부산대학교 도서관

Consider the problem of finding a zero of a nonlinear function $f(x)$. Let $u(x)=f(x)/f\sp \prime(x)$. It is well known that the Newton algorithm $x\sb {n+1}=x\sb n-u(x\sb n)$ $(n=0, 1,\bf\cdots)$ converges only linearly to a root of multiplicity $m$ while the quasi-Newton algorithm $x\sb {n+1}=x\sb n-mu(x\sb n)$ $(n=0,1,\bf\cdots)$ converges quadratically. \par Various authors have discussed methods for estimating the multiplicity of a root so that the quasi-Newton method can be used. For example, see the reviewer and M. L. Patrick [Numer. Math. 27 (1976/1977), no. 1, 121--131; MR0433857 (55 \#6828)] and the references therein. In this paper, the author compares three algorithms of this sort. In the first algorithm, $m$ is approximated using a $\delta\sp 2$-process. The second algorithm is one due to H. Van de Vel [Computing 14 (1975), no. 1-2, 167--171; MR0403205 (53 \#7017)]. The third algorithm is a modification of Van de Vel's derived by the author. It can be written as $m\sb {n+1}= m\sb nu\sb n/(u\sb n-u\sb {n+1})$, $x\sb {n+2}=x\sb {n+1}-m\sb {n+1}u\sb {n+1}$ for $n=0,1,2,\bf\cdots $, where $u\sb n$ denotes $u(x\sb n)$. The initial approximations $x\sb 0$ and $m\sb 0$ can be chosen arbitrarily. \par The efficiencies of the three methods are discussed and a numerical example is given which indicates that the author's method is superior.