Deepmala

A family of generalized modulus-based iterative schemes incorporating an acceleration mechanism is proposed for large-scale sparse linear complementarity problems. The proposed approach is based on a generalized matrix splitting strategy involving an acceleration parameter, which extends several existing accelerated modulus-based methods as special cases. Convergence of the proposed schemes is guaranteed under certain conditions when the coefficient matrix belongs to the class of $P$-matrices or $H_{+}$-matrices, and several numerical experiments are included to illustrate their computational performance. The numerical results show that the generalized accelerated methods significantly reduce the number of iterations and CPU time compared with existing approaches, thereby improving convergence performance for large-scale sparse problems.

The complementarity problems is getting a lot of attention because it is connected to real-world problems in scientific computing and engineering. It shows up in various situations like linear and quadratic programming, two person games, circuit simulation, optimal stop-ping in Markov chains, contact problems with friction, finding a Nash-equilibrium in bima-trix games. The linear complementarity problems (LCP) and absolute value equations (AVE) have an equivalence relation; that is, the AVE can be transformed into an LCP and vice versa. The relationship between LCP and AVE enables the conversion of one problem into another, offering different perspectives for analysis and solution. This equivalence aids in theoretical understanding and the development of numerical methods applicable to both mathematical formulations. In the present study, we discuss the unique solvability of the LCP and the hori-zontal linear complementarity problems (HLCP). Some superior unique solvability conditions are obtained for LCP and HLCP. The unique solvability of the n-absolute value equations 𝐴𝑛𝑥−𝐵𝑛|𝑥| = 𝑏 is also discussed. Some examples are highlighted for improving the current conditions of unique solutions for absolute value equations.

In this article, we investigate the existence and uniqueness of solutions to a generalized nonlinear functional integral equation (G‐NLFIE) associated with certain epidemic models of infectious diseases, defined within the Banach space . Our existence results include several specific cases of nonlinear functional integral equations that commonly occur in nonlinear sciences. We then introduce an iterative algorithm that combines Adomian's decomposition method (ADM) with the modified homotopy perturbation method (mHPM) to approximate solutions to the G‐NLFIE. The paper addresses the convergence properties and error analysis of this method. Finally, we present numerical examples to demonstrate the effectiveness and efficiency of our proposed approach.