Numerical Methods I
FSI-SN1Acad. year: 2020/2021
The course represents the first systematic explanation of selected basic methods of numerical mathematics. Passing this course, students obtain basic knowledge necessary for further study of more specialised areas of numerical mathematics.
Main topics: Direct and iterative methods for linear systems. Interpolation. Least squares method. Numerical differentiation and integration. Nonlinear equations. The students will demonstrate the acquinted knowledge by elaborating at least two semester assignements.
Learning outcomes of the course unit
Students will be made familiar with the basic collection of numerical methods, namely with direct and iterative methods for linear systems, with interpolation, least squares, numerical derivation and integration and with methods for nonlinear equations. Students will demonstrate the acquinted knowledge by elaborating of several semester assignements.
Differential and integral calculus for functions of one and more variables. Ordinary differential equations. Fundamentals of linear algebra. Programming in MATLAB.
Recommended optional programme components
Recommended or required reading
M.T. Heath: Scientific Computing. An Introductory Survey. Second edition. McGraw-Hill, New York, 2002.
L. Čermák, R. Hlavička: Numerické metody, CERM, Brno, 2008.
C.B. Moler: Numerical Computing with Matlab, Siam, Philadelphia, 2004.
L. Čermák: Vybrané statě z numerických metod. [on-line], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-I/sc-1150-sr-1-a-141/default.aspx.
G. Dahlquist, A. Bjork: Numerical Methods, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1974.
J.H. Mathews, K.D. Fink: Numerical Methods Using MATLAB, Pearson Prentice Hall, New Jersey, 2004.
A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, Springer, Berlin, 2000
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of a semester assignments, where the students prove their knowledge acquired. Students, who gain course-unit credits, will also obtain 0--30 points, which will be included in the final course classification.
FORM OF EXAMINATIONS: The exam is oral. 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.
If we measure the exam success in percentage points, then the classification grades are: 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.
Language of instruction
The aim of the course is to familiarise students with some basic numerical methods. Substantial emphasis is also put on a computer implementation of individual methods. Students ought to understand the essence of particular methods and to realise their advantages and drawbacks. Attention is paid also to the stability and conditioning of numerical methods considered. The development of individual semester assignements constitutes an important experience enabling to verify how the subject matter was managed.
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 for by the agreement with the teacher supervising the seminars.
Type of course unit
26 hours, optionally
Teacher / Lecturer
1. Introduction to computing: error analysis, computer arithmetic, conditioning of problems, stability of algorithms.
2. Gaussian elimination method. LU decomposition. Pivoting.
3. Solution of special linear systems. Stability and conditioning. Error analysis.
4. Classical iterative methods: Jacobi, Gauss-Seidel, SOR, SSOR.
5. Generalized minimum rezidual method, conjugate gradient method.
6. Lagrange, Newton and Hermite interpolation polynomial. Piecewise linear and piecewise cubic Hermite interpolation.
7. Cubic interpolating spline. Least squares method: data fitting, solving overdetermined systems.
8. QR decomposition and singular value decomposition in the least squares method.
9. Orthogonalization methods (Householder transformation, Givens rotations, Gram-Schmidt orthogonalization)
10. Numerical differentiation: basic formulas, Richardson extrapolation.
11. Numerical integration: Newton-Cotes formulas, Romberg's method, Gaussian formulas, adaptive integration.
12. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
13. Solving nonlinear systems: Newton's method, fixed point iteration.
26 hours, compulsory
Teacher / Lecturer
Students create elementary programs in MATLAB related to each subject-matter delivered at lectures and verify how the methods work. Furthermore students individually elaborate semester assignemets.
eLearning: currently opened course