15,079 research outputs found
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
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
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