A Basic Family of Iteration Functions for Polynomial Root Finding and Its Characterizations

Let p(x) be a polynomial of degree n 2 with coe cients in a sub eld K of the complex numbers. For each natural number m 2, let L m (x) be the m m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 1, its j-th subdiagonal entries are p (j) (x)=j!. For i = 1; 2, let L (i) m (x) be the matrix obtained from L m (x) by deleting its rst i rows and its last i columns. L (1) 1 (x) 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 m, S(m;M) is the set of all rational iteration functions such that for all roots of p(x) , g(x) = + P M i=m i (x)( x) i , with i (x)'s also rational and well-de ned at . Given g 2 S(m;M), and a simple root of p(x), g (i) ( ) = 0, i = 1; : : : ; m 1, and m ( ) = ( 1) m g (m) ( )=m!. For B m (x) we obtain m ( ) = ( 1) m det(L (2) m+1 ( ))=det(L (1) m ( )). 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 = P n i=0 [p (i) (x)=i!]( 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 m ( ).