A new algorithm using L-BFGS for two-parameter eigenvalue problems from Lamé's system and simulation

We deal with the challenges and solutions for two-parameter eigenvalue problems (TPEPs) involving large-scale various dense coefficient matrices using several numerical methods. We propose a new method, via fused parameter optimization ($fusedparopt$) algorithm using limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) having several variations, for TPEPs. We combine the advantages of the tensor Rayleigh quotient (RQ), Newton (N) and L-BFGS methods, while avoiding the disadvantages of each method. They are designed for certain TPEPs having real eigenvalue tuples. We test the performance of our algorithm and compare them with state-of-art algorithms such as twopareigs from MultiParEig toolbox in Matlab, tensor Rayleigh quotient-Newton (RQ\_N) and L-BFGS alone, by using the coefficient matrices coming from Lamé system and simulations via randomly generated matrices. We also obtain convergence diagrams for the $fusedparopt\_LBFGS$ to understand the convergence behavior for the number of iterations and computational times via Monte Carlo simulation. We observe that our algorithms can reduce computation time, diminish divergence problems, and give more stable solutions for our data set including various matrices for TPEPs. To the best of our knowledge, we perform the first study including $fusedparopt\_LBFGS$ method in this type of eigenvalue problem.