The hypersecant Jacobian approximation for quasi-Newton solves of sparse nonlinear systems

A new Jacobian approximation is developed for use in quasi-Newton methods for solving systems of nonlinear equations. The new hypersecant Jacobian approximation is intended for the special case where the evaluation of the functions whose roots are sought dominates the computation time, and additionally the Jacobian is sparse. One example of such a case is the solution of the discretized transport equation to calculate particle and energy fluxes in a fusion plasma. The hypersecant approximation of the Jacobian is calculated using function values from previous Newton iterations, similarly to the Broyden approximation. Unlike Broyden, the hypersecant Jacobian converges to the finite-difference approximation of the Jacobian. The calculation of the hypersecant Jacobian elements requires solving small, dense linear systems, where the coefficient matrices can be ill-conditioned or even exactly singular. Singular-value decomposition (SVD) is therefore used. Convergence comparisons of the hypersecant method, the standard Broyden method, and the colored finite differencing of the PETSc SNES solver are presented.