Course detail

# Higly Sophisticated Computations

The course aims to show practical methods of solving problems encountered in science and engineering. The numerical solution of systems of differential equations is discussed, and parallel cooperation of microprocessors based on differential calculus is evaluated.  The course mainly examines VSVO (variable step, variable order) methods. During the course, students are going to use an original method based on a direct calculation of the Taylor series terms. The TKSL (FOS) simulation language is also available, which allows specifying problems directly using equations.
The close correspondence of block and equation representations of differential equations is also analysed, using block representation as input. The course discusses the following technical problems: solution of large systems of differential equations, algebraic equations, partial differential equations, stiff systems, problems in automatic control, electric circuits, VLSI circuits, modelling of mechanical systems, electrostatic and electromagnetic fields. The analysis of the parallel algorithms and design of a specialised architecture for the solution of differential equations is also part of the course. Most of the technical problems lead to the matrix/vector representation. Individual problems are also going to be solved in MATLAB/Simulink.

Areas of interest for SDZ

1. Analytical solution of differential equations.
2. Numerical solution of differential equations.
3. Extremely accurate solution of differential equations using the Taylor series, libraries for accurate calculations.
4. Parallel properties of the Taylor series, the basis of the programming of the specialised parallel tasks using differential calculus (close correspondence of block and equation representation).
5. The adjunct differential operators and parallel solution of the differential equations with time-variable coefficients.
6. The solution of large systems of algebraic equations by converting them to the ordinary differential equations..
7. Fourier series, finite integrals.
8. Simulation of electric circuits.
9. Solution of practical problems described by partial differential equations.
10. The concept of a basic specialized processor for parallel computational system.

Language of instruction

Czech

Number of ECTS credits

0

Mode of study

Not applicable.

Entry knowledge

Numerical Mathematics

Rules for evaluation and completion of the course

During the semester, there will be evaluated computer laboratories. Any laboratory should be replaced in the final weeks of the semester.

Aims

To provide overview and basics of practical use of selected methods for numerical solutions of differential equations (based on the Taylor Series Method) for extremely exact and fast solutions of sophisticated problems.
Ability to analyse the selected methods for numerical solutions of differential equations (based on the Taylor Series Method) for extremely exact and fast solutions of sophisticated problems.
• An individual solution of a nontrivial system of diferential equations.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Kozubek, T., Brzobohatý, T., Jarošová, M., Hapla, V., Markopoulos, A.: Lineární algebra s MATLABem, učební text MI21 VŠB-TU Ostrava, 2012 (elektronicky dostupné z http://mi21.vsb.cz/sites/mi21.vsb.cz/files/unit/linearni_algebra_s_matlabem.pdf)
Přednášky ve formátu PDF

• Hairer, E., Norsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I, vol. Nonstiff Problems. Springer-Verlag Berlin Heidelberg, 1987.
Meurant, G.: Computer Solution of Large Linear System, North Holland, 1999
Burden, R. L.: Numerical analysis,  Cengage Learning, 2015
Golub, G. H.: Matrix computations, Hopkins Uni. Press, 2013
Duff, I. S.: Direct Methods for Sparse Matrices (Numerical Mathematics and Scientific Computation), Oxford University Press, 2017
Corliss, G. F.: Automatic differentiation of algorithms, Springer-Verlag New York Inc., 2002
Vavřín, P.: Teorie automatického řízení I (Lineární spojité a diskrétní systémy). VUT, Brno, 1991.
Vitásek,E.: Základy teorie numerických metod pro řešení diferenciálních rovnic. Academia, Praha, 1994.
Šebesta, V.: Systémy, procesy a signály I. VUTIUM, Brno, 2001.

eLearning

Classification of course in study plans

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, summer semester, elective

• Programme DIT-EN Doctoral, any year of study, summer semester, compulsory-optional
• Programme DIT Doctoral, any year of study, summer semester, compulsory-optional
• Programme DIT Doctoral, any year of study, summer semester, compulsory-optional
• Programme DIT-EN Doctoral, any year of study, summer semester, compulsory-optional

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Methodology of sequential and parallel computation (feedback stability of parallel computations)
2. Extremely precise solutions of differential equations by the Taylor series method
3. Parallel properties of the Taylor series method
4. Basic programming of specialised parallel problems by methods using the calculus (close relationship of equation and block description)
5. Parallel solutions of ordinary differential equations with constant coefficients
6. Adjunct differential operators and parallel solutions of differential equations with variable coefficients
7. Methods of solution of large systems of algebraic equations by transforming them into ordinary differential equations
8. Parallel applications of the Bairstow method for finding the roots of high-order algebraic equations
9. Fourier series  and finite integrals
10. Simulation of electric circuits
11. Solution of practical problems described by partial differential equations
12. Library subroutines for precise computations
13. Conception of the elementary processor of a specialised parallel computation system.

Guided consultation in combined form of studies

26 hours, optionally

Teacher / Lecturer

Exercise in computer lab

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Simulation system TKSL
2. Exponential functions test examples
3. First order homogenous differential equation
4. Second order homogenous differential equation
5. Time function generation
6. Arbitrary variable function generation