Ridders' method: Difference between revisions

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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 }}</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 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==