| Project/Area Number |
12480065
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
計算機科学
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| Research Institution | University of Tsukuba |
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Principal Investigator |
SASAKI Tateaki Tsukuba Univ., Inst Math., Professor, 数学系, 教授 (80087436)
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| Co-Investigator(Kenkyū-buntansha) |
KAI Hiroshi Ehime Univ., Dept. Comp. Sci., Lecturer, 工学部, 講師 (10274341)
NODA Matu-tarou Ehime Univ., Dept. Comp. Sci., Professor, 工学部, 教授 (10036402)
TERUI Akira Tsukuba Univ., Inst. Math., Assistant, 数学系, 助手 (80323260)
FUKUI Tetsuo Mukogawa Women's College, Assoc. Prof., 生活環境学部, 助教授 (70218890)
KAKO Fujio Nara Wo. Univ., Dept. Inf. Sci., Professor, 理学部, 教授 (90152610)
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| Project Period (FY) |
2000 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥9,100,000 (Direct Cost: ¥9,100,000)
Fiscal Year 2002: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2001: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2000: ¥3,200,000 (Direct Cost: ¥3,200,000)
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| Keywords | approximate algebra / algebraic computation / algebraic-numeric computation / computer algebra / computer algebra system / formula manipulation / 近似的代数計算法 / 誤差解析と安定化 / 代数的算法 / 代数的算法の誤差解析 / 誤差解析 |
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| Research Abstract |
The purposes of this research are, A) to develop approximate algebraic algorithms for many algebraic operations, B) error analysis and stabilization of approximate algebraic algorithms, C) further improvement of NSL-GAL system, and D) to seek for applications of approximate algebra. We performed the following researches for each purpose.
Algorithm study : drastic improvement of approximate factorization algorithm (Sasaki), development of multivariate approximate GCD algorithm using Hensel construction (Zhi, Kai & Noda), method of rational function approximation of bivariate polynomials (Kai, Noda & Kihara), development of certified method of analytic continuation of algebraic functions (Sasaki & Inaba), and so on. Error analysis and Stabilization : theory of subresultant of polynomials having mutually close roots and analysis of cancellation errors in the Euclidean algorithm (Sasaki), clarification of ill-conditionedness in rational function approximation and stabilization of the approximation (Kai, Noda & Murakami), derivation of a formula for separating small roots from others of univariate polynomial (Terui & Sasaki), stabilization of Wu's method for coupled algebraic equations (Kai, Noda, Zhi & Notake), and so on. System improvement : Usefulness and notice point of the efloat (effective floating-point number) are clarified : the efloat is very useful for estimating cancellation errors in many approximate algebraic computations, but it over-estimates the errors in iteratively approximating algorithms such as Newton's method (Kako, Fukui, Sasaki & Oyoshi). Application : As for image reconstruction from the data given as 2-dimensional matrix, it was found that the image can be reconstructed pretty well if the generalized inverse matrix method is stabilized by the technique of Shirayanagi-Sweedler (Kai, Noda & Mizukuchi).
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