The method of 'external spiral" for solving a large system of linear equations

Abstract

Solving a linear system of n × n equations can be very difficult for the computer, especially if one needs the exact solution, even when the number n -of equations and of unknown variables is relatively small (a few thousands). All existing methods have to overcome at least one of the following problems: 1. Computational complexity, which is expressed with the number of arithmetic operations required in order to determine a 400 VOJNOTEHNIČKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 2 solution; 2. The possibility of overflow and underflow problems; 3. Causing variations in the values of some coefficients in the initial system, which may be leading to instability of the solution; 4. Requiring additional conditions for convergence; 5. In cases of a large number of equations and unknown variables it is often required that the systems matrix be: either sparse, or symmetrical, or diagonal, etc. This paper presents a method for solving a system of linear equations of arbitrary order (any number of equations and unknown variables) to which the problems listed above do not reflect.

"Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns

“…The program code specifies the solution of each system that has a unique solution which the reader can check for themselves. 6. Conclusion.…”

“…In [5] and [6] methods are described that, with the help of computers, can approximately, and in [7] accurately, solve a linear system of any number of equations with any number of unknowns. Any point can be taken for initial approximation.…”

“…For simplicity, it can be a point whose coordinates are all zero. Then, the following points are taken: a point in which the normal line cuts through the next hyperplane, in paper [5]; a point in which the normal line cuts through the intersection of two neighboring hyperplanes, in paper [6]; a vertex of some trihedral in which the point is located, in paper [7].…”