8. Quasi-Newton-Raphson Methods

TL;DR

Introducing Broyden's method, damped Newton-Raphson method, and finite difference approximation for solving nonlinear equations.

Transcript

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Key Insights

• 😜 Broyden's method approximates the Jacobian matrix through a rank one update, providing a simpler but effective solution.
• 🧑‍🏭 The damped Newton-Raphson method introduces a damping factor to ensure a reduction in the function value during each iteration.
• ❓ Finite difference approximation offers an alternative to directly calculating the Jacobian matrix, reducing computational complexity.

Summary & Key Takeaways

• Broyden's method is a rank one update approximation for the Jacobian matrix, allowing for an iterative approach to solving nonlinear equations.

• The damped Newton-Raphson method introduces a damping factor to the Newton-Raphson method, improving convergence and ensuring a reduction in the function value.

• Finite difference approximation is a method of approximating the Jacobian matrix by calculating the difference in function values for small perturbations in the input vector.