Ridders' method: Difference between revisions

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{{short description|Root-finding algorithm in numerical analysis}}

In [[numerical analysis]], '''Ridders' method''' is a [[root-finding algorithm]] based on the [[false position method]] and the use of an [[exponential function]] to successively approximate a root of a continuous function <math>f(x)</math> . The method is due to C. Ridders.<ref>{{Cite journal | last1 = Ridders | first1 = C. | doi = 10.1109/TCS.1979.1084580 | title = A new algorithm for computing a single root of a real continuous function | journal = IEEE Transactions on Circuits and Systems | volume = 26 | pages = 979–980| year = 1979 | pmidissue = | pmc =11 }}</ref><ref>{{cite book|title=Numerical Methods in Engineering with Python|first=Jaan |last=Kiusalaas| publisher=Cambridge University Press| year=2010| isbn=978-0-521-19132-6 | edition=2nd| pages=146–150| url=https://books.google.com/books?id=9SG1r8EJawIC&pg=PT156}}</ref>

Ridders' method is simpler than [[Muller's method]] or [[Brent's method]] but with similar performance.<ref>{{cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=[[Numerical Recipes]]: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 9.2.1. Ridders' Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=452}}</ref> The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall [[order of convergence]] of the method with respect to function evaluations rather than with respect to number of iterates is <math>\sqrt{2}</math> . If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed.

==Method==

Given two values of the independent variable, <math>x_0</math> and <math>x_2</math>, which are on two different sides of the root being sought, i.e.,so that<math>f(x_0)f(x_2) < 0</math>.The, the method begins by evaluating the function at the midpoint <math>x_1 = (x_0 +x_2)/2 </math>. One then finds the unique exponential function <math>e^{ax}</math> such that function <math>h(x)=f(x)e^{ax}</math> satisfies <math>h(x_1)=(h(x_0) +h(x_2))/2 </math>. Specifically, parameter <math>a</math> is determined by

:<math>e^{a(x_1 - x_0)} = \frac{f(x_1)+-\operatorname{sign}[f(x_2x_0)]\sqrt{f(x_1)^2 - f(x_0)f(x_2)}}{f(x_2)} .</math>

The false position method is then applied to the points <math>x_1(x_0, h(x_1x_0))</math> and <math>(x_0x_2,h(x_0x_2))</math>, leading to a new value ''x''<submath>3x_3 </submath>, between ''x''<submath>0x_0 </submath> and ''x''<submath>2x_2 </submath>, which can be used as one of the two bracketing values in the next step of the iteration.

:<math>x_3 = x_1 + (x_1 - x_0)\frac{\operatorname{sign}[f(x_0)]f(x_1)}{\sqrt{f(x_1)^2 - f(x_0)f(x_2)}} .,</math>

which will be used as one of the two bracketing values in the next step of the iteration. The other bracketing value is taken to be ''x''<submath>3x_1</submath> if <math>f(''x''x_1)f(x_3) <sub>30</submath>) has(which thewill oppositebe signtrue toin f(''x''₄the well-behaved case), or otherwise whichever of ''x''<submath>1x_0</submath> and ''x''<submath>2x_2</submath> has f(x)a function value of opposite sign to <math>f(''x''<sub>4x_3).</submath>) The iterative procedure can be terminated when a target accuracy is obtained.

The method can be summarized by the formula

:<math>x_3 = x_1 + (x_1 - x_0)\frac{\operatorname{sign}[f(x_0)]f(x_1)}{\sqrt{f(x_1)^2 - f(x_0)f(x_2)}} .</math>

==References==

{{reflist}}

{{root-finding algorithms}}

[[Category:Root-finding algorithms]]

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