Sometimes Newton's Method Always Cycles

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.

Additional information

Notes on contributors

Joe Latulippe

Joe Latulippe ([email protected]) received his B.S. from Sonoma State University and an M.S. and Ph.D. from Montana State University. He is an Assistant Professor at Norwich University in Vermont. His scholarly interests are in mathematical biology, modeling, and perturbation methods. Outside of academia he practices Aikido, is an assistant coach for the Norwich Men's Lacrosse team, and enjoys painting landscapes.

Jennifer Switkes

Jennifer Switkes ([email protected]) received a B.S. from Harvey Mudd College and a M.S. and Ph.D. from Claremont Graduate University. She is a Professor of Mathematics at California State Polytechnic University, Pomona, where she has taught many numerical methods courses. Her non-mathematical interests include spending time in Uganda.