View source for Saddle point

{{short description|Critical point on a surface graph which is not a local extremum}}

[[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}}}} (hyperbolic paraboloid)]]
[[File:Saddle_Point_between_maxima.svg|thumb|300px|right|Saddle point between two hills (the intersection of the figure-eight {{mvar|z}}-contour)]]

In [[Mathematics|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|graph of a function]] where the [[Slope|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. |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 at a [[Maxima and minima|relative 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.
 
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 or a [[Earth:Mountain pass|mountain pass]] between two peaks forming a [[Earth:Saddle (landform)|landform saddle]]. In terms of [[Contour line|contour line]]s, a saddle point in two dimensions gives rise to a contour map with a pair of lines intersecting at the point.  Such intersections are rare in actual 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: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|curve]], [[Surface (mathematics)|surface]] or [[Hypersurface|hypersurface]]) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the [[Tangent space|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|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|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|monkey saddle]].<ref>{{cite book |first = R. Creighton |last=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 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 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|unit circle]] when computed at the point. Then
a ''saddle point'' is a hyperbolic [[Periodic point|periodic point]] whose [[Stable manifold|stable]] and unstable manifolds have a [[Physics:Dimension|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]]
* [[Monkey saddle]]
* [[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 |last2=Cohn-Vossen |first2=Stephan |title = Geometry and the Imagination |publisher=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. |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
|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 ==

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

{{Sourceattribution|Saddle point}}