
In mathematics, a saddle point is a point in the domain 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 realvalued 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 $z=x^2y^2$ at the stationary point $(0,\; 0)$ is the matrix : $\backslash begin$ 2 & 0\\ 0 & 2 \\ \end 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 $z=x^4y^4,$ 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. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「saddle point」の詳細全文を読む スポンサード リンク
