Polynomiography

Jonathan ChoateGroton School

jchoate@groton.orgwww.zebragraph.com

VCTAL

Iteration

• Pick a function f[x]• Pick a seed x(0)• Create a sequence {x(0) , x(1) = f[x(0]], x(2)=f[(x(1)], … ,x(n) = f[x(n-1)], ….}

This sequence is the ORBIT of x(0) under iteration of f[x]

An Example• Iterate f[x] = (1/2)x + 4 for different seeds.• Two Interesting Questions 1) For a given seed does the orbit

converge to a value S within a given accuracy?

2) If it does converge, how many iterations does it take?

The set of seeds that converge to S is called the BASIN OF ATTRACTION for S and will be denoted by B(S)

Solving Polynomial Equations

• Formulas exist for Quadratics, Cubics and Quartics

• Have to use numerical methods for polynomials of degree higher than 4.

The Babylonian Method for Finding Square Roots

Iterate f[x] =(1/2)(x + a/x)If a>0 orbits converge to andWhat is B( )? B( )?What happens if a < 0?What happens if a is complex? Babylonian Method

Newton’s Method

Given a function f(x) to solve f(x) =0 iterate the function N[x] = x – f(x)/f’(x)1. Use the spreadsheet Newton’s Method for

Quadratics - Doesn’t handle the complex roots

- Doesn’t give much info about the Basins of attraction for the roots.Newton’s Quadratic Analysis

Cayley’s Discovery

• Use a spreadsheet to apply Newton’s Method to Cubics

• Basins of Attraction get complicated!!Newton’s Cubic Analysis

Polynomiography

Polynomioigraphy

The Fundamental Theorem of Algebra: Version 1

An n-th degree polynomial equation with realco-efficients has n roots with complex roots occuring in pairs.Good Exercise:1. What are the possible solutions for a quadratic

equation with real co-efficients?2. What are the possible solutions for a cubic

equation with real co-efficients?

• Good Group Activity Create Poly Images for all the possible 5th

degree equations?

ChallengeWhat is the equation of the 6th degree polynomial whose polynomiograph is shown below?

Solutions

Real Roots z = 3 , z =4Complex Rootsz = 3 + 3i , z = 3 – 3i , z = 1 + I , z = 1 – I

(z -3)(z-4)( z- [3+3i] )(z –[ 3 – 3i ]} )( z- [1+i] )(z –[ 1 – i ]} = 0

DeMoivre’s Theorem: A Visual Approach

Playz12 + z2 – 4z + 54

A Final Linear Thought

We teach about arithmetic sequences x(n+1) = x(n) + d / x(0) = cOr x(n) = c + n*d We teach about geometric sequences x(n+1) = r x(n) / x(0) = cOr x(n) = c rn

How about teaching about mixed sequences?

A mixed sequence x(n+1) = r*x(n) + d / x(0) = cOr x(n) = c rn[ c – (d/(1-r) ] + (d/(1-r)

Use the Language of Dynamical Systems

Let f(x) = ax + b. F has a fixed point F = b/(1—a)The orbit of x(0) under iteration of f has an n-th iterate equal to x(n) = an( x(0) – F) + F