Root-finding algorithm: Difference between revisions

Revision as of 20:00, 16 December 2024

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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 |title=Iterative solution of nonlinear equations in several variables |url=https://dl.acm.org/doi/abs/10.5555/335947 |url-access=subscription |date=2000 |publisher=Society for Industrial and Applied Mathematics |isbn=978-0-89871-461-6 }}</ref>{{Page needed}} 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.