Glossary

Jacobian matrix

The matrix of partial derivatives that generalizes f′f' to systems of several variables.

Definition

For F:Rn→RnF:\mathbb{R}^n\to\mathbb{R}^n it is JF(x)ij=∂Fi/∂xjJ_F(x)_{ij}=\partial F_i/\partial x_j. In Newton's method for systems it replaces the scalar derivative: each step solves the linear system JF(xk) Δx=−F(xk)J_F(x_k)\,\Delta x=-F(x_k) and updates xk+1=xk+Δxx_{k+1}=x_k+\Delta x.

How it is used

Its cost dominates Newton's method for systems: evaluating n2n^2 partials and factorizing JJ at each step is what quasi-Newton variants try to make cheaper.