Development of practical methods for rigorous calculation with guaranteed accuracy

开发可保证精度的严格计算的实用方法

• 批准号: 09640278
• 负责人: YAMAMOTO Nobito
• 金额: $ 2.05万
• 依托单位: The University of Electro-Communications (1999-2001)Kyushu University (1997-1998)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640278/
• 关键词: computation with guaranteed accuracy; numerical verification; numerical analysis; Newton method; eigenvalue problem; 精度保証付き計算法; 有限要素法; 誤差評価; 丸め誤差; Navier-Stokes方程式; 誤差解析

Our objective in this study which is fonded by Grant-in-Aid for Scientific Research is development of practical methods for rigorous calculation with, guaranteed accuracy. Through the period of this study over 4 years, we have obtained some results on the following.1. Verified computation of the maximum eigenvalue of Newton operators in infinite dimensional spaces2. Verified computation methods for eigenvalues of symmetric band matrices together with their indices3. Extension of the above methods to general eigenvalue problems4. Methods for verification of uniqueness of solutions to fixed point equations5. Research on a bifurcation diagram of Perturbed Gelfand Equation with guaranteed accuracy6. Rigorous calculation of constants appearing in error estimations of FEM7. Research on methods for transaction of rounding errors using Fortran 90 and quadruple-precision floating point numbers8. Numerical verification of solutions to the Navier-Stockes equation using spectral methods9. Estimation methods for influence of rounding error by interval arithmetic10. Estimation of ability of approximation of FEM.Consequently we can conclude that practical methods for verified computation of eigenvalue problems. are developed. On the methods for PDEs, they are also developed but there are some difficulties concerning mathematical matters in practical use for non-professional users.

这项研究是由科学研究助学金资助的，我们的目标是开发实用的方法，以保证精确的严格计算。通过4年多的研究，我们在以下几个方面取得了一些成果。验证了无限维空间中牛顿算子最大本征值的计算。验证了对称带状矩阵的特征值及其指标的计算方法。将上述方法推广到一般特征值问题。不动点方程解的唯一性的验证方法5。保证精度的摄动Gelfand方程分岔图的研究6。FEM7误差估计中出现的常数的严格计算。用Fortran 90和四精度浮点数处理舍入误差的方法研究8.用谱方法数值验证Navier-Stockes方程的解。区间算法对舍入误差影响的估计方法10。有限元逼近能力的估计。因此，我们可以得出实用的方法来验证特征值问题的计算。都是开发出来的。关于偏微分方程组的方法也得到了发展，但非专业用户在实际应用中涉及到的数学问题存在一些困难。

项目成果

Yamamoto,N.: "A simple method for error bounds of eigenrilues of symmetric matrices"Linear Algebra and its Applications.

Yamamoto,N.：“对称矩阵特征值误差界的一种简单方法”线性代数及其应用。

Nakao, M.T.: "On the best constant in the error bound for the H^1_0-projection into piecewise polynomial spaces" Journal of Approximation Theory. 93. 491-500 (1998)

Nakao, M.T.：“关于 H^1_0 投影到分段多项式空间的误差范围中的最佳常数”《近似理论杂志》。

Kimura, S., Yamamoto, N.: "On explicit bounds in the error for H^1_0-projection into piecewise polynomial spaces"Bulletin of Informatics and Cybernetics. Vol.31,No.2. 109-115 (1999)

Kimura, S., Yamamoto, N.：“关于 H^1_0 投影到分段多项式空间的误差的显式界限”信息学和控制论通报。

Nakao, M.T., Yamamoto, N., Nagatou, K.: "Numerical verifications of eigenvalues of second-order elliptic operators"Japan Journal of Industrial and Applied Mathematics. Vol.16. 307-320 (1999)

Nakao, M.T.、Yamamoto, N.、Nagatou, K.：“二阶椭圆算子特征值的数值验证”日本工业与应用数学杂志。

Nakao,M.T.,Yamamoto,N.: "A guaranteed bound of the optimal constant in the error estimations for linear triangular element"Computing Supplementurn.

Nakao，M.T.，Yamamoto，N.：“线性三角形单元误差估计中最佳常数的保证界限”计算补充。