Newton’s Method: Complex Numerics and Complex Dynamics
Professor Paul Blanchard
Boston University
Newton’s method is an iterative root-finding algorithm that is both simple and surprisingly efficient. We start with an initial guess for the root and apply the algorithm repeatedly until we obtain the desired approximation. Unfortunately, a random initial guess does not always lead to a root. In this talk, we use the theory of complex dynamics along with some computer graphics to explain the difficulties that might arise, and we suggest ways to avoid these pitfalls. As the story unfolds, we encounter both chaos and fractals.
Professor Paul Blanchard grew up in Sutton, Massachusetts, spent his undergraduate years at Brown University, and received his Ph.D. from Yale University. He has taught college mathematics for more than thirty years, mostly at Boston University. In 2001, he won the Northeastern Section of the Mathematical Association of America’s Award for Distinguished Teaching of Mathematics, and in 2011, the conference “Differential Equations Across the Collegiate Curriculum” was held to celebrate his 60th birthday. He is a Fellow of the American Mathematical Society.
Professor Blanchard’s main area of mathematical research is complex analytic dynamical systems and the related point sets—Julia sets and the Mandelbrot set. For many of the last twenty years, his efforts have focused on modernizing the traditional sophomore-level differential equations course. When he becomes exhausted fixing the errors made by his two coauthors, he heads for the golf course to enjoy a different type of frustration.
