Science Fair Project Encyclopedia
In mathematics, a saddle point is a point of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction (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
- 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.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details