 | The course is divided into 10 parts. The first part acquaints the reader with the Matlab® software and is the necessary theoretical basis for mastering the basics of programming in this system. It is elaborated with emphasis on algorithmic constructions, fundamentals and principles of structured programming in implementation of mathematical algorithms. The second part is devoted to the approximation of roots, i.e. the task of solving the nonlinear equation, which occurs in virtually all engineering fields. In this section, we discuss three basic convergent methods for root approximation, namely bisection, the tangent method and the method of strings, which are implemented and tested on several tasks. In the third part, we present the problem of interpolation as an approximation of a function and we present several computational algorithms according to the types of interpolation. The fourth part deals with measure as a certain integral with implementation into the Matlab® computing environment. The fifth part deals with other methods of numerical mathematics for the calculation of a definite integral, which include the rectangular rule and the Simpson's rule. The content of the sixth part is the topic of global optimization. In this part, we introduce the basic test functions and introduce algorithms for the most widely used methods for finding the global optimum, namely Controlled Random Search, Differential Evolution, and the built-in Fminsearch function. Into the seventh part, we have included basic knowledge about prime numbers and we explain their position among numbers. The part contains several key terms related to prime numbers such as Fundamental theorem of arithmetic or canonical decomposition of an integer. The key part of the part is testing prime numbers using the Fermat's Little Theorem. Fibonacci numbers play an important role in cryptography and appear in many places in solving practical problems even nowadays, so we have included the basic algorithms concerning this issue as the eighth part. The content of the ninth part is the solution of linear diophantic equations of two unknowns and the Fermat's Last Theorem. The part also presents the algorithm of division with a remainder, the so called Euclidean algorithm. The last, the tenth part, is a special part of this course dealing with several interesting applications from the number theory. In the first part of this part, we introduce the term “perfect numbers” and introduce an algorithm for their generation. The second part explains the so-called principle of inclusion and exclusion, which is useful in solving various combinatorial tasks. Problems of number systems is an area that should be interconnected with various computing environments, so in the third part of the last part we also offer one useful function from the Matlab® calculation program for converting numbers between systems of different bases. |
