多项式稳定性判据对代数方程求根的应用及其数值试验

一、引言 已有许多方法可以求多项式的根,其中多数属于迭代法.这类方法的缺点是收敛性依赖于初始近似的选择(求复根还没有大范围收敛的迭代法).另一类方法是先求根的模或实部,然后再设法求同模或等实部的根,如根平方——结式法,按分布理论求根等就属于这一类.这类方法不存在收敛性问题,但也有其不足之处.如根平方——结点法的缺点除了有可能将良态多项式变为病态多项式之外,求结式也增加了计算的复杂性.采用单精度运算的数值试验结果表明,这个方法求解的精度比某些迭代法低.

APPLICATION OF THE CRITERION OF POLYNOMIAL STABILITY TO FINDING THE ROOTS OF AN ALGEBRAIC EQUATION AND ITS NUMERICAL TEST

In the paper the criterion of polynomial stability presented in 5] is applied to findingall roots of a real coefficient polynomial. Theoretical analysis and numerical trial results showthat the roots may be rapidly obtained by this method and there is no problem for conver-gence. For well-conditioned or not very ill-conditioned polynomials the results obtained insingle precision operation can be satisfied, and for very ill-conditioned polynomials, the re-sults obtained in single precision operation are still comparable with those obtained by theroot-squaring and resultant method. For some particular problem the highest precision maybe achieved automatically and correct evaluation for absolute precision of the results can bemade in finding the roots by this method.