Alternative algorithms for accelerated symbolic and numeric summation

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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 different kinds 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.