mat 4725 numerical analysis section 2.1 the bisection method

MAT 4725Numerical Analysis

Section 2.1

The Bisection Method

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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

0

( )( )

( ) t

dN tN t

dt

N t N e

N(t) = size of a population = birth rate

Why?

Population Model 2

0

( )( )

( ) ( 1)t t

dN tN t v

dtv

N t N e e

N(t) = size of a population = birth rate v =

Why?

Population Model 2

435,0001,564,000 1,000,000 ( 1)e e

N0 = 1,000,000, N(1) = 1,564,000 = ??? v = 435,000

Population Model 2

( ) 1,564,000P

We want to find = such that

435,000( ) 1,000,000 ( 1)P e e

Population Model 2

( ) 1,564,000 0P

In general

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

( ) 0f x

IVT

IVT: Special Case

The Bisection Method

Idea

2 2 1 1, , ,n np a b a b a b

Theorem 2.1

The bisection method generates a sequence {pn} approximating a zero p of f such that

for 12n n

b ap p n

Theorem 2.1

The bisection method generates a sequence {pn} approximating a zero p of f such that

for 12n n

b ap p n

Thus, the method always converges to a solution

lim nn

p p

Algorithm 2.1

Pseudo code (description) of the algorithm will be given.

Easy to translate it into a program

Algorithm 2.1

Example 13 2( ) 4 10, 1, 2f x x x a b

Example 2 Theoretical Computations

Find the number of iterations n needed such that

3 2( ) 4 10, 1, 2f x x x a b

310np p

2n n

b ap p

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