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.
Citation
A. Melman. "SOME PROPERTIES OF NEWTON’S METHOD FOR POLYNOMIALS WITH ALL REAL ZEROS." Taiwanese J. Math. 12 (9) 2315 - 2325, 2008. https://doi.org/10.11650/twjm/1500405181
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