How is the Jacobian matrix used to solve systems of nonlinear equations?

In numerical analysis, the Jacobian matrix serves as a multidimensional generalization of the derivative, essential for solving systems of nonlinear equations through Newton’s method.1

To solve a system of $n$ nonlinear equations denoted by $𝐟(𝐱)=0$, where $𝐱$ is a vector of $n$ unknowns, the method generates a sequence of approximations ${𝐱}^{(k+1)}$ using the following iterative process:1

• Linearization: At each iteration $k$, the nonlinear system is approximated by a linear system of order $n$: ${𝐟}^{\prime }({𝐱}^{(k)}){\delta }^{(k)}=-𝐟({𝐱}^{(k)})$, where ${𝐟}^{\prime }(𝐱)$ is the Jacobian matrix.1
• Update: The next approximation is calculated as ${𝐱}^{(k+1)}={𝐱}^{(k)}+{\delta }^{(k)}$.1

By solving this linear system for the correction vector ${\delta }^{(k)}$, the method effectively uses the Jacobian to determine the direction and magnitude of the step toward the root of the nonlinear system. This approach is a primary example of treating nonlinear problems by reducing them to a sequence of linear problems.1