Course detail

Numerical Mathematics I

FSI-9NM1Acad. year: 2024/2025

The introductory course of numerical methods deals with following topics: scientific computing, direct and iterative methods for linear systems, interpolation, least squares, differentiation and quadrature, eigenvalues, zeros and roots,.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

doc. Ing. Petr Tomášek, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, vector calculus, differential and integral calculus.

Rules for evaluation and completion of the course

The exam has an oral part only. The student has to answer three questions, one question from the range "numerical linear algebra", second question from the range "solving nonlinear equations" and third question from the range "interpolation, differentiation and integration".

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

Aims

The objective of the course is to make students familiar with numerical methods of linear algebra, with solution methods for nonlinear equations and with methods of interpolation, numerical differentiation and integration.
Students will be made familiar with basic numerical methods of linear algebra, nonlinear equations, interpolation, differentiation and integration. Based on this knowledge they ought to be able to choose suitable software product (exceptionally to write their own program) and then succesfully apply it in solving their specific technical problems.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. Quarteroni, S. Sacco, F. Saleri: Numerical Mathematics, Springer-Verlag, New York, 2000.
C.B. Moler: Numerical Computing with Matlab, Siam, Philadelphia, 2004.
C.F. Van Loan, G.H. Golub: Matrix Computations, 3th ed., the Johns Hopkins University Press, Baltimore, 1996.
G. Dahlquist, A. Bjork: Numerical Methods. Prentice-Hall, 1974
M.T. Heath: Scientific Computing. An Introductory Survey. Second edition. McGraw-Hill, New York, 2002.

Recommended reading

A. R. Ralston: Základy numerické matematiky. Academia, Praha, 1973.
E. Vitásek: Numerické metody. SNTL, Praha, 1987
I. Horová, J. Zelinka: Numerické metody, učební text Masarykovy univerzity, Brno, 2004.
K. Rektorys: Přehled užité matematiky. Prometheus, Praha, 1995
L. Čermák, R. Hlavička: Numerické metody. Učební text FSI VUT Brno, CERM, 2016.
L. Čermák: Vybrané statě z numerických metod. https://mathonline.fme.vutbr.cz/Numericke-metody-I/sc-1150-sr-1-a-141/default.aspx

Classification of course in study plans

  • Programme D-ENE-P Doctoral 1 year of study, winter semester, recommended course
  • Programme D-ENE-K Doctoral 1 year of study, winter semester, recommended course

Type of course unit

 

Lecture

20 hod., optionally

Teacher / Lecturer

doc. Ing. Petr Tomášek, Ph.D.

Syllabus

The course has 10 two-hours lessons.
1. Introduction to numerical mathematics: foundation of matrix analysis, errors, conditionning of problems and algorithms.
2. Direct methods for solving linear systems: Gaussian elimination method, pivoting, LU decomposition, Cholesky decomposition, conditioning.
3. Iterative methods for solving linear systems: classical iterative methods (Jacobi, Gauss-Seidel, SOR, SSOR), generalized minimum rezidual method, conjugate gradient method.
4. Interpolation: Lagrange, Newton and Hermite interpolation polynomial, interpolating splines.
5. Least squares method: data fitting, solving overdetermined systems (QR factorization, pseudoinverse, orthogonalization methods).
6. Numerical differentiation: basic formulas, Richardson extrapolation.
7. Numerical integration: Newton-Cotes formulas, Gaussian formulas, adaptive integration.
8. Solving nonlinear equations in one dimension (bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, Brent method); solving nonlinear systems (Newton's method and its variants, fixed point iteration).
9. Eigenvalues and eigenvectors: power method, QR method.
10. Eigenvalues and eigenvectors: Arnoldi method, Jacobi method, bisection method, computing the singular value decomposition.