Open AccessThe paper reviews prerequisites to creating a variety of the iterative methods to solve a system of linear equations (SLE). It considers the splitting methods, variation-type methods, projection-type methods, and the methods of relaxation.
A new iterative method based on mechanical analogy (the movement without resistance of a material point, that is connected by ideal elastically-linear constraints with unending guides defined by equations of solved SLE). The mechanical system has the unique position of stable equilibrium, the coordinates of which correspond to the solution of linear algebraic equation. The model of the mechanical system is a system of ordinary differential equations of the second order, integration of which allows you to define the point trajectory. In contrast to the classical methods of relaxation the proposed method does not ensure a trajectory passage through the equilibrium position. Thus the convergence of the method is achieved through the iterative stop of a material point at the moment it passes through the next (from the beginning of the given iteration) minimum of potential energy. After that the next iteration (with changed initial coordinates) starts.
A resource-intensive process of numerical integration of differential equations in order to obtain a precise law of motion (at each iteration) is replaced by defining its approximation. The coefficients of the approximating polynomial of the fourth order are calculated from the initial conditions, including higher-order derivatives. The resulting approximation enables you to evaluate the kinetic energy of a material point to calculate approximately the moment of time to reach the maximum kinetic energy (and minimum of the potential one), i.e. the end of the iteration.
The software implementation is done. The problems with symmetric positive definite matrix, generated as a result of using finite element method, allowed us to examine a convergence rate of the proposed method. The proposed method has shown the comparable results with the most commonly used method of conjugate gradients. There are prospects for development of the proposed method, in particular through the preconditioned use.