1. Liz Balsam Advisor: Bahman Kalantari

2.  Term coined by Dr. Kalantari  Polynomial + graph  Definition: the art and science of visualization in the approximation of zeroes of complex polynomials  Each image is called a “polynomiograph”

4.  How do we find the solutions to a complex polynomial equation?  Classic question  Not at all an easy question  We only have closed formulas for polynomials of degree n < 5  The rest is left up to approximations

5.  Best known method for approximating roots  Formula:  Where z can be either a real or complex input, in which case z = x + iy  Makes use of an iteration function

6.  Machine into which you input the output and eventually hope that the terms converge to some value  Definition: if θ is a root of the polynomial p, then it is a fixed point of the iteration function F  If p(θ) = 0, then F(θ) = θ

7.  Want to approximate √2  i.e. solve f(x) = x 2 – 2  Newton’s formula for this f(x) results in an iteration function nxnxn 01 1N(x 0 )1.50000000 2N(x 1 )1.41666667 3N(x 2 )1.41421568 4N(x 3 )1.41421356 5N(x 4 )1.41421356

8.  Did it work?  True value of √2 = 1.41421356  Indeed our expectations hold:  Newton’s method seems to converge at 1.41421356 i.e. it is a fixed point of N(x)  1.41421356 is a root of f(x) = x 2 – 2

10.  Formally: Given a set of n Euclidean points, find a point in their convex hull that maximizes the product of the distances to the n given points.  Informally: In an art gallery with, say, 3 paintings, what is the optimal position for a camera?

11.  Consider three points in the plane  Their convex hull (the minimal set containing all the points) is naturally a triangle  Would the camera be somewhere along the edges of the triangle or inside the triangle?

12.  Answer: the point that would maximize the product of the distances is at the boundary

14.  Why is the optimal point at the boundary?  Is it unique?  How do you find the optimal point?

15.  We will use geometry, polynomial root- finding methods, and polynomiography to solve these problems  How you find roots of polynomial equations ≈ how you find the optimal point  Why and how this is so… To Be Discovered  Consider the Algebraic Art Gallery Problem in 3D  Explore other geometric problems related to root-finding