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| [[File:Saddle point.png|thumb|right|200px|A saddle point on the graph of z=x<sup>2</sup>−y<sup>2</sup> (in red)]] | | Catrina Le is what's designed on her birth records though she doesn't truly like being called such as that. Her job is probably a [http://Www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=cashier cashier] but very quickly her husband and the actual will start their own [http://Imgur.com/hot?q=business business]. To drive is something that she's been doing the population. For years she's been living located in Vermont. Go to her website to find out doors more: http://prometeu.net<br><br> |
| [[File:Saddle pt.jpg|thumb|150px|right|Saddle point between two hills (the intersection of the figure-eight <math>z</math>-contour)]] | |
| In [[mathematics]], a '''saddle point''' is a point in the [[domain (mathematics)|domain]] of a [[function (mathematics)|function]] that is a [[stationary point]] but not a [[local extremum]]. The name derives from the fact that the prototypical example in two dimensions is a [[surface]] that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a [[saddle]] or a [[mountain pass]]. In terms of [[contour line]]s, a saddle point in two dimensions gives rise to a contour that appears to intersect itself.
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| == Mathematical Discussion ==
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| 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 matrix]], 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>(0, 0)</math> is the matrix
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| : <math>\begin{bmatrix}
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| 2 & 0\\
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| 0 & -2 \\
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| \end{bmatrix}
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| </math>
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| which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point <math>(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.
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| In the most general terms, a '''saddle point''' for a [[smooth function]] (whose [[graph of a function|graph]] is a [[curve]], [[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.
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| [[File:x cubed plot.svg|thumb|150px|The plot of ''y'' = ''x''<sup>3</sup> with a saddle point at 0]]
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| In 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]].
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| == Other uses ==
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| In [[dynamical systems]], a ''saddle point'' is a [[periodic point]] whose [[stable manifold|stable]] and [[unstable manifold]]s have a [[dimension]] that is not zero. If the dynamic is given by a [[differentiable map]] ''f'' then a point is hyperbolic if and only if the differential of ''ƒ'' <sup>''n''</sup> (where ''n'' is the period of the point) has no eigenvalue on the (complex) [[unit circle]] when computed at the point.
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| 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.
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| A saddle point is an element of the matrix which is both the largest element in its column and the smallest element in its row.
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| For a second-order linear autonomous systems, 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>
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| == See also ==
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| * [[Saddle-point method]] is an extension of [[Laplace's method]] for approximating integrals
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| * [[Extremum]]
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| * [[First derivative test]]
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| * [[Second derivative test]]
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| * [[Higher-order derivative test]]
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| * [[Saddle surface]]
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| * [[Hyperbolic equilibrium point]]
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| * [[Sion's minimax theorem]]
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| * [[Mountain pass]]
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| * [[Max–min inequality]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{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 |pages= page 375|isbn=0-387-97388-5 |oclc= |doi=}}
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| * {{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 | location=New York | edition=2nd | isbn=978-0-8284-1087-8 | year=1952 }}
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| * {{citation|first=Tobias|last=von Petersdorff|url=http://www2.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)}}
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| * {{citation |last=Widder|first=D. V. |title=Advanced calculus |publisher=Dover Publications |location=New York |year=1989 |pages=page 128 |isbn=0-486-66103-2 |oclc= |doi=}}
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| * {{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}}
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| {{DEFAULTSORT:Saddle Point}}
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Multivariable calculus]]
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| [[Category:Stability theory]]
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| [[Category:Analytic geometry]]
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Catrina Le is what's designed on her birth records though she doesn't truly like being called such as that. Her job is probably a cashier but very quickly her husband and the actual will start their own business. To drive is something that she's been doing the population. For years she's been living located in Vermont. Go to her website to find out doors more: http://prometeu.net
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