  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.
