Learning saddle points with powers
Build and perturb classic saddle-point examples. Drag the coefficients continuously, and choose integer powers to see how the local geometry and gradient-descent field change.
Function value
f(0,0) = 0
Gradient at (0,0)
∇f = (0, 0)
Hessian at (0,0)
Origin regularity
Gradient and Hessian are finite
Surface plot
f(x, y) = x² − y²
x, y ∈ [−2, 2] · z ∈ [−8, 8]
Contours and gradient-descent field
f(x, y) = x² − y²
arrows point along −∇f; length is log-scaled
Powers are restricted to integers so the objective remains real across the full x–y plane. At power 0, this applet uses t⁰ = 1, including at t = 0.