Rosenbrock explorer
f(x,y) = (1−x)² + 100(y−x²)² · min at (1, 1)
Contour · gradient field
Drag the point to explore the surface — gradient and Newton step update live
Position
x1.000
y0.000
f(x, y)100.00
log₁₀ f2.00
Gradient
∂f/∂x400.0
∂f/∂y−200.0
‖∇f‖447.2
Hessian eigenvalues
λ₁ (min)86.7
λ₂ (max)1315.3
κ(H) = λ₂/λ₁15.2
Newton step
p₁0.000
p₂1.000
f(x+p)0.00
Display
Gradient field
Newton step
Steepest descent
−∇f (steepest descent)
Newton step p*
Minimum (1, 1)
Gradient arrows are orthogonal to contour lines. Their length is proportional to ‖∇f‖ — watch them shrink near the valley floor.
The condition number κ(H) measures how elongated the local quadratic bowl is. High κ means steepest descent zigzags; Newton's method doesn't care.
Compare the two arrows. Steepest descent points downhill but overshoots across the valley. Newton aims straight for the basin floor.

From Rosenbrock Explorer, part 1 of a series on trust regions.