Convergence Theorems for Least-Change Secant Update Methods

The purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means. The theorems and proofs given here can be viewed as generalizations of those given by Broyden–Dennis–Moré [J. Inst. Math. Appl. 12 (1973), pp. 223–246] and by Dennis–Moré [Math. Comp., 28 (1974), pp. 549–560]. The analysis is done in the orthogonal projection setting of Dennis–Schnabel [SIAM Rev., 21(1980), pp. 443–459] and many readers might feel that it is easier to understand. The theorems here readily imply local and q-superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of Marwil and Toint and the nonlinear least-squares method of Dennis–Gay–Welsch.