Saddle point

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In the most general terms, a saddle point for a smooth function (curve, surface or hypersurface) is a point such that the curve/surface/etc. in the neighborhood of this point lies on different sides of the tangent at this point. In certain contexts the definition may vary.

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

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.