Introduction by numerical methods

Many counting problems can be solved only with the help of modern computers.

It is customary to single out the following main stages of solving a problem on a computer:

1. Physical formulation of the problem.

2. Mathematical formulation of the problem. Record of a physical problem in terms of one or another mathematical model.

3. The choice of a numerical method for solving the problem.

4. Realization of the method in one or another programming language or with the help of one or another package of solving applied problems (Matcad, Matlab, Exel, etc.).

5. Conduct test calculations and comparison with experimental data.

A simple mathematical model is a collection of algebraic formulas by which the desired quantities are explicitly calculated. However, most often the behavior of parameters is described by complex algebraic or partial differential equations. Find a solution to these complex problems is possible only with the use of modern high-speed computers.

Even to use the standard , i.e. already ready program, you need to have an idea about the existing methods of solution, their advantages, disadvantages and features of use.

All methods for solving equations can be divided into two classes: exact and approximate . In exact methods, the solution is obtained in the form of formulas for a finite number of operations, but they can only be used to solve equations of a special type. In the general case, the problem can be solved only approximately. Approximate methods make it possible to obtain a solution in the form of an infinite sequence converging to an exact solution.

The use of computers imposes additional requirements on the algorithm for finding both exact and approximate solutions: it must be stable, realizable, and economical . Stability means that small errors introduced into the solution process do not lead to large errors in the final result. Errors occur due to inaccurate initial data (fatal errors), due to rounding numbers , which always takes place in calculations on a computer, and are also associated with the accuracy of the method used . The feasibility of the algorithm means that the solution can be obtained in a valid time. It should be borne in mind that the time of the approximate solution depends on the accuracy with which we want to obtain a solution. In practice, accuracy is chosen taking into account the feasibility of the algorithm on the computer that is supposed to be used for the solution. Economical is the algorithm that allows you to get a solution with a given accuracy for the minimum number of operations, and therefore, for the minimum estimated time.

In the course we are studying, we will become familiar with the basic methods used to solve various mathematical problems.