Lecture 4
 Bisection method for root finding
 Binary search
 fzero
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1
Drawbacks of Newton method
x=linspace(-5,5);plot(x,x.^2-2*x-2);
hold on;plot(x,-3,'r')
Tangent line with zero slope
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2
Perturb current guess
if zero derivative is detected
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3
Failure
f=inline('x.^3-2*x+2');
x=linspace(-2,2);plot(x,f(x));hold on
plot(x,0,'g')
plot([0 1],[0 f(1)],'r');
plot([1 0],[0 f(0)],'r');
The tangent lines of x^3 2x + 2 at 0 and 1 intersect
the x-axis at 1 and 0,
respectively, illustrating
wh y Ne wt on's method
oscillates between these
values for some starting
points.
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4
Bisection method
A root is within an interval [L,R]
 The bisection method


Cut [L,R] into equal-size subintervals.
such as [L,R] to [L,M] U [M,R]
Determine which interval contains a root
 Then select it as the searching interval
 Repeat the same process until halting
condition holds.

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Different signs
Let f be continuous in the interval [L,R]
 These exists at least one root in [L,R]
if f(L) and f(R) have different signs,
equivalently
f(L) f(R) < 0

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Bisection Method
L
M=(L+R)/2
R
f(L) and f(M) have the same sign: L  M
f(R) and f(M) have the same sign: R  M
RM
LM
L M
R
L
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M
R
7
Bisection method
1.
2.
3.
4.
5.
6.
7.
Create an inline function, f
Input two guesses, L < R
If f(L)f(R) > 0, return
Set M to the middle of L and R
If f(L)f(M) < 0, R = M
If f(R)f(M) < 0, L = M
If the halting condition holds, exit, otherwise
goto step 4.
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8
Flow chart
c=bisection(f,L,R)
Input L,R,f
with f(L)f(R) < 0
M=0.5*(L+R)
f(L)f(M)<0
T
R=M
L=M
F
abs(f(M)) < epslon
T
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9
Flow chart
M=bisection(f,L,R)
if f(L)f(R) > 0 return
M=0.5*(L+R)
T
return
abs(f(M)) < epslon
M=0.5*(L+R)
f(L)f(M)<0
T
R=M
數值方法2008 Applied Mathematics,
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L=M
10
Flow chart
M=bisection(f,L,R)
if f(L)f(R) > 0 return
M=0.5*(L+R)
return
abs(f(M)) > epslon
T
f(L)f(M)<0
T
R=M
L=M
M=0.5*(L+R)
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Halting condition
abs(f(M)) < epslon
 Absolute f(M) is small enough to
approach zero.
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12
Non-decreasing sequence

y
x=
2
18
23
44
46
49
61
62
74
76
79
82
89
92
95
R
L
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13
Binary search
Binary search is a general searching
approach
 Let x represent a non-decreasing
sequence



x(i) is less or equal than x(j) if i < j
Let y be an instance in sequence x

Determine i such that
x(i) = y
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Random sequence

>> x=round(rand(1,15)*100)
x=
Columns 1 through 9
95
23
61
49
89
76
46
2
82
Columns 10 through 15
44
62
79
92
74
18
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Searching for non-decreasing sequence

y
x=
2
18
L=1
23
44
46
49
61
62
74
76
M=8
數值方法2008 Applied Mathematics,
NDHU
79
82
89
92
95
R=15
16
Searching for non-decreasing sequence

y
x=
2
18
23
44
46
49
61
62
74
76
L=8
數值方法2008 Applied Mathematics,
NDHU
79
82
89
M=12
92
95
R=15
17
Searching for non-decreasing sequence

y
x=
2
18
23
44
46
49
61
62
74
76
79
82
89
92
95
R=12
L=8
M=10
Halt since x(c) equals y
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Searching
Ex. y=76
 The answer i=10 indicates an index
where x(i) equals y

>> x(10)
ans =
76
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19
Flow chart
Input x,y
L=1;R=length(x)
M=ceil(0.5*(L+R))
x(M)<y
L=M
R=M
F
Halting condition
T
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20
Flow chart
c=bi_search(x,y)
L=1;R=length(x)
M=ceil(0.5*(L+R))
T
return
Halting condition
x(M)<y
T
L=M
R=M
M=ceil(0.5*(L+R))
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21
Discussions
What is a reasonable halting condition?
 Index L must be less than index R
 Halt if L >= R or x(M) == y

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22
MATLAB: fzero
x=linspace(-5,5);plot(x,sin(x.*x));
hold on;
plot(x,0,'r')
x = fzero(@(x)sin(x*x),-0.1);
plot(x,0,'or')
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fzero
x=linspace(-5,5);plot(x,sin(x.*x));
hold on;
plot(x,0,'r')
x = fzero(@(x)sin(x*x),0.1);
plot(x,0,'or')
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24
Exercise 4 due to 10/22
1.
2.
3.
Draw a flow chart to illustrate the
bisection method for root finding
Implement the flow chart
Give two examples to verify the matlab
function
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