Applying Taylor Mode AD in Nonlinear Equations, ODEs and More

Explore advanced applications of Taylor mode automatic differentiation in scientific computing through TaylorDiff.jl, a Julia package that transforms first-order AD rules to higher-order AD rules using symbolic-numeric techniques. Learn how this powerful tool enhances numerical methods for solving nonlinear equations and ordinary differential equations with improved efficiency, stability, and scalability. Discover implementations of Householder's method and Halley's method for large-scale nonlinear equation solving, where Halley's method achieves cubic convergence using second-order directional derivatives computed efficiently through Taylor-mode AD, avoiding costly full Hessian evaluations. Examine performance comparisons showing how these higher-order methods outperform traditional Newton's method in both dense and sparse Jacobian settings, particularly for ill-conditioned systems arising from PDE discretizations. Understand how Taylor-mode AD reduces computational time while maintaining accuracy, and see performance improvements when applying Halley's method to nonlinear equation solving steps within implicit ODE solvers. Investigate the extension of TaylorDiff.jl to develop efficient implicit ODE solvers for stiff systems, incorporating advanced features like adaptive order, adaptive step size, and solution extrapolation by leveraging the fact that local ODE solutions can be obtained through Taylor series expansion in time.