Table of Links
2. Mathematical Description and 2.1. Numerical Algorithms for Nonlinear Equations
2.4. Matrix Coloring & Sparse Automatic Differentiation
3.1. Composable Building Blocks
3.2. Smart PolyAlgortihm Defaults
3.3. Non-Allocating Static Algorithms inside GPU Kernels
3.4. Automatic Sparsity Exploitation
3.5. Generalized Jacobian-Free Nonlinear Solvers using Krylov Methods
4. Results and 4.1. Robustness on 23 Test Problems
4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model
4.3. Large Ill-Conditioned Nonlinear Brusselator System
2. Mathematical Description
This section introduces the mathematical framework for numerically solving nonlinear problems and demonstrates the built-in support for such problems in NonlinearSolve.jl. A nonlinear problem is defined as:
where J๐ is the Jacobian of ๐ (๐ข, ๐) with respect to ๐ข, evaluated at ๐ข๐. This method exhibits rapid convergence [23, Theorem 11.2] when the initial guess is sufficiently close to a root. Furthermore, it requires only the function and its Jacobian, making it computationally efficient for many practical applications. Halleyโs method enhances the Newton-Raphson method, leveraging information from the second total derivative of the function to achieve cubic convergence. It refines the initial guess ๐ข0 using:
This method provides higher convergence orders without reliance on higher-order derivatives. [25] summarizes other multi-step schemes that provide higher-order convergence using only first-order derivatives. These methods are local algorithms, and their convergence relies on having a good initial guess. We will discuss some techniques to facilitate the global convergence of these methods in the following section.
This paper is available on arxiv under CC BY 4.0 DEED license.
[5] For Halleyโs method, we additionally assume twice-differentiability.
Authors:
(1) AVIK PAL, CSAIL MIT, Cambridge, MA;
(2) FLEMMING HOLTORF;
(3) AXEL LARSSON;
(4) TORKEL LOMAN;
(5) UTKARSH;
(6) FRANK SCHรFER;
(7) QINGYU QU;
(8) ALAN EDELMAN;
(9) CHRIS RACKAUCKAS, CSAIL MIT, Cambridge, MA.