In mathematics, a saddle point is a point in the domain of a 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 lines, a saddle point in two dimensions gives rise to a contour that 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 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 is an element of the matrix 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
- First derivative test
- Second derivative test
- Higher-order derivative test
- Saddle surface
- Hyperbolic equilibrium point
- Sion's minimax theorem
- Mountain pass
- Max–min inequality
- Gray, Lawrence F.; Flanigan, Francis J.; Kazdan, Jerry L.; Frank, David H; Fristedt, Bert (1990), Calculus two: linear and nonlinear functions, Berlin: Springer-Verlag, pp. page 375, ISBN 0-387-97388-5
- Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: 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: Dover Publications, pp. page 128, ISBN 0-486-66103-2
- Agarwal, A., Study on the Nash Equilibrium (Lecture Notes)