The Seidel-Gauss method. International method

The Seidel method (the second name is Gauss-Seidel) is a classical international method by which various systems of linear equations can be solved . Now we will talk about this in more detail.

The essence of the work

This method is a kind of simplified modification of the Jacobi method. The innovation is that the new value ^{( i)} It is used immediately after receiving, and not after the next iteration. In addition, the conditions for convergence and termination are clearly defined, the violation of which leads to an incorrect answer of the equation. The Seidel method, the example of which we provided in the picture, not only simplifies the decision process, but also accelerates it. Therefore, it is actively used by programmers to create and solve complex systems.

Seidel's method. "Pascal"

No programmer can do without mathematical formulas and equations. And this means that Seidel's method is actively used in the program "Pascal" to gain experience with robots with basic elements. It looks pretty simple: a new document is created in the program sheet, from the very beginning, the condition of the equation and its boundaries are introduced, then additional removable elements are explained (subject to availability), then a compatibility test is prescribed. If it is positive, then the solution algorithm itself is output, and only then the derivation of the roots. Equations can include several decision steps, each part of which has its own algorithm, mandatory composite, replaceable elements and basic formulas. All this is recorded exclusively in English, without possible analogues. The solution of the equation will be output as a finished formula or number after saving all the data.

"With ++"

Seidel's method is also widely used in the "C ++" program, but here everything is quite different than in "Pascal". The equation in "C ++" does not begin with the condition of the entire task, but with the condition of the end, which is prescribed in three or four stages with a final output of the result. Further, the course of the solution itself is prescribed using this method, describing in detail all the unknowns, after which a formula is derived in order to prove the equality between the two results of the equation. The condition is that each value of the previous one is necessary to solve the next one. Accounts are also maintained in English, which can not be replaced. "C ++" is much more complicated than "Pascal", therefore, without basic knowledge, it should not be used initially.

Let's sum up the results

So, Seidel's method is a special way, thanks to which it is possible to solve systems of linear equations of any complexity. Most often it is basic for programs such as "Pascal" and "C ++". This is a kind of improved modification of the Jacobi method, which eliminates the use of additional formulas, but it has clear convergence and termination conditions. Strictly established criteria simplify the whole process of work, because if one of the conditions is not met, the program, either "Pascal" or "C ++", simply refuses to solve the problem further.