Root-finding algorithm: Difference between revisions

Revision as of 20:22, 2 February 2024

Revision as of 02:22, 20 February 2024

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== Finding roots in higher dimensions ==

The [[bisection method]] has been generalized to higher dimensions; these methods are called '''generalized bisection methods'''.<ref name=":0">{{Cite journal |last1=Mourrain |first1=B. |last2=Vrahatis |first2=M. N. |last3=Yakoubsohn |first3=J. C. |date=2002-06-01 |title=On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree |journal=Journal of Complexity |language=en |volume=18 |issue=2 |pages=612–640 |doi=10.1006/jcom.2001.0636 |issn=0885-064X|doi-access=free }}</ref><ref>{{Cite book |last=Vrahatis |first=Michael N. |date=2020 |editor-last=Sergeyev |editor-first=Yaroslav D. |editor2-last=Kvasov |editor2-first=Dmitri E. |chapter=Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions |chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-40616-5_17 |journaltitle=Numerical Computations: Theory and Algorithms |series=Lecture Notes in Computer Science |language=en |___location=Cham |publisher=Springer International Publishing |volume=11974 |pages=223–238 |doi=10.1007/978-3-030-40616-5_17 |isbn=978-3-030-40616-5 |s2cid=211160947}}</ref> At each iteration, the ___domain is partitioned into two parts, and the algorithm decides - based on a small number of function evaluations - which of these two parts must contain a root. In one dimension, the criterion for decision is that the function has opposite signs. The main challenge in extending the method to multiple dimensions is to find a criterion that can be computed easily, and guarantees the existence of a root.

The [[Poincaré–Miranda theorem]] gives a criterion for the existence of a root in a rectangle, but it is hard to verify, since it requires to evaluate the function on the entire boundary of the triangle.

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A third criterion is based on a ''characteristic polyhedron''. This criterion is used by a method called Characteristic Bisection.<ref name=":0" />{{Rp|page=19--}} It does not require to compute the topological degree - it only requires to compute the signs of function values. The number of required evaluations is at least <math>\log_2(D/\epsilon)</math>, where ''D'' is the length of the longest edge of the characteristic polyhedron.<ref name=":2">{{Cite journal |last1=Vrahatis |first1=M. N. |last2=Iordanidis |first2=K. I. |date=1986-03-01 |title=A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations |url=https://doi.org/10.1007/BF01389620 |journal=Numerische Mathematik |language=en |volume=49 |issue=2 |pages=123–138 |doi=10.1007/BF01389620 |s2cid=121771945 |issn=0945-3245}}</ref>{{Rp|page=11|___location=Lemma.4.7}} Note that <ref name=":2" /> prove a lower bound on the number of evaluations, and not an upper bound.

A fourth method uses an intermediate-value theorem on simplices.<ref>{{Cite journal |last=Vrahatis |first=Michael N. |date=2020-04-15 |title=Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros |url=https://www.sciencedirect.com/science/article/pii/S0166864119304420 |journal=Topology and Its Applications |language=en |volume=275 |pages=107036 |doi=10.1016/j.topol.2019.107036 |s2cid=213249321 |issn=0166-8641}}</ref> Again, no upper bound on the number of queries is given.

== See also ==