Crash Course in Scientific Computing

XVI. Higher Dimensions

We have dealt with so far one-dimensional problems in one variable. Today, we will extend some of the techniques we have described in this case to their higher-dimensional counterpart, starting with root-finding, i.e., we generalize Newton-Raphson to $n$ equations in $n$ unknowns. Letting $J(\mathbf{x})$ be the Jacobian of the vector-valued function $f(\mathbf{x})$, Newton-Raphson for $f(\mathbf{x})=0$ reads simply:

$$\mathbf x_{i+1}=\mathbf x_i-\mathbf J(\mathbf x_i)^{-1}\mathbf f(\mathbf x_i)$$

where $\mathbf J(\mathbf x_i)^{-1}$ is the inverse matrix of the Jacobian.