Alternative algorithms for accelerated symbolic and numeric summation

尽管CPU设计取得了无可争议的进步，但不同应用领域中的一些计算任务仍然可以归类为繁重或耗时的。对于此类问题实现的任何类型的加速都受到最终用户的欢迎。这里提出的研究目标长期存在的性能相关的问题，符号，符号数字和数字求和。对于符号求和，目标是开发一套新的算法，将克服长期存在的标准和广泛采用的计算机代数系统算法的缺点。这种方法的新奇在于使用（直接或间接）积分表示和不同类型的积分变换的思想来以算法的方式解决求和问题。本文的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，本文的目标是研究一种基于模计算的快速收敛超几何级数的高精度计算方法，该方法具有特殊的模（Cunningham数）选择。这种方法的新奇是在实际使用中的快速重建有理数从特殊的模集的模图像。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，应用于优化代码生成和优化电路（VHDL代码）生成。这种方法的区别特征将是在电路发生器的实现中，其将生成以每个时钟周期的项的速率对任意复杂度的闭合形式表达式执行数字求和的专用浮点单元。

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.