High precision numerical algorithms by finite fields

• Project/Area Number: 26610039
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kobe University
• Principal Investigator: Takayama Nobuki 神戸大学, 理学研究科, 教授 (30188099)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 数値解析 / 有限体 / 高精度計算 / 常微分方程式 / modular method / 超幾何多項式 / Runge-Kutta 法 / 超幾何関数

Outline of Final Research Achievements

We give numerical analysis algorithms and implementations for the following problems over the rational number fields, which give high precision numerical outputs. (1) fast evaluation method of iterations of linear transformations by a modular arithmetic and a distributed computation. (2) an efficient variation of the Bulirsch-Stoer method to solve linear ordinary differential equations numerically over rational numbers. An application of (1) is an exact evaluation of A-hypergeometric polynomial. An application of (2) is a numerical analysis near singular points of linear ordinary differential equations.

Research Products

• [Journal Article] 2元分割表に対する差分ホロノミック勾配法の実装 (2017)
  • Author(s): 橘義仁, 後藤良彰, 高山信毅
  • Journal Title: 数理研考究録 Volume: 印刷中
• [Journal Article] 2元分割表に対する差分ホロノミック勾配法の実装 (2016)
  • Author(s): 後藤良彰, 橘義仁, 高山信毅
  • Journal Title: 数理研講究録 Volume: 未定
• [Presentation] 常微分(差分)方程式用有理数対応数値解析パッケージ (2017)
  • Author(s): 高山信毅
  • Organizer: Risa/Asir Conference 2017
  • Place of Presentation: 金沢大学
  • Year and Date: 2017-03-28
• [Presentation] 2元分割表の条件付き確率の差分HGMによる計算 (2016)
  • Author(s): 高山信毅
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学(茨城県、つくば市)
  • Year and Date: 2016-03-17
• [Presentation] Sylvester 型行列による超幾何関数の数値評価 --- グレブナー基底を使わない方法 (2015)
  • Author(s): 高山信毅
  • Organizer: Risa/Asir Conference 2015
  • Place of Presentation: 金沢大学
  • Year and Date: 2015-03-18
• [Presentation] 多変数超幾何関数の数値計算とその応用 (2014)
  • Author(s): 高山信毅
  • Organizer: 第八回玉原特殊多様体研究集会
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2014-09-17
• [Remarks] 常微分（差分）方程式用有理数対応数値解析パッケージ
  • URL: http://www.math.kobe-u.ac.jp/HOME/taka/2016/20170328-ohp-takayama.pdf
• [Remarks] Risa/Asir
  • URL: http://www.math.kobe-u.ac.jp/Asir