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Numerical Methods

Course developer

Prior knowledge

Mate1003, Mathematics I

Mate1037, Mathematics II

Mate1038, Mathematics III

Course abstract

The aim of the study course is to create an understanding of the more frequently used numerical methods in solving mathematical problems, to teach how to implement the simplest numerical methods with the help of the computer program MatLab.
This course analysed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations, iterative methods for solving systems of linear and nonlinear equations. The study course is foreseen for knowledge accumulation of the numerical analysis and application of the programming skills to solve different mathematical problems of numerical methods.
During the course, students acquire and apply skills in working with appropriate software applications such as Matlab.

Learning outcomes and their assessment

By the successful completion of this study course:
1. Students are able to manage and demonstrate knowledge and critical understanding on the use of numerical methods for the numerical solving of nonlinear equations and systems of equations, numerical integration, data approximation and interpolation, solving the Cauchy problem for ordinary first order differential equations. – defence of laboratory work.
2. Students are able to demonstrate understanding of relevant concepts and regularities, to perform necessary calculations and operations. Students able to use the appropriate software. – laboratory works.
3. Students are able to apply mathematical calculations corresponding to the specialty problem situation, to make a professional assessment and interpretation of the intermediate result of the calculations and the final results by working in the group or performing the work independently.- independent studies

Course Content(Calendar)

1. Mathematical modelling and model. Numerical experiment. Numerical methods. – 2 h
2. Numerical solving of nonlinear equations with dichotomy, regula falsi, Newton-Rapson and fixed-point iteration methods. – 7 h
3. Iterative methods for solving systems of linear equations (Jacobi and Gauss-Seidel methods)– 3h
4. Iterative methods for solving systems of nonlinear equations (Jacobi and Newton methods). - 4 h
5. Numerical integration with rectangles, trapezoidal, Simpson’s rules. – 4 h
6. Approximation of data. Method of least squares. Choosing the best empirical formula. – 4 h
7. Interpolation of data with polynomials and splines. – 4 h
8. Solving of first-order ordinary differential equations with Euler's, midpoint, Runge-Kutta’s methods. – 4 h

Requirements for awarding credit points

The course is assessed through an examination.

Description of the organization and tasks of students’ independent work

The following laboratory works must be completed using the appropriate application software:
Laboratory work 1: Numerical solving of nonlinear equations.
Laboratory work 2: Iterative methods for solving systems of linear equations.
Laboratory work 3: Iterative methods for solving systems of nonlinear equations.
Laboratory work 4: Numerical integration.
Laboratory work 5: Approximation of data.
Laboratory work 6: Interpolation of data.
Laboratory work 7: Numerical solving of first order ordinary differential equations.
In order to improve the assessment, the student can prepare a report on the chosen numerical method, which is not considered in the study course program and it can be implemented using the appropriate application software, pre-coordinating the choice of method with the teaching staff.

Criteria for Evaluating Learning Outcomes

During the semester the student has to develop and defend 7 laboratory works. Each developed and defended laboratory work is graded (max. 8 points possible). Laboratory work is passed only if it is defended successfully, so at least 4 points are scored.
The student can receive the accumulative exam score if All independent works are defended successfully until the beginning of period of individual studies and examinations.
The mark of the accumulative exam consists of the average mark (80%) of all the defended laboratory works and the mark of the prepared report (20%).
Laboratory work that has not been passed on time can be completed in a time allotted by the academic staff, but not more than two per one visit.
The exam can be arranged at the time indicated by the academic staff if all independent works are successfully defended.
The mark of the examination consists of the average mark of all laboratory works (50%), the evaluation of the examination work (50%: 10% theoretical part and 40% practical tasks).

Compulsory reading

1. R. Kalniņš, G. Hiļkeviča, E.Vītola. Skaitliskās metodes. Ventspils: Ventspils augstskola, 2009. – 107 lpp.
2. M. Iltiņa, I. Iltiņš. Skaitliskās metodes: mācību līdzeklis.– Rīga: Rīgas Tehniskā Universitāte, 2005. – 93 lpp.
3. I. Meirāns. Skaitliskās metodes: metodiskie norādījumi un uzdevumi praktisko darbu izpildei.- Rēzekne: RA Izdevniecība, 2003. – 44 lpp.
4. A. Zviedris. Datorrealizācijas matemātiskās metodes: Lekciju konspekts.– Rīga: Rīgas Tehniskā Universitāte, 2001. – 93 lpp.
5. K. A. Ansari, B. Dichone. Introduction to Numerical Methods Using MATLAB, SDC Publications, 2019. - 368 p. (pieejama Matemātikas katedrā/ available in the Department of Mathematics)
6. A. Kharab, R. Guenther. Introduction to Numerical Methods: A MATLAB (R) Approach, 4th edition. Taylor & Francis Ltd, 2021. - 632 p. (pieejama Matemātikas katedrā/ available in the Department of Mathematics)

Further reading

1. H. Kalis, I. Kangro. Matemātiskās metodes inženierzinātnēs: mācību līdzeklis.- Rēzekne: RA Izdevniecība, 2004. – 292 lpp.
2. K. Šteiners, S. Baliņa. Augstākā matemātika III. Lekciju konspekts inženierzinātņu studentiem.- Rīga: Zvaigzne ABC, 1998. – 192 lpp.
3. K. Šteiners, S. Baliņa. Augstākā matemātika IV. Lekciju konspekts inženierzinātņu studentiem.- Rīga: Zvaigzne ABC, 1999. – 168 lpp.
4. M. Brāzma, A. Brigmane, A.Krastiņš, J.Rāts. Augstākā matemātika.- Rīga: Zvaigzne ABC, 1970. – 547 lpp.
5. Elnashaie, S. S. E. H. Numerical techniques for chemical and biological engineers using MATLAB: a simple bifurcation approach / Said Elnashaie, Frank Uhlig, with the assistance of Chadia Affane. - New York: Springer, 2007. – 590 p.

Periodicals and other sources

International Journal for Numerical Methods in Engineering. Pieejams: http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0207 [tiešsaiste]. [skatīts 05.12.2018.]

Notes

Compulsory course for Faculty of Information Technologies Bachelor’s study programmes “Computer Control and Computer Science” and „Information Technologies for Sustainable Development”.