Course detail

Numerical Methods

FSI-2NUAcad. year: 2007/2008

The course provides an introduction to basic numerical methods frequently used in technical computations. Emphasis is placed on understanding of how numerical methods work. Shorter numerical exercises are carried out with a pocket calculator, but others can be done more efficiently by computers. Main topics: Scientific computing. Systems of linear equations. Interpolation. Least squares method. Numerical differentiation and integration. Nonlinear equations. Optimization.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. RNDr. Libor Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

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.

Prerequisites

Linear algebra, vector calculus, differential and integral calculus of functions of one variable. Basics of programming.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practisals. Students have to pass two check tests and to work out semester assignment (the form will be specified by the teacher supervising seminars, the assignment ought to contain several tasks, preferably elaborated on computer. It is desirable to elaborate at least two of the tasks using the MATLAB, MAPLE, SCILAB, MATHCAD, DELPHI and the like, by mutual agreement with the teacher). Students, who gain course-unit credits, will also obtain 0--30 points, which will be included in the final course classification.
FORM OF THE EXAMINATIONS: The exam is written and has a practical and a theoretical part. In the practical part students solve numerical examples by hand using pocked calculator, in the theoretical part they answer several questions to basic notions in order to check up how they understand the subject. As a result of the exam students will obtain 0--70 points.
FINAL ASSESSMENT: The final point course classicifation is the sum of points obtained from both the practisals (0--30) and the exam (0--70).
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.

Course curriculum

Not applicable.

Work placements

Not applicable.

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.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. Absence from lessons may be compensated by the agreement with the teacher supervising the seminars.

Recommended optional programme components

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)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme B3901-3 Bachelor's

    branch B3904-00 , 1. year of study, summer semester, elective (voluntary)
    branch B3940-00 , 1. year of study, summer semester, compulsory

  • Programme B2341-3 Bachelor's

    branch B2339-00 , 1. year of study, summer semester, compulsory

  • Programme N2301-2 Master's

    branch N2370-BS , 1. year of study, summer semester, compulsory
    branch N2335-BS , 1. year of study, summer semester, compulsory
    branch N2303-BS , 1. year of study, summer semester, compulsory

  • Programme N2301-3 Master's

    branch N2300-00 , 1. year of study, summer semester, compulsory

Type of course unit

 

Lecture

13 hours, optionally

Teacher / Lecturer

doc. RNDr. Libor Čermák, CSc.
RNDr. Rudolf Hlavička, CSc.
doc. PaedDr. Dalibor Martišek, Ph.D.
doc. Ing. Luděk Nechvátal, Ph.D.
doc. RNDr. Libor Žák, Ph.D.

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.
Numerical integration: Basic quadrature rules (midpoint, trapezoidal and Simpson's rule). Gaussian quadrature. Composite quadrature. Adaptive quadrature.
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

doc. RNDr. Libor Čermák, CSc.
RNDr. Jiří Dočkal, CSc.
RNDr. Milana Faltusová
RNDr. Rudolf Hlavička, CSc.
Ing. Jiří Jánský, Ph.D.
doc. PaedDr. Dalibor Martišek, Ph.D.
doc. Ing. Luděk Nechvátal, Ph.D.
doc. Ing. Petr Tomášek, Ph.D.
doc. RNDr. Libor Čermák, CSc.

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.