Location of stable and unstable equilibrium configurations using a model trust region quasi-Newton method and tunnelling

A hybrid method consists of a quasi-Newton method and a homotopy method for locating multiple equilibrium configurations has been proposed recently. The hybrid method combined the efficiency of a quasi-Newton method capable of locating stable and unstable equilibrium solutions with a robust homotopy method capable of tracking equilibrium paths with turning points and exploiting sparsity of the Jacobian matrix at the same time. A quasi-Newton method in conjunction with a deflation technique is proposed here as an alternative to the hybrid method. The proposed method not only exploits sparsity and symmetry, but also represents an improvement in efficiency. Limit points and nearby equilibrium solutions, either stable or unstable, can be accurately located with the use of a modified pseudoinverse based on the singular value decomposition. This pseudoinverse modification destroys the Jacobian matrix sparsity, but is invoked only rarely (at limit arid bifurcation points where the Jacobian matrix is singular).