TY - JOUR
TI - A basic family of iteration functions for polynomial root finding and its characterizations
DO - https://doi.org/doi:10.7282/t3-hkp5-pp95
AU - Kalantari, Bahman
AU - Kalantari, Iraj
AU - Zaare-Nahandi, Rahim
PY - 1996-06
AB - Let $p(x)$ be a polynomial of degree $n geq 2$ with coefficients in a subfield $K$ of the complex numbers. For each natural number $m geq 2$, let $L_m(x)$ be the $m imes m$ lower triangular matrix whose diagonal entries are $p(x)$ and for each $j=1, dots, m-1$, its $j$-th subdiagonal entries are $p^{(j)}(x)/ j!$. For $i=1,2$, let $L_m^{(i)}(x)$ be the matrix obtained from $L_m(x)$ by deleting its first $i$ rows and its last $i$ columns. $L^{(1)}_1(x) equiv 1$. Then, the function $B_m(x)=x-p(x)~{{det(L^{(1)}_{m-1}(x))}/ {det(L^{(1)}_{m}(x))}}$ is a member of $S(m,m+n-2)$, where for any $M geq m$, $S(m,M)$ is the set of all rational iteration functions such that for all roots $heta$ of $p(x)$ , $g(x)=heta+ sum_{i=m}^{M} gamma_i(x)(heta-x)^i$, with $gamma_i(x)$'s also rational and well-defined at $heta$. Given $g in S(m,M)$, and a simple root $heta$ of $p(x)$, $g^{(i)}(heta)=0$, $i=1, dots, m-1$, and $gamma_m(heta)=(-1)^mg^{(m)}(heta)/m!$. For $B_m(x)$ we obtain $gamma_m(heta)=(-1)^{m} det (L^{(2)}_{m+1}(heta)) /det (L^{(1)}_m(heta))$. For $m=2$ and $3$, $B_m(x)$ coincides with Newton's and Halley's, respectively. If all roots of $p(x)$ are simple, $B_m(x)$ is the unique member of $S(m,m+n-2)$. By making use of the identity $0 = sum_{i=0}^n {{p^{(i)}(x)} / {i!}} (heta- x)^i$, we arrive at two recursive formulas for constructing iteration functions within the $S(m,M)$ family. In particular the $B_m$'s can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schr"oder, whose $m$-th order member belong to $S(m,mn)$, $m >2$. The iteration functions within $S(m,M)$ can be extended to arbitrary smooth functions $f$, with the automatic replacement of $p^{(j)}$ with $f^{(j)}$ in $g$ as well as $gamma_m(heta)$.
LA - English
ER -