| Co-Investigator(Kenkyū-buntansha) |
MOTOYOSHI Fumio Electro-Technical Laboratory, Chief, 知能情報部, 室長
SUZUKI Masayuki Iwate Univ., Dept.Inf.Eng., Assoc.Prof., 工学部, 助教授 (20143365)
KAKO Fujio Nara Wo.Univ., Dept.Inf.Sci., Professor, 理学部, 教授 (90152610)
NODA Matu-tarou Ehime Univ., Dept.Comp.Sci., Professor, 工学部, 教授 (10036402)
KITAMOTO Takuya Tsukuba Univ., Inst.Math., Assistant, 数学系, 助手 (30241780)
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| Research Abstract |
The purposes of this research were, 1) to develop an approximate algebra system which allows us to monitor cancellation errors of the floating-point numbers, 2) to study algorithms of various algebraic operations, and 3) to seek for applications of approximate algebra.
As for approximate algebra system, effective floating-point number which we have proposed has been implemented in NSL (Nara Standard Lisp) (by Kako), and proved to be quite useful for approximate algebraic computation (by Sasaki & Kako). Furthermore, effective complex floating-point number and interval have been implemented in NSL (by Kako). On the basis of these facilities, numeric types of effective-float and interval have been equipped in formula manipulation system GAL,and arithmetic of approximate polynomials with coefficients of these numbers have been implemented in GAL (by Sasaki & Kako).
Algorithm studies done are, improvement of approximate GCD algorithm (by Sasaki et al., Noda & Kai), factor separation of univariate polynomial and its application to multiple/close root problem (Sasaki et al.), relationship between approximate GCD and Pade approximation & continued fraction expansion (by Noda & Kai), Puiseux series expansion method for eigenvalues and eigenvectors of matrices of univariate polynomial entries (by Kitamoto), primary decomposition method using approximate power series expansion of polynomial roots (by Kitamoto), fast algorithm for polynomial GCD (by Motoyoshi), and so on.
As for application, approximate GCD was utilized for numerical integration of rational functions (by Noda & Kai), multivariate power series expansions of polynomial roots was utilized for algebraically solving control system with parameters (by Kitamoto), and so on.
We think that effectiveness of approximate algebra has been proved almost sufficiently by these studies.
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