41,236 research outputs found
On the unique solvability and numerical study of absolute value equations
The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), , when the condition holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where .
Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition , that the algorithm converges linearly global to the unique solution of the AVE.
Numerical results are reported to show the efficiency of the proposed method and to compare with an available method
A Parameterized multi-step Newton method for solving systems of nonlinear equations
We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft
Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations
The SOR-like iteration method for solving the absolute value equations~(AVE)
of finding a vector such that with is investigated. The convergence conditions of the SOR-like iteration method
proposed by Ke and Ma ([{\em Appl. Math. Comput.}, 311:195--202, 2017]) are
revisited and a new proof is given, which exhibits some insights in determining
the convergent region and the optimal iteration parameter. Along this line, the
optimal parameter which minimizes with and the approximate optimal parameter which
minimizes are explored.
The optimal and approximate optimal parameters are iteration-independent and
the bigger value of is, the smaller convergent region of the iteration
parameter is. Numerical results are presented to demonstrate that the
SOR-like iteration method with the optimal parameter is superior to that with
the approximate optimal parameter proposed by Guo, Wu and Li ([{\em Appl. Math.
Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method
with the optimal parameter performs better, in terms of CPU time, than the
generalized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009])
for solving the AVE.Comment: 23 pages, 7 figures, 7 table
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