The local convergence of ABS methods for nonlinear algebraic equations

Abstract

Summary. In this paper we consider an extension to nonlinear algebraic systems of the class of algorithms recently proposed by Abaffy, Broyden and Spedicato for general linear systems. We analyze the convergence properties, showing that under the usual assumptions on the function and some mild assumptions on the free parameters available in the class, the algorithm is locally convergent and has a superlinear rate of convergence (per major iteration, which is computationally comparable to a single Newton's step). Some particular algorithms satisfying the conditions on the free parameters are considered.

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

“…6 We use the NFA in preference to other methods for solving underdetermined systems of nonlinear equations. However, one could use instead a suitable version of the ABS method [1], the Huang method [16], or the augmented Jacobian algorithm [38,39]. If one wishes to avoid the computation of the Jacobian matrix H (z t ) at each iteration, then one may consider a quasi-Newton formulation of (19) as in Martinez [23].…”

Truncated nonlinear ABS algorithm and its convergence property

“…The local convergence property of nonlinear ABS algorithm is discussed in [1] and [2] under the following assumptions.…”

“…To solve (1.1) Abaffy et al [1] proposed nonlinear ABS algorithm and proved that the sequence {xi} generated by the algorithm is quadratically convergent (see also [2]). The major iteration of nonlinear ABS algorithm consists of minor iterations.…”

A ABS algorithm for solving singular nonlinear system with space transformation

“…There are several publications dealt with solving nonlinear systems of F with full rank by the ABS algorithm, see for instance, Abaffy, Broyden and Spedicato [1], Abaffy, Galantai and Spedicato [2], Abaffy and Spedicato [3], Galantai [5,6], Huang [11], Spedicato, Chen and Deng [15,16], Spedicato and Huang [17]. For the survey on general progress of the ABS algorithms it also can be referred to Spedicato [13,14].…”

“…, p}, where s k , z k are defined by Frank and Schnabel [4]. Solution of the tensor model is to find d ∈ R N such that d is a solution of min d∈R N M T (x c + d) 2 . Under the assumptions that F (x * ) is singular with only one zero singular value and u T F (x * )vv = 0, the sequence of iterations generated by the tensor method based on an ideal tensor model converges locally and two-step Q-superlinearly to the solution with Q-order 3/2, and the sequence of iterates generated by the tensor method based on a practical tensor model converges locally and three-step Q-superlinearly to the solution with Q-order 3/2.…”