SADDLE POINT

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. It is most frequently used at critical points.
The term comes from the two dimension picture of the indefinite quadratic form $x^2\; -\; y^2$, which curves up on the ''x''-axis and down on the ''y''-axis, like a saddle (see below); one dimensional saddle points do not look like saddles though.

In one dimension, 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.

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.

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
:$x^2\; -\; y^2,,$
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.

Contents

See also

See also

★ Hyperbolic point, a special case of a saddle point.

★ Saddle-point method

★ Stationary point

★ Extremum'

★ First derivative test

★ Second derivative test

★ Higher order derivative test

★ Saddle surface