High-performance code generation for polynomials and power series

Newton iteration is a versatile tool. In this thesis, we investigate its applications to the computation of power series solutions of first-order non-linear differential equations. To speed-up such computations, we first focus on improving polynomial multi­ plication and its variants: plain multiplication, transposed multiplication and short multiplication, for Karatsuba’s algorithm and its generalizations. Instead of rewriting code for different multiplication algorithms, a general approach is designed to output computer-generated code based on multiplication graph representations. Next, we investigate the existing Newton iteration algorithms for differential equa­ tion solving problems. To improve their efficiency, we recall how one can reduce the amount of useless computations by using transposed multiplication and short mul­ tiplication. We provide an optimized code generator that applies these techniques automatically to a given differential equation

Implementation techniques for fast polynomial arithmetic in a high-level programming environment

Though there is increased activity in the implementation of asymptotically fast polynomial arithmetic, little is reported on the details of such effort. In this paper, we discuss how we achieve high performance in implementing some well-studied fast algorithms for polynomial arithmetic in two high-level programming environments, AXIOM and Aldor. Two approaches are investigated. With Aldor we rely only on high-level generic code, whereas with AXIOM we endeavor to mix high-level, middle-level and low-level specialized code. We show that our implementations are satisfactory compared with other known computer algebra systems or libraries such as Magma v2.11-2 and NTL v5.4. Categories and Subject Descriptors