Accelerating the convergence of Newton's method for the Yang-Baxter like matrix equation

Area of Science:

• Numerical analysis
• Matrix theory
• Computational mathematics

Background:

• The Yang-Baxter matrix equation is crucial in quantum groups and integrable systems.
• Efficient numerical methods are needed for nontrivial solutions.
• Newton's method offers a framework but requires convergence enhancement.

Purpose of the Study:

• To apply and compare exact line search and successive over-relaxation for Newton's method.
• To analyze the condition numbers (normwise, mixed, componentwise) of the Yang-Baxter equation.
• To assess the numerical stability and convergence properties of the proposed methods.

Main Methods:

• Implementation of exact line search with Newton's method.
• Application of successive over-relaxation to Newton's method.
• Derivation and computation of normwise, mixed, and componentwise condition numbers.

Main Results:

• Exact line search significantly accelerates convergence compared to successive over-relaxation.
• The Yang-Baxter equation demonstrates good conditioning with mixed and componentwise measures (close to one).
• Normwise condition numbers indicate a higher sensitivity to perturbations.

Conclusions:

• Exact line search is a superior technique for accelerating Newton's method for the Yang-Baxter equation.
• The equation is generally well-posed numerically, but normwise sensitivity should be considered.
• The findings provide valuable insights for solving complex matrix equations in scientific computing.