Systems of linear algebraic equations arise when solving a number of applied problems described by differential, integral or systems of nonlinear (transcendent) equations. They can also appear in problems of mathematical programming, statistical data processing, approximation of functions, when discretizing boundary value differential problems by the method of finite differences or the method of finite elements.
Exact methods of solving systems of linear equations Methods for solving systems of linear equations can be divided into two groups. The first includes accurate methods. With the help of such methods, in principle, it is possible to obtain exact values of unknowns due to a finite number of steps. At the same time, it is assumed that both the coefficients in the right part and the elements of the column of free terms are exact numbers, and all calculations are performed without rounding. However, in practice this may happen in exceptional cases or may be related to the solution of a special class of problems (for example, when the solutions are only integers). Such methods include:
- Kramer's method (method of determinants) is well known from the algebra course;
- matrix method (inverse matrix method);
- different variants of the method of excluding unknowns (Gauss method, Gauss method with selection of the main element, Jordan-Gauss method).
Note that the practical application of the first two methods may be ineffective or impossible.
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