Gradskillnaden

Gradskillnaden is a term used in mathematical contexts to denote the difference between the gradient vectors of a differentiable scalar function at two points, x and y. It is defined as Δ∇f = ∇f(y) − ∇f(x).

If f is twice differentiable, the gradient difference is governed by the Hessian. For small steps h = y − x, we have ∇f(y) ≈ ∇f(x) + Hf(x) h, so Δ∇f ≈ Hf(x) h. If the gradient is L-Lipschitz, then ||∇f(y) − ∇f(x)|| ≤ L ||y − x||.

Example: take f(z) = 1/2 z^T A z, where A is a constant matrix. Then ∇f(z) = A z, and the gradient difference is Δ∇f = A(y − x). This illustrates how the gradient change is governed by the Hessian (or by A in this quadratic case).

Uses and context: gradskillnaden is relevant in optimization and numerical analysis for analyzing local curvature, stability of gradient-based methods, and the accuracy of finite-difference gradient estimates. It provides a concise way to describe how rapidly the slope of f changes between two points.

Notes and variation: gradskillnaden is not a universally standardized formal concept in all mathematical texts; it is often used descriptively in Swedish-language materials to discuss the change in gradient across a region. When encountered, its precise meaning is typically clarified by the surrounding context and conventions for gradients and Hessians.

See also: gradient, Hessian, Lipschitz continuity, finite difference method.