Abstract
The core of most algebraic manipulation systems is a polynomial arithmetic package. There are a number of good reasons for this. First, a large number of problems in pure and applied mathematics can be expressed as problems solely involving polynomials. Second, polynomials provide a natural foundation on which to build more complex structures like rational functions, algebraic functions, power series and rings of transcendental functions. And third, the algorithms for polynomial arithmetic are well understood, efficient and relatively easy to implement.
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Notes
A survey of asymptotically fast algorithms for polynomials is contained in [191]. Some experiences with parallel implementations of polynomial arithmetic algorithms are given in [196].
§7.1 The “vectorized subscript” notation is fairly commonly used now, although some authors use capital letters in the subscript to indicate the vector, i.e., instead of they write XÎII believe this notation was first introduced in Laurent Schwartz’s work on distributions.
§7.4 ALTRAN was the first computer algebra system to use heap structures to represent polynomials [32]. Despite the performance improvement demonstrated in ALTRAN [216], most other systems have not chosen to follow ALTRAN’S lead.
§7.5 When raising a polynomial to a large power modulo another polynomial a variant of the “repeated squaring technique” can be used effectively. This technique is described in Section 18.2 on page 296.
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© 1993 Springer Science+Business Media New York
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Zippel, R. (1993). Polynomial Arithmetic. In: Effective Polynomial Computation. The Springer International Series in Engineering and Computer Science, vol 241. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3188-3_7
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DOI: https://doi.org/10.1007/978-1-4615-3188-3_7
Publisher Name: Springer, Boston, MA
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