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MAT 4725Numerical Analysis
Section 2.1
The Bisection Method
http://myhome.spu.edu/lauw
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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