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- W610303706 abstract "Symbolic-numeric algorithms have been studied by several researchers since early 1990s in the world. With recent studies of approximate factorizations, approximate GCDs and others for numerical or empirical polynomials, we can operate such computations with non-exact coefficients. Those algorithms work well on several Computer Algebra Systems (Mathematica and Maple, for example). Especially on Maple, we can use SNAP (Symbolic Numeric Algorithms for Polynomials) package to compute approximate GCDs. In various implementations of such algorithms, bounding errors on coefficients are provided by polynomial norms (1-norm, 2-norm and∞-norm, for example) in general. Because operating with non-exact coefficients involves computing numerical properties (eg. singular values) and polynomial norms have high affinities for matrix norms and operations. In fact, lots of algorithms for empirical polynomials use the singular value decomposition. However, from a practical point of view, tracking or treating errors as polynomial norms is not applicable as much as floating-point numbers, since for floating-point numbers, we have a few kind of significance arithmetics (interval numbers, for example). In this paper, we review known symbolic numeric algorithms in terms of significance arithmetics, and introduce an absolute irreducibility testing for bivariate polynomials and a relatively prime testing for univariate polynomials. Basically, there is no difference from those original algorithms, but they would be more natural for the people who use significance arithmetics and give more efficient error bounds. We note that this paper is involved with the authors’ presentation at the 14th Workshop of JSSAC (Japan Society for Symbolic and Algebraic Computation), 2005." @default.
- W610303706 created "2016-06-24" @default.
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- W610303706 date "2006-03-01" @default.
- W610303706 modified "2023-09-24" @default.
- W610303706 title "新人賞受賞論文 Using Coefficient-wise Tolerance in Symbolic-Numeric Algorithms for Polynomials" @default.
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