DIFFERENTIAL EQUATION SYSTEMS MATRIX SERIES
More convergent matrix series were presented in which can be applied to approximate the solution of the system of equations. These differential equations were obtained using a quadratic functional related to the system of algebraic equations and describe the orthogonal trajectories of the family of hypersurfaces representing the functional. A smaller amount of computation is needed to obtain an approximate solution of large linear systems as compared to existing methods.Įxact analytic expressions in the form of infinite series of matrices for the solution of linear systems of algebraic equations were derived in by integrating associated ordinary differential equations. Efficient computation of a rough approximation of the solution, applicable even to poorly conditioned systems, is also performed based on the alternate application of two different types of minimization of associated functionals. The solution methods are flexible, permitting various intervening parameters to be changed whenever necessary in order to increase their efficiency. A related iterative method is also proposed. The solution procedure allows to easily control and monitor the magnitude of the residual vector at each step of integration. For a more efficient computation of approximate solutions of the algebraic systems, we consider new differential equations which are solved by simple techniques of numerical integration. Obviously, this helps the numerical computations. In these matrix series, each new term is obtained from the preceding one by multiplication with a matrix which becomes better and better conditioned tending to the identity matrix. Exact solutions were derived in terms of a new kind of infinite series of matrices which are truncated and applied repeatedly to approximate the solution. Various ordinary differential equations of the first order have recently been used by the author for the solution of general, large linear systems of algebraic equations.
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