Editing Saddle point

{{short description|Critical point on a surface graph which is not a local extremum}}
{{About|the mathematical property|the peninsula in the Antarctic|Saddle Point|the type of landform and general uses of the word "saddle" as a technical term|Saddle (landform)}}

[[File:Saddle point.svg|thumb|right|300px|A saddle point (in red) on the graph of {{math|1=''z'' = ''x''{{sup|2}} − ''y''{{sup|2}}}} ([[Paraboloid#Hyperbolic_paraboloid|hyperbolic paraboloid]])]]

In [[mathematics]], a '''saddle point''' or '''minimax point'''<ref>Howard Anton, Irl Bivens, Stephen Davis (2002): ''Calculus, Multivariable Version'', p.&nbsp;844.</ref> is a [[Point (geometry)|point]] on the [[surface (mathematics)|surface]] of the [[graph of a function]] where the [[slope]]s (derivatives) in [[orthogonal]] directions are all zero (a [[Critical point (mathematics)|critical point]]), but which is not a [[local extremum]] of the function.<ref>{{cite book |last = Chiang |first = Alpha C. |author-link=Alpha Chiang |title = Fundamental Methods of Mathematical Economics |url = https://archive.org/details/fundamentalmetho0000chia_h4v2 |url-access = registration |location=New York |publisher = [[McGraw-Hill]] |edition=3rd |year=1984 |isbn = 0-07-010813-7 |page = [https://archive.org/details/fundamentalmetho0000chia_h4v2/page/312 312] }}</ref>  An example of a saddle point is when there is a critical point with a relative [[minimum]] along one axial direction (between peaks) and a relative [[maxima and minima|maximum]] along the crossing axis.  However, a saddle point need not be in this form.  For example, the function <math>f(x,y) = x^2 + y^3</math> has a critical point at <math>(0, 0)</math> that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the <math>y</math>-direction.
 
[[File:Gordena Jackson, Saddle, 1935-1942, NGA 26642.jpg|thumb|right|150px|A riding saddle]]
The name derives from the fact that the prototypical example in two dimensions is a [[surface (mathematics)|surface]] that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding [[saddle]]. In terms of [[contour line]]s, a saddle point in two dimensions gives rise to a contour map with, in principle, a pair of lines intersecting at the point. Such intersections are rare in contour maps drawn with discrete contour lines, such as ordnance survey maps, as the height of the saddle point is unlikely to coincide with the integer multiples used in such maps. Instead, the saddle point appears as a blank space in the middle of four sets of contour lines that approach and veer away from it. For a basic saddle point, these sets occur in pairs, with an opposing high pair and an opposing low pair positioned in orthogonal directions. The critical contour lines generally do not have to intersect orthogonally.

[[File:Saddle_Point_between_maxima.svg|thumb|300px|right|Saddle point between two hills (the intersection of the figure-eight {{mvar|z}}-contour)]]
[[File:Julia set for z^2+0.7i*z.png|thumb|right|300px|Saddle point on the contour plot is the point where level curves cross ]]

== Mathematical discussion ==
A simple criterion for checking if a given [[stationary point]] of a real-valued function ''F''(''x'',''y'') of two real variables is a saddle point is to compute the function's [[Hessian matrix]] at that point: if the Hessian is [[Positive-definite matrix#Indefinite|indefinite]], then that point is a saddle point. For example, the Hessian matrix of the function <math>z=x^2-y^2</math> at the stationary point <math>(x, y, z)=(0, 0, 0)</math> is the matrix
: <math>\begin{bmatrix}
2 & 0\\
0 & -2 \\
\end{bmatrix}
</math>
which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point <math>(0, 0, 0)</math> is a saddle point for the function <math>z=x^4-y^4,</math> but the Hessian matrix of this function at the origin is the [[null matrix]], which is not indefinite.

In the most general terms, a '''saddle point''' for a [[smooth function]] (whose [[graph of a function|graph]] is a [[curve]], [[surface (mathematics)|surface]] or [[hypersurface]]) is a stationary point such that the curve/surface/etc. in the [[neighborhood (mathematics)|neighborhood]] of that point is not entirely on any side of the [[tangent space]] at that point.
[[File:x cubed plot.svg|thumb|150px|The plot of ''y''&nbsp;=&nbsp;''x''<sup>3</sup> with a saddle point at 0]]

In a domain of one dimension, a saddle point is a [[Point (geometry)|point]] which is both a [[stationary point]] and a [[Inflection point|point of inflection]]. Since it is a point of inflection, it is not a [[local extremum]].

== Saddle surface {{anchor|Surface}} ==
[[Image:HyperbolicParaboloid.svg|thumb|right|[[Hyperbolic paraboloid]]]]
[[Image:Ruled hyperboloid.jpg|thumb|right|A model of an [[elliptic hyperboloid]] of one sheet]]
[[Image:Monkey_saddle_surface.svg|thumb|right|A [[monkey saddle]]|300px]]

A '''saddle surface''' is a [[smooth surface]] containing one or more saddle points.

