Development of practical methods for rigorous calculation with guaranteed accuracy

这项研究由科学研究资助基金资助，我们的目标是开发实用的严格计算方法，保证准确性。通过4年多的研究，我们在以下方面取得了一些成果：1。无限维空间中牛顿算子最大特征值的验证计算2。验证了对称带矩阵特征值及其指标的计算方法。上述方法在一般特征值问题上的推广。不动点方程解唯一性的验证方法[j]。保证精度的摄动Gelfand方程的分岔图研究[j]。对误差估计中出现的常数进行了严格的计算。用Fortran 90和四精度浮点数处理舍入误差的方法研究[j]。用谱法对Navier-Stockes方程解的数值验证[j]。基于区间算术的舍入误差影响估计方法[j]。有限元逼近能力的估计。由此得出了特征值问题验证计算的实用方法。开发。关于偏微分方程的方法，也有发展，但在非专业用户实际使用的数学问题上有一些困难。

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.