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 function|root]] of a [[Function (mathematics)|function]] ''f''. 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 | pmid = | pmc = }}</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=httphttps://books.google.co.ukcom/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 √2. 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. The algorithm also makes use of square roots, which are slower than basic floating point operations.