Classical examples of two-dimensional saddle surfaces in the [[Euclidean space]] are second order surfaces, the [[hyperbolic paraboloid]] <math>z=x^2-y^2</math> (which is often referred to as "''the'' saddle surface" or "the standard saddle surface") and the [[hyperboloid of one sheet]]. The [[Pringles]] potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.

Saddle surfaces have negative [[Gaussian curvature]] which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is the [[monkey saddle]].<ref>{{cite book |first = R. Creighton |last=Buck |author-link=Robert Creighton Buck |year=2003 |title = Advanced Calculus |location = Long Grove, IL |publisher=[[Waveland Press]] |edition=3rd |isbn = 1-57766-302-0 |page=160 |url = https://books.google.com/books?id=7cYQAAAAQBAJ&pg=PA160 }}</ref>

==Examples==
In a two-player [[zero-sum (game theory)|zero sum]] game defined on a continuous space, the [[Nash equilibrium|equilibrium]] point is a saddle point.

For a second-order linear autonomous system, a [[critical point (mathematics)|critical point]] is a saddle point if the [[Characteristic equation (calculus)|characteristic equation]] has one positive and one negative real [[Eigenvalues and eigenvectors|eigenvalue]].<ref>{{harvnb|von Petersdorff|2006}}</ref>

In optimization subject to equality constraints, the first-order conditions describe a saddle point of the [[Lagrange multiplier|Lagrangian]].

== Other uses ==
In [[dynamical systems]], if the dynamic is given by a [[differentiable map]] ''f'' then a point is hyperbolic if and only if the differential of ''&fnof;'' <sup>''n''</sup> (where ''n'' is the period of the point) has no eigenvalue on the (complex) [[unit circle]] when computed at the point. Then
a ''saddle point'' is a hyperbolic [[periodic point]] whose [[stable manifold|stable]] and [[unstable manifold]]s have a [[dimension]] that is not zero.

A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.

== See also ==
* [[Saddle-point method]] is an extension of [[Laplace's method]] for approximating integrals
* [[Maximum and minimum]]
* [[Derivative test]]
* [[Hyperbolic equilibrium point]]
* [[Hyperbolic geometry]]
* [[Minimax theorem]]
* [[Max–min inequality]]
* [[Mountain pass theorem]]

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
* {{citation |last1=Gray |first1=Lawrence F. |last2=Flanigan |first2=Francis J. |last3=Kazdan |first3=Jerry L. |last4=Frank |first4=David H. |last5=Fristedt |first5=Bert |title=Calculus two: linear and nonlinear functions |publisher=Springer-Verlag |location=Berlin |year=1990 |page=[https://archive.org/details/calculustwolinea00flan/page/375 375] |isbn=0-387-97388-5 |url-access=registration |url=https://archive.org/details/calculustwolinea00flan/page/375 }}
* {{citation |last1=Hilbert |first1=David |author1-link = David Hilbert |last2=Cohn-Vossen |first2=Stephan |author2-link = Stephan Cohn-Vossen |title = Geometry and the Imagination |publisher=[[Chelsea Publishing|Chelsea]] |location = New York, NY |edition=2nd |isbn = 978-0-8284-1087-8 |year=1952 }}
* {{citation |first=Tobias |last = von Petersdorff |chapter-url = https://www.math.umd.edu/~petersd/246/stab.html |chapter=Critical Points of Autonomous Systems |year=2006 |title = Differential Equations for Scientists and Engineers (Math 246 lecture notes) }}
* {{citation |last = Widder |first = D. V. |author-link=David Widder |title = Advanced calculus |publisher=Dover Publications |location = New York, NY |year=1989 |page=128 |isbn = 0-486-66103-2 }}
* {{citation |last = Agarwal |first = A. |title = Study on the Nash Equilibrium (Lecture Notes) |url = http://www.cse.iitd.ernet.in/~rahul/cs905/lecture3/nash1.html#SECTION00041000000000000000 }}
{{refend}}

== Further reading ==
* {{cite book
|last1 = Hilbert |first1 = David
|author1-link = David Hilbert
|last2 = Cohn-Vossen |first2 = Stephan
|title = Geometry and the Imagination
|url = https://archive.org/details/geometryimaginat00davi_0 |url-access = registration |edition = 2nd 
|year = 1952
|publisher = Chelsea
|isbn = 0-8284-1087-9
}}

== External links ==
* {{Commons category-inline|Saddle point}}

{{DEFAULTSORT:Saddle Point}}
[[Category:Differential geometry of surfaces]]
[[Category:Multivariable calculus]]
[[Category:Stability theory]]
[[Category:Analytic geometry]]