Saddle point

A saddle point on the graph of z=x²−y² (in red)

Saddle point between two hills (the intersection of the figure-eight $z$-contour)

In mathematics, a saddle point is a point in the range 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.

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 at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function $z=x^{2}-y^{2}$ at the stationary point $(0,0)$ is the matrix

${\begin{bmatrix}2&0\\0&-2\\\end{bmatrix}}$

which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point $(0,0)$ is a saddle point for the function $z=x^{4}-y^{4},$ 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.

The plot of y = x³ with a saddle point at 0

In 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.

Other uses

In dynamical systems, a saddle point is a periodic point whose stable and unstable manifolds have a dimension that is not zero. 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.

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

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.

For a second-order linear autonomous systems, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.

See also

• Saddle-point method is an extension of Laplace's method for approximating integrals
• Extremum
• First derivative test
• Second derivative test
• Higher-order derivative test
• Saddle surface
• Hyperbolic equilibrium point
• Sion's minimax theorem
• Mountain pass
• Max–min inequality

References

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