On the Numerical Solution of System of Linear Algebraic Equations with Positive Definite Symmetric Ill-Posed Matrices

In this paper we propose a so-called variational sigmoidal transformation, containing two additional parameters, which inherits principal features of the traditional sigmoidal transformations. The transformation is used for numerical evaluation of singular integrals appearing inevitably in the numerical techniques for engineering problems, for example, the boundary element method. The principal role of the transformation is to weaken or remove the original singularity of the considered integrand. Purpose of this work is to enhance the accuracy of the numerical evaluation techniques such as the Gauss-Legenre quadrature rule for the weakly integrals and the EulerMaclaurin formula for the Cauchy-principal value and Hadamard finite part integrals. Based on the asymptotic analysis of the transformed integrands, it is proved that the presented transformation combined with the existing quadrature rules will be effective in improving the approximation errors, thanks to the parameters. Availability of the proposed method is verified by the results of some numerical examples. In numerical fulfillment we explore the approximation method, with respect to various values of the parameters included in the presented transformation, for each case of the singularity of the integrand. It is demonstrated that most numerical results of the proposed method are consistent with those of the theoretical analysis.