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| {{about|searching continuous function values|searching a finite sorted array|binary search algorithm}}
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| [[Image:Bisection method.svg|250px|thumb|A few steps of the bisection method applied over the starting range [a<sub>1</sub>;b<sub>1</sub>]. The bigger red dot is the root of the function.]]
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| The '''bisection method''' in [[mathematics]] is a [[root-finding method]] that repeatedly bisects an [[Interval (mathematics)|interval]] and then selects a subinterval in which a [[Root of a function|root]] must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods.<ref>{{Harvnb|Burden|Faires|1985|p=31}}</ref> The method is also called the '''interval halving''' method,<ref>http://siber.cankaya.edu.tr/NumericalComputations/ceng375/node32.html</ref> the '''binary search method''',<ref>{{Harvnb|Burden|Fairies|1985|p=28}}</ref> or the '''dichotomy method'''.<ref>[http://www.encyclopediaofmath.org/index.php/Dichotomy_method Encyclopedia of Mathematics]</ref>
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| == The method == | |
| The method is applicable when we wish to solve the equation ''f''(''x'') = 0 for the [[Real number|real]] variable ''x'', where ''f'' is a [[continuous function]] defined on an interval [''a'', ''b''] and ''f''(''a'') and ''f''(''b'') have opposite signs. In this case ''a'' and ''b'' are said to bracket a root since, by the [[intermediate value theorem]], the ''f'' must have at least one root in the interval (''a'', ''b'').
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| At each step the method divides the interval in two by computing the midpoint ''c'' = (''a''+''b'') / 2 of the interval and the value of the function ''f''(''c'') at that point. Unless ''c'' is itself a root (which is very unlikely, but possible) there are now two possibilities: either ''f''(''a'') and ''f''(''c'') have opposite signs and bracket a root, or ''f''(''c'') and ''f''(''b'') have opposite signs and bracket a root. The method selects the subinterval that is a bracket as a new interval to be used in the next step. In this way the interval that contains a zero of ''f'' is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.
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| Explicitly, if ''f''(''a'') and ''f''(''c'') are opposite signs, then the method sets ''c'' as the new value for ''b'', and if ''f''(''b'') and ''f''(''c'') are opposite signs then the method sets ''c'' as the new ''a''. (If ''f''(''c'')=0 then ''c'' may be taken as the solution and the process stops.) In both cases, the new ''f''(''a'') and ''f''(''b'') have opposite signs, so the method is applicable to this smaller interval.<ref>{{Harvnb|Burden|Faires|1985|p=28}} for section</ref>
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| == Example: Finding the root of a polynomial ==
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| Suppose that the bisection method is used to find a root of the polynomial
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| :<math> f(x) = x^3 - x - 2 \,.</math>
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| First, two numbers <math> a </math> and <math> b </math> have to be found such that <math>f(a)</math> and <math>f(b)</math> have opposite signs. For the above function, <math> a = 1 </math> and <math> b = 2 </math> satisfy this criterion, as
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| :<math> f(1) = (1)^3 - (1) - 2 = -2 </math>
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| and
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| :<math> f(2) = (2)^3 - (2) - 2 = +4 \,.</math>
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| Because the function is continuous, there must be a root within the interval [1, 2].
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| In the first iteration, the end points of the interval which brackets the root are <math> a_1 = 1 </math> and <math> b_1 = 2 </math>, so the midpoint is
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| :<math> c_1 = \frac{2+1}{2} = 1.5 </math>
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| The function value at the midpoint is <math> f(c_1) = (1.5)^3 - (1.5) - 2 = -0.125 </math>. Because <math> f(c_1) </math> is negative, <math> a = 1 </math> is replaced with <math> a = 1.5 </math> for the next iteration to ensure that <math> f(a) </math> and <math> f(b) </math> have opposite signs. As this continues, the interval between <math> a </math> and <math> b </math> will become increasingly smaller, converging on the root of the function. See this happen in the table below.
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| {| width="500" border="1" cellpadding="2"
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| ! Iteration !! <math>a_n</math> !! <math>b_n</math> !! <math>c_n</math> !! <math>f(c_n)</math>
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| |- align="right"
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| | 1|| 1 || 2 || 1.5 || −0.125
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| |- align="right"
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| | 2|| 1.5|| 2|| 1.75|| 1.6093750
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| |- align="right"
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| | 3|| 1.5|| 1.75|| 1.625|| 0.6660156
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| |- align="right"
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| | 4|| 1.5|| 1.625|| 1.5625|| 0.2521973
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| |- align="right"
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| | 5|| 1.5|| 1.5625|| 1.5312500|| 0.0591125
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| |- align="right"
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| | 6|| 1.5|| 1.5312500|| 1.5156250|| −0.0340538
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| |- align="right"
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| | 7 || 1.5156250|| 1.5312500|| 1.5234375|| 0.0122504
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| |- align="right"
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| | 8|| 1.5156250|| 1.5234375|| 1.5195313|| −0.0109712
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| |- align="right"
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| | 9 || 1.5195313|| 1.5234375|| 1.5214844|| 0.0006222
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| |- align="right"
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| | 10|| 1.5195313|| 1.5214844|| 1.5205078|| −0.0051789
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| |- align="right"
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| | 11|| 1.5205078|| 1.5214844|| 1.5209961|| −0.0022794
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| |- align="right"
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| | 12|| 1.5209961|| 1.5214844|| 1.5212402|| −0.0008289
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| |- align="right"
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| | 13|| 1.5212402|| 1.5214844|| 1.5213623|| −0.0001034
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| |- align="right"
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| | 14|| 1.5213623|| 1.5214844|| 1.5214233|| 0.0002594
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| |-align="right"
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| | 15|| 1.5213623|| 1.5214233|| 1.5213928|| 0.0000780
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| |}
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| After 13 iterations, it becomes apparent that there is a convergence to about 1.521: a root for the polynomial.
