Studies on fast numerical calculation with verification

快速数值计算及验证研究

• 批准号: 12650399
• 负责人: OISHI Shinichi
• 金额: $ 2.05万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12650399/
• 关键词: Verified Numerical Computation; Fast Verification; Rounding Mode Controlled Computation; IEEE754; クラスター上の高速計算; 連立一次方程式の精度保証; 固有値問題の精度保証; 一般化固有値問題の精度保証; 大規模問題; PCクラスター; 丸め誤差; 条件数; 数値計算ツール; LAPACK; 連立一次方程式の数値解; 固有値問題の数値解; インタプリタ; 事後誤差評価; 数

The present researcher has shown that the addition and the product of two matrices can be calculated with verification via tow times changes of rounding mode of floating point arithmetic. Namely he has proposed a vector interval arithmetic. Furthermore, utilizing perturbation theory, which gives a posteriori error estimate, it has shown that a rigorous error bound of an approximate solution of a system of linear equations can be calculated with the same cost as that of calculating such an approximate solution. It is around from 1,000 to 10,000 speed up compared with the previous method. This method can be extended to many problems of numerical linear algebra including matrix eigenvalue problems and singular value problems. As examples, from 1,000 to 30,000 dimensional full matrix systems have been solved via PC cluster system.

本研究人员证明，通过两次浮点运算舍入方式的改变，可以计算出两个矩阵的加法和乘积，并得到验证。即他提出了向量区间算法。此外，利用提供后验误差估计的摄动理论，表明可以用与计算线性方程组近似解相同的成本来计算线性方程组近似解的严格误差界。与之前的方法相比，速度提高了1000到10000左右。该方法可以扩展到数值线性代数的许多问题，包括矩阵特征值问题和奇异值问题。例如，从 1,000 到 30,000 维的全矩阵系统已通过 PC 集群系统解决。

项目成果

T.Ogita, S.Oishi, Y.Ushiro: "Fast Verification of Solutions for Sparse Monotone Matrix Equations"Computing [Supple]. 15. 175-187 (2001)

T.Ogita、S.Oishi、Y.Ushiro：“稀疏单调矩阵方程解的快速验证”计算 [补充]。

Shin'ichi Oishi, Siegfried M.: "Rump, Fast verification of solutions of matrix equations"Numerische Mathematik. 90. 755-773 (2002)

Shinichi Oishi，Siegfried M.：“Rump，矩阵方程解的快速验证”数值数学。

T.Ogita, S.Oishi: "Fast Inclusion of Interval Matrix Multiplication"submitted to Reliable Computing. Apr. (2004)

T.Ogita、S.Oishi：“区间矩阵乘法的快速包含”提交给可靠计算。

R.B.Kearfott, M.Neher, S.Oishi, F.Rico: "Libraries, Tools, and Interactive Systems for Verified Computations : Four Case Studies, in Numerical Software with Result Verification"Lecture Notes in Computer Science (ed. R.Alt, A.Frommer, R.B.Kearfott, W.Luthe

R.B.Kearfott、M.Neher、S.Oishi、F.Rico：“用于验证计算的库、工具和交互式系统：具有结果验证的数值软件中的四个案例研究”计算机科学讲义（R.Alt 编，

T.Ogita, S.Oishi, Y.Ushiro: "Computation of Sharp Rigorous Componentwise Error Bounds for the Approximate Solutions of Systems of Linear Equations"Reliable Computing. vol.9, no.3. 229-239 (2003)

T.Ogita、S.Oishi、Y.Ushiro：“线性方程组近似解的尖锐严格分量误差界的计算”可靠计算。