Liz Balsam Advisor: Bahman Kalantari
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”
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
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
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
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 ) N(x 1 ) N(x 2 ) N(x 3 ) N(x 4 )
Did it work? True value of √2 = Indeed our expectations hold: Newton’s method seems to converge at i.e. it is a fixed point of N(x) is a root of f(x) = x 2 – 2
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?
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?
Answer: the point that would maximize the product of the distances is at the boundary
Why is the optimal point at the boundary? Is it unique? How do you find the optimal point?
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