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 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.
Highlights
If we perceive mathematics as a science oriented primarily towards a man as a subject of its application, the problem of solving large systems of linear equations is not a mathematical one
Let us suppose you need to solve the full system of linear equations having a very large number of equations and unknowns, e.g. n 100,000 or more
Assume that the computer memory contains a system of linear equations n × n
Summary
If we perceive mathematics as a science oriented primarily towards a man as a subject of its application, the problem of solving large systems of linear equations is not a mathematical one. It is essentially the problem steming from computer science since the very forming of such a system is impossible without the help of computers. The number of coefficients is of order n2 This implies that a minimum number of operations required to obtain the exact solution of the system of n × n is proportional to the number n3, in general. Want to reach a solution using fewer operations, it is necessary to seek the approximate methods or approximate solution
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