  In this pedagogical article, we present a simple direct matrix method for
analytically computing the Jacobian of nonlinear algebraic equations that arise
from the discretization of nonlinear integro-differential equations. The method
is based on a formulation of the discretized equations in vector form using
only matrix-vector products and component-wise operations. By applying simple
matrix-based differentiation rules, the matrix form of the analytical Jacobian
can be calculated with little more difficulty than that required when computing
derivatives in single-variable calculus. After describing the direct matrix
method, we present numerical experiments demonstrating the computational
performance of the method, discuss its connection to the Newton-Kantorovich
method, and apply it to illustrative 1D and 2D example problems. MATLAB code is
provided to demonstrate the low code complexity required by the method.
