Liz BalsamAdvisor: 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) = x2 – 2

Newton’s formula for this f(x) results in an iteration function

n xn

0 11 N(x0) 1.500000002 N(x1) 1.416666673 N(x2) 1.41421568 4 N(x3) 1.41421356 5 N(x4) 1.41421356

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) = x2 – 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 problemsHow 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