In
mathematics, a saddle point or minimax point is a
point on the
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
of the
graph of a function where the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
s (derivatives) in
orthogonal directions are all zero (a
critical point), but which is not a
local extremum of the function. An example of a saddle point is when there is a critical point with a relative
minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
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.
The name derives from the fact that the prototypical example in two dimensions is a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding
saddle
The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not k ...
or a
mountain pass between two peaks forming a
landform saddle. In terms of
contour line
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional grap ...
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.
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
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 ...
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 In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed b ...
, which is not indefinite.
In the most general terms, a saddle point for a
smooth function (whose
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
,
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
or
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
) 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
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at that point.
In a domain of one dimension, a saddle point is a
point which is both a
stationary point
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" in ...
and a
point of inflection
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
. Since it is a point of inflection, it is not a
local extremum.
Saddle surface
A saddle surface is a
smooth surface
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
containing one or more saddle points.
Classical examples of two-dimensional saddle surfaces in the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
. 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
In mathematics, the monkey saddle is the surface defined by the equation
: z = x^3 - 3xy^2, \,
or in cylindrical coordinates
:z = \rho^3 \cos(3\varphi).
It belongs to the class of saddle surfaces, and its name derives from the observation tha ...
.
Examples
In a two-player
zero sum
Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is e ...
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.
In optimization subject to equality constraints, the first-order conditions describe a saddle point of the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
.
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 ''ƒ''
''n'' (where ''n'' is the period of the point) has no eigenvalue on the (complex)
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
when computed at the point. Then
a ''saddle point'' is a hyperbolic
periodic point In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
whose
stable and
unstable manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repe ...
s have a
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form
:\int_a^b e^ \, dx,
where f(x) is a twice-differentiable function, ''M'' is a large number, and the endpoints ''a'' an ...
for approximating integrals
*
Extremum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
*
Derivative test
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 abo ...
*
Hyperbolic equilibrium 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 hyperbo ...
*
Minimax theorem
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality.
The first theorem in this sense is von Neumann's minimax theorem from 1928, which was c ...
*
Max–min inequality In mathematics, the max–min inequality is as follows:
:For any function \ f : Z \times W \to \mathbb\ ,
::
\sup_ \inf_ f(z, w) \leq \inf_ \sup_ f(z, w)\ .
When equality holds one says that , , and satisfies a strong max–min property (or a ...
*
Monkey saddle
In mathematics, the monkey saddle is the surface defined by the equation
: z = x^3 - 3xy^2, \,
or in cylindrical coordinates
:z = \rho^3 \cos(3\varphi).
It belongs to the class of saddle surfaces, and its name derives from the observation tha ...
*
Mountain pass theorem
References
Citations
Sources
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Further reading
*
External links
*
{{DEFAULTSORT:Saddle Point
Differential geometry of surfaces
Multivariable calculus
Stability theory
Analytic geometry