Truncated nonlinear ABS algorithm and its convergence property

Abstract

--ZusammenfassnngTruncated Nonlinear ABS Algorithm and Its Convergence Property. This paper shows that the nonlinear ABS algorithm [1] can be viewed as inexact Newton method [3]. From this point of view, the truncated nonlinear ABS algorithm is established and its convergence property is discussed.

A class of difference ABS‐type algorithms for a nonlinear system of equations

“…The algorithms corresponding to the scaled ABS class and to the block ABS class can also be established in a similar way. In addition, a truncated form of these algorithms can be developed similarly to the truncated nonlinear ABS class of Deng and Chen [6]. These extensions will be discussed in other papers.…”

“…where U ( x , 6) is an open ball of center x and radius 6. Proof It is not difficult to obtain the desired conclusion by induction if there are two positive scalars 81 and M1 such that…”

“…(A2) The Jacobian J ( x * ) is nonsingular. (A3) There exist two positive scalars So and Lo such that So) where U ( x , 6) is an open ball of center x and radius 6.…”