Parallel Algorithms for Polynomials

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This research proposal is about designing and implementing algorithms for computing with mathematical formulas.It focusses on computing with large polynomials in more than one variable and with large formulas involving algebraicnumbers like sqrt(2) and algebraic functions and sqrt(1+y^2).Such formulas arise in many applications in Science, Engineering and Mathematics.This research is part of a field known as Computer Algebra. It is also known as Symbolic Computation.Researchers in this field design algorithms and software systems, called Computer Algebra Systems,for doing algebra and calculus on the computer. Computer Algebra Systems like Maple and Mathematica are used byScientists, Engineers and Mathematicians in both industry and academia for their work. The researcher on thisproposal has been and continues to be involved with the design and development of Maple which is a Canadian product.A focus of the proposal is the development and implementation of parallel algorithms.This is because today's laptops, desktops and servers are all multi-core computers andthe only way now to improve performance of software is to exploit their multi-coreprocessing capability. We propose to use Cilk for implementing parallel algorithms.Cilk was developed at MIT and has now been adopted by Intel. We also propose to beginexperimenting with using Graphics Processing Units (GPUs) for implementing algorithms.A second focus of the research is how to interpolate, that is, how to reconstructpolynomials and rational functions in more than one variable from their values.Often formulas involving many variables are sparse and have structure.We propose to try to automate the detection of this structure so we can interpolate them rapidly.Some of the software we develop in this research will be written in Maple and some will be written in C and/or Cilk.We do plan to make it available to the Maple user community so that Scientists, Engineers and Mathematicians can use it.For the students working on these projects, it is very encouraging for them to see other people using their software.Students who work on the research projects in this proposal will learn how to design and implement algorithmsfor solving various problems in computational mathematics like factoring polynomials and fast algorithms for solvingproblems involving matrices. They will learn how to design and implement parallel algorithms using Cilk.They will be learning mathematics (e.g. about algebraic numbers) and computer science(e.g. how to analyze the efficiency of algorithms.)