Studies on fast numerical calculation with verification

通过两次改变浮点运算的舍入方式，证明了两个矩阵的加法和乘积都可以通过验证计算出来。也就是说，他提出了一种向量区间算法。此外，利用摄动理论给出了后验误差估计，证明了可以用与计算近似解相同的代价来计算线性方程组的近似解的严格误差界。与以前的方法相比，速度提高了1000到10000左右。该方法可以推广到许多数值线性代数问题，包括矩阵特征值问题和奇异值问题。作为例子，用PC集群系统求解了从1,000到30,000维的全矩阵系统。

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.