可交换Freché导数及其应用

Exchangeable Frechét Derivatives and Its Applications

在矩阵和矩阵增量的乘法满足交换律的条件下,讨论了Frechét导数及其应用。所得结果不但可以较为精确地估计因对矩阵进行某些处理而形成的误差,而且可以对上述误差进行校正。对著名的Hilbert病态线性方程组的数值计算表明,当阶数n=500～2 000,解的各元素为1时约可以有4～9位有效数字。

An improved Frechét calculus for matrix functions was proposed based on the assumption that exchange law holds true for the multiplication of matrix and its increment. The results can not only be used to estimate the error produced in some treatment on matrix, but also to correct the error. The calculation for ill-conditioned linear systems such as Hilbert linear equations shows that satisfied result can be obtained at least up to n = 2 000.