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

A. Melman
Taiwanese Journal of Mathematics
Vol. 12, No. 9 (December 2008), pp. 2315-2325
Stable URL: http://www.jstor.org/stable/43834636
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
SOME PROPERTIES OF NEWTON'S METHOD FOR POLYNOMIALS WITH ALL REAL ZEROS
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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 kth 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.

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