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This paper presents a method for solving nonlinear system with singular Jacobian at the solution. The convergence rate in the case of singularity deteriorates and one way to accelerate convergence is to form bordered system. A local algorithm, with finite-difference approximations, for forming and solving such system is proposed in this paper. To overcome the need that initial approximation has to be very close to the solution, we also propose a method which is a combination of descent method with finite-differences and local algorithm. Some numerical results obtained on relevant examples are presented. [ABSTRACT FROM AUTHOR]

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton's and Shamanski's method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained. [ABSTRACT FROM AUTHOR]

Abstract: A new smoothing quasi-Newton method for nonlinear complementarity problems is presented. The method is a generalization of Thomas’ method for smooth nonlinear systems and has similar properties as Broyden''s method. Local convergence is analyzed for a strictly complementary solution as well as for a degenerate solution. Presented numerical results demonstrate quite similar behavior of Thomas’ and Broyden''s methods. [Copyright &y& Elsevier]

A new approach to non-additive set functions is presented. The notion of non-additive measure is extended using non-commutative and non-associative operations with two parameters and the cumulative distribution function of a random variable. One new non-additive measure, generalized pseudo-probability measure, is defined and some of its basic properties are proven. [ABSTRACT FROM AUTHOR]

This paper presents the Chebyshev type of pseudo-integral inequalities, with the particular emphasis on the g -semirings. With the suitable choice of the generator g , the pseudo-expectation and pseudo-moments exist for the Cauchy distribution (in general case the standard expectation and moments for the Cauchy distribution are not defined). The computation of the pseudo-expectation for the Cauchy distribution and some examples that illustrate the applicability of the introduced generalization are also given in the paper. [ABSTRACT FROM AUTHOR]