In this paper we study the solution of a system of equations Ax=b with singular and nearly singular, symmetric positive definite coefficient matrix A. Our algorithm based on, the Divide and Conquer strategy leading to the Divide-andConquer Algorithm (D&C algorithm) with, Cholesky's factorization algorithm. The Cholesky's factorization will be used to convert the matrix into a product of the form LL T , where L is a lower triangular matrix. The algorithm will be implemented on MATLAB and simulated as a user-subroutine. The user-subroutine is considering MATLAB features for reducing the round-off error especially for sensitive systems. Numerical examples will be given of a non-singular matrix and another for illconditioned matrix. The effect of round-off error will be analyzed. Results will be compared with previous ones, where LU factorization is used.

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We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p+1 (GCFP). We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the hyper-log-logistic and the shifted hyper-log-logistic functions. The limiting case of the interval-valued Heaviside step function is also discussed which imposes the use of Hausdorff metric. Numerical examples are presented using CAS MATHEMATICA.

The Hausdorff approximation of the shifted Heaviside function by Log-logistic and quadratic transmuted Log-logistic sigmoid functions is investigated and an expression for the error of the best approximation is obtained. The results of numerical examples performed in the programming environment Mathematica confirm our theoretical conclusions. Some applications in the field of biochemical processes and debugging theory are also explored.

We will explore the interesting methodological task for constructing new activation functions using "correcting amendments" of "Gompertz-Makehamtype" (GMAF). We also define the new family of recurrence generated activation functions based on "Gompertz-Makeham correction" -(RGGMAF).We prove upper and lower estimates for the Hausdorff approximation of the sign function by means of this new class of parametric activation functions -(RGGMAF). Numerical examples, illustrating our results are given.