In mathematics, a **saddle point** or **minimax point**^{ [1] } is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function.^{ [2] } 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 relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function has a critical point at 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 -direction.

- Mathematical discussion
- Saddle surface
- Examples
- Other uses
- See also
- References
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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 riding saddle or a mountain pass between two peaks forming a landform saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour graph or trace in which the contour corresponding to the saddle point's value appears to intersect itself.

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 indefinite, then that point is a saddle point. For example, the Hessian matrix of the function at the stationary point is the matrix

which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point is a saddle point for the function 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 is a curve, surface or 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 at that point.

In a domain of one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.

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 (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.^{ [3] }

In a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point.

For a second-order linear autonomous system, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.^{ [4] }

In optimization subject to equality constraints, the first-order conditions describe a saddle point of the Lagrangian.

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 *ƒ*^{n} (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 and unstable manifolds 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.

- Saddle-point method is an extension of Laplace's method for approximating integrals
- Extremum
- Derivative test
- Hyperbolic equilibrium point
- Minimax theorem
- Max–min inequality
- Monkey saddle
- Mountain pass theorem

In mathematics, an **integral** assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called **integration**. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

In mathematics, **differential calculus** is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

**Mathematical optimization** or **mathematical programming** is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In geometry, a **paraboloid** is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.

In mathematical optimization, the **method of Lagrange multipliers** is a strategy for finding the local maxima and minima of a function subject to equality constraints. It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the **Lagrangian function**.

In vector calculus, the **Jacobian matrix** of a vector-valued function in several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the **Jacobian determinant**. Both the matrix and the determinant are often referred to simply as the **Jacobian** in literature.

In differential geometry, the **Gaussian curvature** or **Gauss curvature**Κ of a surface at a point is the product of the principal curvatures, *κ*_{1} and *κ*_{2}, at the given point:

In mathematical analysis, the **maxima** and **minima** of a function, known collectively as **extrema**, are the largest and smallest value of the function, either within a given range, or on the entire domain. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

In differential calculus and differential geometry, an **inflection point**, **point of inflection**, **flex**, or **inflection** is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave to convex, or vice versa.

In mathematics, the **Hessian matrix** or **Hessian** is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".

In mathematics, the **mean curvature** of a surface is an *extrinsic* measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.

In calculus, a **derivative test** uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function.

In mathematics, particularly in calculus, a **stationary point** of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing.

In calculus, **Newton's method** is an iterative method for finding the roots of a differentiable function *F*, which are solutions to the equation *F* (*x*) = 0. As such, Newton's method can be applied to the derivative *f* ′ of a twice-differentiable function *f* to find the roots of the derivative, also known as the critical points of *f*. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function *f*.

**Critical point** is a wide term used in many branches of mathematics.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

In mathematics, the **second partial derivative test** is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.

In the study of dynamical systems, a **hyperbolic equilibrium point** or **hyperbolic fixed point** is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz notes that "hyperbolic is an unfortunate name—it sounds like it should mean 'saddle point'—but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably

In mathematics, the **method of steepest descent** or **saddle-point method** is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point, in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.

*Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.*

- ↑ Howard Anton, Irl Bivens, Stephen Davis (2002):
*Calculus, Multivariable Version*, p. 844. - ↑ Chiang, Alpha C. (1984).
*Fundamental Methods of Mathematical Economics*(3rd ed.). New York: McGraw-Hill. p. 312. ISBN 0-07-010813-7. - ↑ Buck, R. Creighton (2003).
*Advanced Calculus*(3rd ed.). Long Grove, IL: Waveland Press. p. 160. ISBN 1-57766-302-0. - ↑ von Petersdorff 2006

- Gray, Lawrence F.; Flanigan, Francis J.; Kazdan, Jerry L.; Frank, David H.; Fristedt, Bert (1990),
*Calculus two: linear and nonlinear functions*, Berlin: Springer-Verlag, p. 375, ISBN 0-387-97388-5 - Hilbert, David; Cohn-Vossen, Stephan (1952),
*Geometry and the Imagination*(2nd ed.), New York, NY: Chelsea, ISBN 978-0-8284-1087-8 - von Petersdorff, Tobias (2006), "Critical Points of Autonomous Systems",
*Differential Equations for Scientists and Engineers (Math 246 lecture notes)* - Widder, D. V. (1989),
*Advanced calculus*, New York, NY: Dover Publications, p. 128, ISBN 0-486-66103-2 - Agarwal, A.,
*Study on the Nash Equilibrium (Lecture Notes)*

- Hilbert, David; Cohn-Vossen, Stephan (1952).
*Geometry and the Imagination*(2nd ed.). Chelsea. ISBN 0-8284-1087-9.

- Media related to Saddle point at Wikimedia Commons

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