QUASI-NEWTON, OR MODIFICATION METHODS**This work has done while the author was Visiting Research Professor, Department of Computer Science, Cornell University, and was supported by NSF Grant #GJ-27528.

Publisher Summary This chapter discusses modification methods. Of the many algorithms in existence for solving a large variety of problems, most of the successful ones require the calculation of a sequence {xk} together with the associated sequences {fk} and {Jk}, where Jk is the Jacobian of f evaluated at xk . Examples of these are Newton's method for nonlinear equations, the Gauss–Newton method for least-squares problems, and the various methods inspired by the work of Levenberg and Marquardt. In all of these methods, it is frequently quite costly to compute Jk. The most obvious way of implementing a quasi-Newton method is merely to replace Jk whenever it occurs by Bk. Only two basic quasi-Newton methods have been seriously proposed to solve general systems of nonlinear simultaneous equations, and these are the secant method, and Broyden's method. In the chapter, good local convergence properties are observed, although the algorithms developed might not be optimal for the solution of the particular problems quoted. They do illustrate, however, one way in which special-purpose quasi-Newton methods might be developed and the way underlying philosophy of these methods might be applied to a larger range of problems than hitherto.