Saddle point

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Plot of y = x³ with a saddle-point at (0,0).

In mathematics, a saddle point is a point of a function (of one or more variables) which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.

For a function of a single variable, such a point is one where the first derivative is zero, and the second derivative changes sign. For example, the function y = x³ has such a point at the origin.

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

For a function of two or more variables, the surface at a saddle-point resembles a saddle that curves up in one or more directions, and curves down in one or more other directions (like a mountain pass). In terms of contour lines, a saddle point can be recognised, in general, by a contour that appears to intersect itself. For example, two hills separated by a high pass will show up a saddle point, at the top of the pass, like a figure-eight contour line.

Saddle point in the graph of z=x²-y²

More formally, given a real function F(x,y) of two real variables, the Hessian matrix H of F is a 2×2 matrix. If it is indefinite (neither H nor −H is positive definite) then in general it can be reduced to the Hessian of the function

x² − y²,

at the point (0,0). This function has a saddle point there, curving up along the line y = 0 and down along the line x = 0.

In fact if H is a non-singular matrix (general case) and F is smooth enough, this is the correct local model for a stationary point of F that is not a local maximum nor a local minimum. If H has rank < 2 one cannot be certain in the same way about the local behaviour.

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

A saddle point is an element of the matrix which is both the smallest element in its row and the largest element in its column is called saddle point.

Saddle point

See also