Basic numerical methods - Numerical analysis

It's well known that the methods used in classical mathematics are incapable of solving all problems. For example, we can't give a formula for calculating exactly the single number x that verifies x = exp (– x); nor can we find the analytical solution to certain differential equations, or calculate certain definite integrals. In these cases, the exact mathematical resolution of the problem is replaced by its numerical resolution, which is usually approximate. Numerical analysis is the branch of mathematics that studies numerical problem-solving methods, known as constructive methods. By constructive method, we mean a set of rules (also known as an algorithm) that enables us to obtain the numerical solution of a problem with a desired accuracy after a finite number of arithmetic operations.

Numerical analysis is an ancient branch of mathematics. In the past, mathematicians developed the tools they needed to solve the problems posed by the natural sciences. Newton was first and foremost a physicist, Gauss an astronomer... They soon realized that the practical problems they faced were too complicated for their tools, and so, little by little, the techniques of numerical analysis were developed. However, it was only with the advent of computers, from 1945-1947, that these methods achieved their present-day heyday.

The following is not a theoretical course in numerical analysis. There are excellent books for that. Nor is it a catalog of methods and recipes. To be used correctly, and for their results to be interpreted correctly, numerical analysis methods require a knowledge of the basic principles that have guided mathematicians; it is very difficult, if not impossible, to use a numerical analysis algorithm as a black box. For these reasons, a middle way has been chosen, and algorithms are always placed in their theoretical context; readers concerned with demonstrations can refer to the relevant literature.

Numerical analysis methods are designed to be programmed on a computer. Computer arithmetic has only limited precision (due to the technology involved), which often poses extremely significant problems that need to be analyzed and avoided. This is why the first paragraph is dedicated to this issue.