Saddlepunkter

Saddlepunkter, or saddle points, are critical points of a real-valued function f: R^n -> R where the gradient vanishes but the point is not a local extremum. In the neighborhood of a saddle point, f increases along some directions and decreases along others, producing a saddle-shaped surface in two dimensions.

To identify saddlepoints, first solve ∇f(x0) = 0 for candidate critical points. Then examine the Hessian matrix H at x0. If H has both positive and negative eigenvalues (indefinite), x0 is a saddle point. If H is positive definite, x0 is a local minimum; if negative definite, a local maximum. If H is semidefinite, higher-order derivatives or other criteria are needed to determine the nature.

An example is f(x,y) = x^2 − y^2, which has a saddle point at (0,0). Here ∇f = (2x, −2y) vanishes at (0,0) and the Hessian diag(2, −2) is indefinite, reflecting opposite curvatures along the x and y axes.

Saddle points arise in multiple fields. In optimization and numerical analysis they pose challenges for algorithms that seek minima or maxima. In game theory and optimization, the concept relates to minimax equilibria, where a saddle point represents a point that minimizes over one variable and maximizes over another. In dynamical systems, saddle points are equilibria with both stable and unstable directions.