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| == Analysis ==
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| The method is guaranteed to converge to a root of ''f'' if ''f'' is a [[continuous function]] on the interval [''a'', ''b''] and ''f''(''a'') and ''f''(''b'') have opposite signs. The [[approximation error|absolute error]] is halved at each step so the method [[Rate of convergence|converges linearly]], which is comparatively slow.
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| Specifically, if ''c''<sub>1</sub> = (''a''+''b'')/2 is the midpoint of the initial interval, and ''c''<sub>''n''</sub> is the midpoint of the interval in the ''n''th step, then the difference between ''c''<sub>''n''</sub> and a solution ''c'' is bounded by<ref>{{Harvnb|Burden|Faires|1985|p=31}}, Theorem 2.1</ref>
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| :<math>|c_n-c|\le\frac{|b-a|}{2^n}.</math>
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| This formula can be used to determine in advance the number of iterations that the bisection method would need to converge to a root to within a certain tolerance.
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| The number of iterations needed, ''n'', to achieve a given error (or tolerance), ε, is given by:
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| <math>n = \log_2\left(\frac{\epsilon_0}{\epsilon}\right)=\frac{\log\epsilon_0-\log\epsilon}{\log2} , </math>
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| where <math>\epsilon_0 = \text{initial bracket size} = b-a .</math>
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| Therefore, the linear convergence is expressed by <math>\epsilon_{n+1} = \text{constant} \times \epsilon_n^m, \ m=1 .</math>
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| == Pseudocode ==
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| The method may be written in [[Pseudocode]] as follows:<ref>{{Harvnb|Burden|Faires|1985|p=29}}</ref>
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| INPUT: Function ''f'', endpoint values ''a'', ''b'', tolerance ''TOL'', maximum iterations ''NMAX''
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| CONDITIONS: ''a'' < ''b'', either ''f''(''a'') < 0 and ''f''(''b'') > 0 or ''f''(''a'') > 0 and ''f''(''b'') < 0
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| OUTPUT: value which differs from a root of ''f''(''x'')=0 by less than ''TOL''
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| ''N'' ← 1
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| '''While''' ''N'' ≤ ''NMAX'' ''# limit iterations to prevent infinite loop''
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| ''c'' ← (''a'' + ''b'')/2 ''# new midpoint''
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| '''If''' ''f''(''c'') = 0 or (''b'' – ''a'')/2 < ''TOL'' '''then''' ''# solution found''
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| Output(''c'')
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| '''Stop'''
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| '''EndIf'''
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| ''N'' ← ''N'' + 1 ''# increment step counter''
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| '''If''' sign(''f''(''c'')) = sign(''f''(''a'')) '''then''' ''a'' ← ''c'' '''else''' ''b'' ← ''c'' ''# new interval''
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| '''EndWhile'''
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| Output("Method failed.") ''# max number of steps exceeded''
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| == See also ==
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| *[[Secant method]]
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| *[[Newton's method]]
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| *[[Root-finding algorithm]]
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| *[[Binary search algorithm]]
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| *[[Lehmer–Schur algorithm]], generalization of the bisection method in the complex plane
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| *[[Nested intervals]]
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| *[[Brent's method]]
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| == References ==
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| {{reflist|30em}}
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| * {{Citation| last1=Burden | first1=Richard L. | last2=Faires | first2=J. Douglas | title=Numerical Analysis | publisher=PWS Publishers | edition=3rd | isbn=0-87150-857-5 | year=1985 | chapter=2.1 The Bisection Algorithm}}.
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| * {{Citation | last1=Corliss | first1=George | title=Which root does the bisection algorithm find? | year=1977 | journal=SIAM Review | issn=1095-7200 | volume=19 | issue=2 | pages=325–327 | doi=10.1137/1019044}}.
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| * {{Citation | last1=Kaw | first1=Autar | last2=Kalu | first2=Egwu | year=2008 | title=Numerical Methods with Applications | edition=1st | publisher= |url=http://numericalmethods.eng.usf.edu/topics/textbook_index.html |isbn= |doi= }}<!-- isbn for 2nd abridged edition: 978-0578057651. Why isn't the website the 2nd edition? -->
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| == External links ==
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| {{wikiversity|The bisection method}}
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| {{wikibooks|Numerical Methods|Equation Solving}}
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| *{{MathWorld|title=Bisection|urlname=Bisection}}
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| * [http://numericalmethods.eng.usf.edu/topics/bisection_method.html Bisection Method] Notes, PPT, Mathcad, Maple, Matlab, Mathematica from [http://numericalmethods.eng.usf.edu Holistic Numerical Methods Institute]
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| *[http://math.fullerton.edu/mathews/n2003/BisectionMod.html Module for the Bisection Method by John H. Mathews]
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| *[http://catc.ac.ir/mazlumi/jscodes/bisection.php Online root finding of a polynomial-Bisection method] by Farhad Mazlumi
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| [[Category:Root-finding algorithms]]
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| [[Category:Articles with example pseudocode]]
| |
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