### DisCont mathematics 2

#### Course Description

Selected topics of discrete mathematics and mathematical analysis, with emphasis on solving complex examples and tasks, based on algorithmic approach.

#### General Competencies

Learning advanced and modern techniques of evaluation of the values of some functions and various forms of its approximations.

#### Learning Outcomes

1. Understand the principles of numerical calculation of some elementary functions.
2. Understand the principles of varyous types of approximations.
3. Use the technique of approximatrion of a function by orthogonal polynomials.
4. Use the technique of fast summing algorithms.
5. Use the technique of acceleration of the convergence of some numerical series.
6. Understzand the asymptotical convergence and its application in calculations of values of some special functions.
7. Learn how to use modern mathematical literature
8. Lear how to use mathematical software in solving of complex problems.

#### Forms of Teaching

Lectures

Lectures are organized through two cycles. The first cycle consists of 7 weeks of classes and mid-term exam, a second cycle of 6 weeks of classes and final exam. Classes are conducted through a total of 15 weeks with a weekly load of 4 hours.

Exams

Mid-term exam in the 8th week of classes and final exam in the 15th week of classes.

Consultations

Consultations are held one hour weekly according to arrangement with students.

Other

Student's seminars.

Continuous Assessment Exam
Homeworks 0 % 20 % 0 % 20 %
Seminar/Project 0 % 10 % 0 % 10 %
Mid Term Exam: Written 0 % 30 % 0 %
Final Exam: Written 0 % 40 %
Exam: Written 0 % 60 %

#### Week by Week Schedule

1. Evaluation of values of functions. Horner's algorithm. Fast summation algorithm. identities.
2. Solving algebraic equations
3. Acceleration of convergence. Manipulation with series.
4. The connection between integrals and sums. Complex techniques of summing.
5. Continued fractions. Basic properties and formulas.
6. Representation of numbers and functions by continued fractions. Rational approximations.
7. Orthogonal polynomials. Polynomials given by reccursive relations. Applications of fast summing algorithms
8. Exam
9. Čebyšev's polynomials and problem of approximations. Fast calculations of Fourier series.
10. Series of functions. Generating functions of functional series. Z-transformation
11. Factorial and gamma functions. Stirling formula. Approximations of binomial coefficients.
12. Asympthotic behaviour. Asymptotic series.
13. Harmonic series and related problems. Euler constant.
14. Student seminar. Solution of advanced problems
15. Exam

#### Study Programmes

Computer Engineering (module)
Courses for exceptionally successful students (5. semester)
Computer Science (module)
Courses for exceptionally successful students (5. semester)
Computing (study)
Courses for exceptionally successful students (3. semester)
Control Engineering and Automation (module)
Courses for exceptionally successful students (5. semester)
Electrical Engineering and Information Technology (study)
Courses for exceptionally successful students (3. semester)
Electrical Power Engineering (module)
Courses for exceptionally successful students (5. semester)
Electronic and Computer Engineering (module)
Courses for exceptionally successful students (5. semester)
Electronics (module)
Courses for exceptionally successful students (5. semester)
Information Processing (module)
Courses for exceptionally successful students (5. semester)
Software Engineering and Information Systems (module)
Courses for exceptionally successful students (5. semester)
Telecommunication and Informatics (module)
Courses for exceptionally successful students (5. semester)
Wireless Technologies (module)
Courses for exceptionally successful students (5. semester)

#### Literature

F. S. Acton (1990.), Numerical Methods That Usually Work, Mathematical Association of America
J. Borwein J, D. Bailey, R. Girgensohn (2004.), Experimentation in Mathematics, Computational Paths to Discovery, A. K. Peters
Z. A. Melzak (1973.), Companion to Concrete Mathematics, Mathematical Techniques and Various Applications, John Wiley & Sons
A. Cuyt et al (2008.), Handbook of Continued Fractions for Special Functions, Springer
G. Boros, V. Moll (2004.), Irresistible Integrals - Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press

#### Lecturers in Charge

ID 90095
Winter semester
6 ECTS
L0 English Level
L1 e-Learning
60 Lectures
0 Exercises
0 Laboratory exercises
0 Project laboratory

80 Excellent
70 Very Good
60 Good
50 Acceptable