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05/03/23http://
numericalmethods.eng.usf.edu 1
Bisection Method
Computer Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
Bisection Method
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Basis of Bisection MethodTheorem
x
f(x)
xu x
An equation f(x)=0, where f(x) is a real continuous function, has at least one root between xl and xu if f(xl) f(xu) < 0.
Figure 1 At least one root exists between the two points if the function is real, continuous, and changes sign.
x
f(x)
xu x
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Basis of Bisection Method
Figure 2 If function does not change sign between two points, roots of the equation may still exist between the two points.
xf 0xf
x
f(x)
xu x
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Basis of Bisection Method
Figure 3 If the function does not change sign between two points, there may not be any roots for the equation between the two points.
x
f(x)
xu x
xf 0xf
x
f(x)
xu x
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Basis of Bisection Method
Figure 4 If the function changes sign between two points, more than one root for the equation may exist
between the two points.
xf 0xf
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Algorithm for Bisection Method
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Step 1Choose x and xu as two guesses for the root such that f(x) f(xu) < 0, or in other words, f(x) changes sign between x and xu. This was demonstrated in Figure 1.
x
f(x)
xu x
Figure 1
x
f(x)
xu x
xm
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Step 2Estimate the root, xm of the equation f (x) = 0 as the mid point between x and xu as
xx
m = xu
2
Figure 5 Estimate of xm
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Step 3Now check the following
a) If , then the root lies between x and xm; then x = x ; xu = xm.
b) If , then the root lies between xm and xu; then x = xm; xu = xu.
c) If ; then the root is xm. Stop the algorithm if this is true.
0ml xfxf
0ml xfxf
0ml xfxf
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Step 4
xx
m = x u
2
100
newm
oldm
new
a xxx
m
root of estimatecurrent newmx
root of estimate previousoldmx
Find the new estimate of the root
Find the absolute relative approximate error
where
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Step 5
Is ?
Yes
No
Go to Step 2 using new upper and lower
guesses.
Stop the algorithm
Compare the absolute relative approximate error with the pre-specified error tolerance .
as
sa
Note one should also check whether the number of iterations is more than the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user about it.
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Example 1To find the inverse of a value, a, one can use the equation
where x is the inverse of a.
Use the bisection method of finding roots of equations to find the inverse of a = 2.5. Conduct three iterations to estimate the root of the above equation.
Find the absolute relative approximate error at the end of each iteration and the number of significant digits at least correct at the end of each iteration.
01x
axf
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Example 1 Cont.
Figure 8 Graph of the function f(x).
01x
axf
0 0.25 0.5 0.75 11
0.5
0
0.5
1
1.5
f(x)
1.5
1
0f x( )
10 x
1 0.5 0 0.5 1 1.54
3
2
1
0
1
2
f(x)xu (upper guess)xl (lower guess)
1.505
3.5
0
f x( )
f x( )
f x( )
1.0021 x x u x l
Entered function on given interval with initial upper and lower guesses
Figure 9 Checking that the bracket is valid.
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Example 1 Cont.Solutio
n
Let us assume015.2)(
01)(
01
xxfaxxf
xa
1 ,0 ul xx
Check if the function changes sign between and .
lx ux 5.1115.21
1105.20
fxffxf
u
l
There is at least one root between the brackets.
05.1110 ffxfxf ul
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Example 1 Cont.Iteration 1The estimate of the root is
5.02
102
ulm
xxx
025.015.00
25.015.05.25.0
ffxfxf
fxf
ml
m
The root is bracketed between and . The lower and upper limits of the new bracket are 5.0 ,0 ul xxThe absolute relative approximate error cannot be calculated as we do not have a previous approximation.
mxlx
aFigure 10 Graph of the estimated root after Iteration 1.
1 0.5 0 0.5 1 1.54
3
2
1
0
1
2
f(x)xu (upper guess)xl (lower guess)new guess
1.505
3.5
0f x( )
f x( )
f x( )
f x( )
1.0021 x x u x l x r
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Example 1 Cont.Iteration 2The estimate of the root
is25.0
25.00
2
ul
mxxx
375.0125.05.225.0 fxf m
The root is bracketed between and . The lower and upper limits of the new bracket are
5.0 x,25.0 ulx
uxmx
Figure 11 Graph of the estimated root after Iteration 2.
1 0.5 0 0.5 1 1.54
3
2
1
0
1
2
f(x)xu (upper guess)xl (lower guess)new guess
1.505
3.5
0f x( )
f x( )
f x( )
f x( )
1.0021 x x u x l x r
025.0375.0
5.025.0
ffxfxf um
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Example 1 Cont.
%100
10025.0
5.025.0
100
newm
oldm
newm
a xxx
None of the significant digits are at least correct in the estimated root of
as the absolute relative approximate error is greater than 5%.
25.0mx
The absolute relative approximate error at the end of Iteration 2 is
a
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Example 1 Cont.Iteration 3The estimate of the
root is375.0
25.025.0
2
ul
mxxx
0625.01375.05.2375.0 fxf m
The root is bracketed between and .
5.0 ,25.0 ul xx
uxmx
025.00625.0
5.0375.0
ffxfxf um
Figure 12 Graph of the estimated root after Iteration 3.
1 0.5 0 0.5 1 1.54
3
2
1
0
1
2
f(x)xu (upper guess)xl (lower guess)new guess
1.505
3.5
0f x( )
f x( )
f x( )
f x( )
1.0021 x x u x l x r
The lower and upper limits of the new bracket are
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Example 1 Cont.
%333.33
100375.0
25.0375.0
100
newm
oldm
newm
a xxx
Still none of the significant digits are at least correct in the estimated root of the equation as the absolute relative approximate error is greater than 5%. Seven more iterations were conducted and these iterations are shown in the table below.
The absolute relative approximate error at the end of Iteration 3 is
a
Iteration xl xu xm12345678910
000.250.3750.3750.3750.390630.398440.398440.39844
10.50.50.50.43750.406250.406250.406250.402340.40039
0.50.250.3750.43750.406250.390630.398440.402340.400390.39941
----------10033.3314.28577.69234.001.96080.970870.487800.24450
0.25−0.375−0.0625 0.09375 0.01563−0.02344−3.90625×10-3
5.8594×10-3
9.7656×10-4
−1.4648×10-3
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Example 1 Cont.Table 1 Root of as function of number of iterations for bisection method.
%a
0xf
mxf
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Advantages Always convergent The root bracket gets halved with
each iteration - guaranteed.
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Drawbacks Slow convergence If one of the initial guesses is close
to the root, the convergence is slower
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Drawbacks (continued) If a function f(x) is such that it just
touches the x-axis it will be unable to find the lower and upper guesses.
f(x)
x
2xxf
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Drawbacks (continued) Function changes sign but root does
not exist
f(x)
x
x
xf 1
Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/bisection_method.html