在于在实际应用中对特殊模集的模映像进行有理数的快速重构。对于数值求和，目标是以适当的方式利用递归链技术的力量，并将其应用于优化代码生成和优化电路（VHDL代码）生成。这种方法的显著特征将是在电路生成器的实现中，它将生成专门的浮点单元，以每个时钟周期的速率对任意复杂性的封闭形式表达式进行数值求和。

In spite of undisputable progress in CPU design some computational tasks in different application domains can still be categorized as heavy-weighted or time-consuming. Any kind of acceleration achieved for such problems is welcomed by the end users. Research proposed here targets long-standing performance related issues in symbolic, symbolic-numeric and numeric summation. For symbolic summation the objective is to to develop a set of new algorithms that will overcome long-standing shortcomings of standard and widely adopted by computer algebra systems algorithms. The novelty of this approach is in the use (direct or indirect) of ideas of integral representation and differentkinds of integral transforms for solving summation problems in an algorithmic fashion. The results will be also extended to the algorithms of solving linear difference equations with polynomial coefficients.For symbolic-numeric summation the objective is to investigate an alternative approaches to the high-precision evaluation of the rapidly convergent hypergeometric series based on modular computations with special choice of moduli (Cunningham numbers). The novelty of this approach is in practical use of fast reconstruction of rational numbers from modular images for special sets of moduli. For numeric summation the objective is to exploit in proper way the power of Chains of Recurrences technique with applications to both optimized code generation and optimized circuit (VHDL code) generation. The distinguishing feature of this approach will be in an implementation of circuit generator, which will generate specialized floating point units performing numeric summation of closed form expressions of arbitrary complexity at the rate of term per clock cycle.