Editing Unimodality (section)

==Unimodal function==
As the term "modal" applies to data sets and probability distribution, and not in general to [[function (mathematics)|functions]], the definitions above do not apply. The definition of "unimodal" was extended to functions of [[real number]]s as well.  

A common definition is as follows: a function ''f''(''x'') is a '''unimodal function''' if for some value ''m'', it is [[monotonic]]ally increasing for ''x''&nbsp;≤&nbsp;''m'' and monotonically decreasing for ''x''&nbsp;≥&nbsp;''m''. In that case, the [[maximum]] value of ''f''(''x'') is ''f''(''m'') and there are no other local maxima.

Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to be suitable for simple functions only. A general method based on [[derivative]]s exists,<ref>{{cite web|url=http://homepage.univie.ac.at/thibaut.barthelemy/METRIC.pdf|title=On the unimodality of METRIC Approximation subject to normally distributed demands.|work=Method in appendix D, Example in theorem 2 page 5|access-date=2013-08-28}}</ref> but it does not succeed for every function despite its simplicity.

Examples of unimodal functions include [[quadratic polynomial]] functions with a negative quadratic coefficient, [[tent map]] functions, and more.

The above is sometimes related to as '''{{visible anchor|strong unimodality}}''', from the fact that the monotonicity implied is ''strong monotonicity''. A function ''f''(''x'') is a '''weakly unimodal function''' if there exists a value ''m'' for which it is weakly monotonically increasing for ''x''&nbsp;≤&nbsp;''m'' and weakly monotonically decreasing for ''x''&nbsp;≥&nbsp;''m''.  In that case, the maximum value ''f''(''m'') can be reached for a continuous range of values of ''x''. An example of a weakly unimodal function which is not strongly unimodal is every other row in [[Pascal's triangle]].

Depending on context, unimodal function may also refer to a function that has only one local minimum, rather than maximum.<ref>{{cite web|url=https://glossary.informs.org/indexVer1.php?page=U.html|title=Mathematical Programming Glossary.|access-date=2020-03-29}}</ref> For example, [[local unimodal sampling]], a method for doing numerical optimization, is often demonstrated with such a function. It can be said that a unimodal function under this extension is a function with a single local [[extremum]].

One important property of unimodal functions is that the extremum can be found using [[search algorithm]]s such as [[golden section search]], [[ternary search]] or [[successive parabolic interpolation]].<ref>{{Cite book |last1=Demaine |first1=Erik D. |last2=Langerman |first2=Stefan |title=Algorithms – ESA 2005 |chapter=Optimizing a 2D Function Satisfying Unimodality Properties |series=Lecture Notes in Computer Science |date=2005 |volume=3669 |editor-last=Brodal |editor-first=Gerth Stølting |editor2-last=Leonardi |editor2-first=Stefano  |chapter-url=https://link.springer.com/chapter/10.1007/11561071_78 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=887–898 |doi=10.1007/11561071_78 |isbn=978-3-540-31951-1}}</ref>