Topics in matrix analysis
An analytical and numerical study of the two-dimensional Bratu equation
Journal of Scientific Computing
A quasi-Newton method for elliptic boundary value problems
SIAM Journal on Numerical Analysis
Applied numerical linear algebra
Applied numerical linear algebra
Iterative solution of nonlinear equations in several variables
Iterative solution of nonlinear equations in several variables
Spectral methods in MatLab
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
Using AD to solve BVPs in MATLAB
ACM Transactions on Mathematical Software (TOMS)
A Pseudospectral Fictitious Point Method for High Order Initial-Boundary Value Problems
SIAM Journal on Scientific Computing
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
Computers & Mathematics with Applications
Computers & Mathematics with Applications
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In this 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 required to compute 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 from electrochemical transport.