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Sometimes Newton’s Method Always Cycles

Joe Latulippe and Jennifer Switkes
The College Mathematics Journal
Vol. 43, No. 5 (November 2012), pp. 365-370
DOI: 10.4169/college.math.j.43.5.365
Stable URL: http://www.jstor.org/stable/10.4169/college.math.j.43.5.365
Page Count: 6
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Sometimes Newton’s Method Always Cycles
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Abstract

Summary Are there functions for which Newton’s method cycles for all non-trivial initial guesses? We construct and solve a differential equation whose solution is a real-valued function that two-cycles under Newton iteration. Higher-order cycles of Newton’s method iterates are explored in the complex plane using complex powers of x. We find a class of complex powers that cycle for all non-trivial initial guesses and present the results analytically and graphically.

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