Critical point on a surface graph which is not a local extremum
This article is about the mathematical property. For the peninsula in the Antarctic, see Saddle Point. For the type of landform and general uses of the word "saddle" as a technical term, see Saddle (landform).
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.
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. In terms of contour lines, 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.
^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.
In mathematics, a saddlepoint or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions...
SaddlePoint (53°1′S 73°29′E / 53.017°S 73.483°E / -53.017; 73.483) is a rocky point separating Corinthian Bay and Mechanics Bay on the north coast...
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...
include: Homoclinic bifurcation in which a limit cycle collides with a saddlepoint. Homoclinic bifurcations can occur supercritically or subcritically....
Stirling's Formula is considered one of the earliest examples of the saddle-point method. In 1990, Philippe Flajolet and Andrew Odlyzko developed the theory...
a stationary point that is not a local extremum is called a saddlepoint. An example of a stationary point of inflection is the point (0, 0) on the graph...
multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddlepoint. Suppose that f(x, y) is a differentiable real...
A 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...
the mass of the counter-Earth. The Sun–Earth L3, however, is a weak saddlepoint and exponentially unstable with time constant of roughly 150 years. Moreover...
least one saddlepoint. The historical meaning is a synonym for a gable roof particularly a dual-pitched roof on a tower, also called a pack-saddle roof....
one for the tail. The point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} on the monkey saddle corresponds to a degenerate critical point of the function z ( x...
(mathematics) Fermat's theorem Derivative test Fixed point (mathematics) Saddlepoint Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics...
(0)} is very close to a saddlepoint. The body would linger near the saddlepoint, then rapidly move to the other saddlepoint, near ω ( T / 2 ) {\displaystyle...
use of a gap, saddle, col or notch. A topographic saddle is analogous to the mathematical concept of a saddle surface, with a saddlepoint marking the minimum...
sufficient, conditions for a local maximum, because of the possibility of a saddlepoint. For use of these conditions to solve for a maximum, the function z must...
like it should mean 'saddlepoint'—but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably A stable manifold...
critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddlepoint). However, more can be said from the point of view...
solution corresponding to the original constrained optimization is always a saddlepoint of the Lagrangian function, which can be identified among the stationary...
Minmax (sometimes Minimax, MM or saddlepoint) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy...
points of a function and determine whether each point is a local maximum, a local minimum, or a saddlepoint. Derivative tests can also give information about...
example of a degenerate critical point is the origin of the monkey saddle. The index of a non-degenerate critical point p {\displaystyle p} of f {\displaystyle...
Sidesaddle riding is a form of equestrianism that uses a type of saddle which allows riders, generally female, to sit aside rather than astride an equine...