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F (X) F (A) F (B) F (A) F (B) X A+b) : 1.2 Bisection Method

The bisection method is used to find roots (or zeroes) of a continuous function between two values a and b where the function values at those points have opposite signs. The interval is bisected and the midpoint x0 is tested. If f(x0) = 0 then a root is found, otherwise the subinterval containing the root is selected based on the sign of f(x0) and the process repeats with bisection until the desired accuracy is reached. Three examples apply this method to find roots of different functions within a specified tolerance of 1 x 10-5.

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Johnlloyd Barreto
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60 views2 pages

F (X) F (A) F (B) F (A) F (B) X A+b) : 1.2 Bisection Method

The bisection method is used to find roots (or zeroes) of a continuous function between two values a and b where the function values at those points have opposite signs. The interval is bisected and the midpoint x0 is tested. If f(x0) = 0 then a root is found, otherwise the subinterval containing the root is selected based on the sign of f(x0) and the process repeats with bisection until the desired accuracy is reached. Three examples apply this method to find roots of different functions within a specified tolerance of 1 x 10-5.

Uploaded by

Johnlloyd Barreto
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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1.

2 Bisection Method
If a function f ( x ) is continuous between a and b, and f ( a )∧f ( b ) are of
opposite signs then there exists at least one root between a and b. Let f ( a ) be
negative and f ( b ) be positive, then the root lies between a and b and the
( a+b)
approximate value is given by x 0=
2

If f ( x 0 ) =0then the x 0is the root, otherwise the root lies between x 0∧b∨a∧x 0 ,
depending on whether f (x¿ ¿ 0) ¿ is negative or positive, respectively. Then as
before, we bisect the interval and repeat the process until the root is known to the
desired accuracy.

Examples:

Find the root of the given functions of x. Use 1 x 10−5 as the tolerance.

1. f(x) = x 3+ 3 x −11
a b x0 f (x¿ ¿ 0) ¿
1.7 1.8 1.75 -0.390625
1.75 1.8 1.775 -0.082640625
1.775 1.8 1.7875 0.073841797
1.775 1.7875 1.78125 -0.004608154
1.78125 1.7875 1.784375 0.034564545
1.78125 1.784375 1.7828125 0.014965137
1.78125 1.7828125 1.78203125 0.005175229
1.78125 1.78203125 1.781640625 0.000282722
1.78125 1.781640625 1.781445313 -0.00216292
1.781445313 1.781640625 1.781542969 -0.000940147
1.781542969 1.781640625 1.781591797 -0.000328726
1.781591797 1.781640625 1.781616211 -2.30052 x 10−5

1.781616211 1.781640625 1.781628418 0.000129857


1.781616211 1.781628418 1.781622315 5.34259 x 10−5

1.781616211 1.781622315 1.781619263 1.52135 x 10−5

1.781616211 1.781619263 1.781617737 -3.89586 x 10−6

2. f(x) = 10 x 5−3 x3 −11 x2 −5


a b x0 f (x¿ ¿ 0) ¿
1.2 1.3 1.25 2.470703125
1.2 1.25 1.225 0.56380166
1.2 1.225 1.2125 -0.312929391
1.2125 1.225 1.21875 0.11922282
1.2125 1.21875 1.215625 -0.098393308
1.215625 1.21875 1.2171875 0.010028093
1.215625 1.2171875 1.21640625 -0.044279066
1.21640625 1.2171875 1.216796875 -0.017149627
1.216796875 1.2171875 1.216992188 -0.003566806
1.216992188 1.2171875 1.217089844 0.003229151
1.216992188 1.21789844 1.217445314 0.027991966
1.216992188 1.217445314 1.217218751 0.012204467
1.216992188 1.217218751 1.21710547 0.004316816
1.216992188 1.21710547 1.217048829 0.000374532
1.216992188 1.217048829 1.217020509 -0.001596246
1.217020509 1.217048829 1.217034669 -0.000610871
1.217034669 1.217048829 1.217041749 -0.000118177
1.217041749 1.217048829 1.217045289 0.000128175
1.217041749 1.217045289 1.217043519 4.99846 x 10−6

3. f(x) = x 2−5 sinx−11


a b x0 f (x¿ ¿ 0) ¿
3.2 3.3 3.25 0.103475673
3.2 3.25 3.225 -0.182821639
3.225 3.25 3.2375 -0.039791827
3.2375 3.25 3.24375 0.031812819
3.2375 3.24375 3.240625 -0.003996856
3.240625 3.24375 3.2421875 0.013906153
3.240625 3.241875 3.24125 0.003163908
3.240625 3.24125 3.2409375 -0.000416548
3.2409375 3.24125 3.24109375 0.001373662
3.2409375 3.24109375 3.241015625 0.000478553
3.2409375 3.241015625 3.240976563 3.10014 x 10−5

3.2409375 3.240976563 3.240957032 -0.00019277


3.240957032 3.240976563 3.240966798 -8.08789 x 10−5

3.240966798 3.240976563 3.240971681 -2.49331 x 10−5

3.240971681 3.240976563 3.240974122 3.03988 x 10−6

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