Development of practical methods for rigorous calculation with guaranteed accuracy

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。

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.