Saddle PointIt refers to a stationary point that is not a local extreme point.

In general, a saddle point of a smooth function (curve, surface, or hypersurface) is a point whose neighborhood of curves, surfaces, or hypersurfaces lies on different sides of the tangent line to that point.

Definition of saddle points in different fields

In differential equations, a singularity that is stable in one direction but unstable in another direction is called a saddle point.

In functional theory, a critical point that is neither a maximum point nor a minimum point is called a saddle point.

In a matrix, a number that is the maximum value in its row and the minimum value in its column is called a saddle point.

In physics, it is broader and refers to a point that is a maximum in one direction and a minimum in another direction.

Saddle point identification

As shown in the figure below, the term saddle point comes from the two-dimensional graph of the indefinite quadratic form z = x^2 – y^2, which looks like a saddle: it curves upward in the x-axis direction and downward in the y-axis direction.

For a function with only one variable, the first derivative of the function at the saddle point is equal to zero, and the second derivative changes signs. For example, the function y = x^3 has a saddle point at the origin.

A simple way to check whether a stationary point of a two-variable real function F(x,y) is a saddle point is to compute the Hessian matrix of the function at this point: if the matrix is indefinite, then the point is a saddle point.

Related words: stationary point, inflection point, extreme value.

References:

【1】https://www.csdn.net/article/2015-11-05/2826132

【2】https://zh.wikipedia.org/wiki/saddle point