Editing Root-finding algorithm (section)

== 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 |title=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 because it requires evaluating the function on the entire boundary of the rectangle.

Another criterion is given by a theorem of [[Leopold Kronecker|Kronecker]].<ref>{{Cite book |last1=Ortega |first1= James M. |last2=Rheinboldt |first2=Werner C. |title=Iterative solution of nonlinear equations in several variables |date=2000 |publisher=Society for Industrial and Applied Mathematics |isbn=978-0-89871-461-6 }}</ref>{{Page needed|date=December 2024}} It says that, if the [[Degree of a continuous mapping|topological degree]] of a function ''f'' on a rectangle is non-zero, then the rectangle must contain at least one root of ''f''. This criterion is the basis for several root-finding methods, such as those of Stenger<ref>{{Cite journal |last=Stenger |first=Frank |date=1975-03-01 |title=Computing the topological degree of a mapping inRn |url=https://doi.org/10.1007/BF01419526 |journal=Numerische Mathematik |language=en |volume=25 |issue=1 |pages=23–38 |doi=10.1007/BF01419526 |s2cid=122196773 |issn=0945-3245}}</ref>  and Kearfott.<ref>{{Cite journal |last=Kearfott |first=Baker |date=1979-06-01 |title=An efficient degree-computation method for a generalized method of bisection |url=https://doi.org/10.1007/BF01404868 |journal=Numerische Mathematik |volume=32 |issue=2 |pages=109–127 |doi=10.1007/BF01404868 |s2cid=122058552 |issn=0029-599X}}</ref> However, computing the topological degree can be time-consuming.

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 computing the topological degree; it only requires computing 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 |issn=0945-3245 |s2cid=121771945}}</ref>{{Rp|page=11|location=Lemma.4.7}} Note that Vrahatis and Iordanidis <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 |journal=Topology and Its Applications |language=en |volume=275 |pages=107036 |doi=10.1016/j.topol.2019.107036 |s2cid=213249321 |issn=0166-8641|doi-access=free }}</ref> Again, no upper bound on the number of queries is given.