Studies on fast numerical calculation with verification

本文通过对浮点运算舍入方式的两次变换，证明了两个矩阵的加法和乘积可以通过验证计算出来。即他提出了一种向量区间算法。此外，利用摄动理论，它给出了一个后验误差估计，它表明，严格的误差界的近似解的线性方程组可以计算与计算这样的近似解的成本相同。与以前的方法相比，速度提高了1,000到10,000左右。该方法可以推广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。作为例子，从1,000到30,000维的全矩阵系统已经通过PC集群系统解决。

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.