MAT 4725 Numerical Analysis Section 2.1 The Bisection Method

MCM Teams Any progress?

Homework Download the homework Read 2.2 (Burden) You may skip all the proofs unless specified

Preview Find the solutions of an equation in one variable. Repeatly cut the intervals that contain the solution in half.

Population Model 1 N(t) = size of a population = birth rate Why?

Population Model 2 N(t) = size of a population = birth rate v = Why?

Population Model 2 N 0 = 1,000,000, N(1) = 1,564,000 = ??? v = 435,000

Population Model 2 We want to find = such that

Population Model 2

In general We want to find the solutions of a equation in one variable.

IVT

IVT: Special Case

The Bisection Method Idea

Theorem 2.1 The bisection method generates a sequence {p n } approximating a zero p of f such that

Theorem 2.1 The bisection method generates a sequence {p n } approximating a zero p of f such that Thus, the method always converges to a solution

Algorithm 2.1 Pseudo code (description) of the algorithm will be given. Easy to translate it into a program

Algorithm 2.1

Example 1

Example 2 Theoretical Computations Find the number of iterations n needed such that

Classwork 1 Write a program to implement the bisection algorithm.

Remark #1 Bisect:=proc(f, aa, bb, tol, N0) local i, p, a, b, FA, FP; a:=aa; b:=bb; The function f is passed into the procedure, not the expression f(x)

Remark #2 Bisect:=proc(f, aa, bb, tol, N0) local i, p, a, b, FA, FP; a:=aa; b:=bb; The values of the parameters passed into a procedure cannot be changed

Remark #3 Use return() to stop the program

Homework From now on… Use the Maple program in your classwork to do all the computations Use Maple to plot all the graphs