Course detail

Numerical Methods

Students will be made familiar with a basic collection of numerical methods. They will make sense of errors in mathematical modelling, learn to find zeros of nonlinear equation and to solve systems of linear equations. They will master the basics of approximation including the least squares method, manage to use quadrature formulas and obtain an initial insight into the unconstrained minimization.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Entry knowledge

Numerical linear algebra, approximation of functions, numerical differentiation and integration, differential and integral calculus, basic Matlab programming.

Rules for evaluation and completion of the course

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in seminars. Students have to pass two check tests successfully and to work out semester assignment solved by means of MATLAB. A student can receive up to 20 points for both tests and up to 10 points for a semester assignment, in total up to 30 points. A necessary condition for course credit acquirement is a gain of at least 15 points, including at least 10 points in both check tests.
FORM OF THE EXAMINATIONS: The exam has a practical and a theoretical part. In the practical part students solve several numerical examples by hand using a scientific calculator. In the theoretical part they answer several questions to basic notions in order to check up how they understand the subject. Students will obtain 0--100 points as a result of the exam.
FINAL COURSE CLASSIFICATION: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.

Attendance at seminars is checked. Lessons are planned according to the week schedules. Absence may be replaced by the agreement with the teacher.

Aims

The aim of the course is to familiarize students with essential methods applied for solving numerical problems and provide them with an ability to solve such problems individually by hand and especially on computer. Students ought to realize that only the knowledge of substantial features of particular numerical methods enables them to choose a suitable method and an appropriate software product.
Students will be made familiar with a basic collection of numerical methods. They will make sense of errors in mathematical modelling, learn to find zeros of nonlinear equation and to solve systems of linear equations. They will master the basics of approximation including the least squares method, manage to use quadrature formulas and obtain an initial insight into the unconstrained minimization.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

HEATH, Michael T. Scientific computing: an introduction survey. 2nd ed. Boston: McGraw-Hill, 2002, 563 s. ISBN 0-07-239910-4. (EN)
MOLER, Cleve B. Numerical computing with MATLAB. Philadelphia: SIAM, 2004, xi, 336 s. : il. ISBN 0-89871-560-1. (EN)
DAHLQUIST, Germund a Ake BJÖRCK. Numerical Methods. New Jersey: Englewood Cliffs, 1974, 573 s. (EN)
MATHEWS, John H. a Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River: Pearson Prentice Hall, 2004, ix, 680 s. : il. ISBN 0-13-191178-3. (EN)

ČERMÁK, Libor a Rudolf HLAVIČKA. Numerické metody. Brno: Akademické nakladatelství CERM, 2016. (CS)
MOLER, Cleve B. Numerical computing with MATLAB. Philadelphia: SIAM, 2004, xi, 336 s. : il. ISBN 0-89871-560-1. (EN)
MATHEWS, John H. a Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River: Pearson Prentice Hall, 2004, ix, 680 s. : il. ISBN 0-13-191178-3. (EN)

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Classification of course in study plans

• Programme B-ZSI-P Bachelor's

specialization STI , 1. year of study, summer semester, compulsory

• Programme B-FIN-P Bachelor's, 1. year of study, summer semester, compulsory
• Programme B-MET-P Bachelor's, 2. year of study, summer semester, compulsory

Type of course unit

Lecture

13 hours, optionally

Teacher / Lecturer

Syllabus

Two-hour lessons take place every other week.
Week 1-2. Introduction to computing: Error analysis. Computer arithmetic. Conditioning of problems, stability of algorithms.
Solving linear systems: Gaussian elimination. LU decomposition. Pivoting.
Week 3-4. Solving linear systems: Effect of roundoff errors. Conditioning. Iterative methods (Jacobi, Gauss-Seidel, SOR method).
Week 5-6. Interpolation: Lagrange, Newton and Hermite interpolation polynomial. Piecewise linear and piecewise cubic Hermite interpolation. Cubic interpolating spline. Least squares method.
Week 7-8. Numerical differentiation: Basic formulas. Richardson extrapolation.
Week 9-10. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
Solving nonlinear systems: Newton's method, fixed point iteration.
Week 11-12. Minimization of a function of one variable: golden ratio, quadratic interpolation.
Minimization methods for multivariable functions: Nelder-Mead method, steepest descent and Newton's method.
Week 13. Teacher's reserve.

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Seminars are organized in biweekly cycles, alternatively in a classical classroom and in a computer lab. The seminar schedule corresponds to the subject of the corresponding lecture.

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