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Sometimes Newton's Method Always Cycles

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Abstract:

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.

Document Type: Research Article

DOI: https://doi.org/10.4169/college.math.j.43.5.365

Publication date: 2012-11-01

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