SOME PROPERTIES OF NEWTON’S METHOD FOR POLYNOMIALS WITH ALL REAL ZEROS

Abstract

We prove an overshooting property of a multistep Newton method for polynomials with all real zeros, a special case of which is a classical result for the double-step Newton method. This result states, in essence, that a double Newton step from a point to the left of the smallest zero of a polynomial with all real zeros never overshoots the first critical point of the polynomial. Our result here states, in essence, that a Newton $(k+1)$-step from a point to the left of the smallest zero never overshoots the $k$th critical point of the polynomial, thereby generalizing the double-step result. Analogous results hold when starting from a point to the right of the largest zero. We also derive a version of the aforementioned classical result that, unlike that result, takes into account the multiplicities of the first or last two zeros.