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F5330 Basic numerical methods

Faculty of Science
Autumn 2010

Extent and Intensity

1/1/0. 3 credit(s). Type of Completion: z (credit).

Teacher(s)

doc. RNDr. Jan Celý, CSc. (lecturer)
doc. RNDr. Jan Celý, CSc. (seminar tutor)

Supervisor

prof. RNDr. Josef Humlíček, CSc.
Department of Condensed Matter Physics - Physics Section - Faculty of Science
Contact Person: doc. RNDr. Jan Celý, CSc.

Timetable

Wed 17:00–17:50 F1,01014, Wed 18:00–18:50 F1,01014

Prerequisites

Knowledge of the programming (Pascal,Fortran, C,C++)

Course Enrollment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

Fields of study the course is directly associated with

Physics (programme PřF, B-FY)
Biophysics (programme PřF, M-FY)
Physics (programme PřF, M-FY)
Physics (programme PřF, N-FY)

Course objectives

The course presents to students knowledge on basic numerical methods: matrix operations, solving systems of linear algebraic equations and regression. Another part of the lecture deals with polynomial interpolation and solution of one-dimensional nonlinear equations.

After successful passing of the course the students should be able to
- list and describe basic numerical methods lectured
- successfully apply these methods for solving a specified problem.

Syllabus

1) Number representation in a computer,precision, accuracy. Errors in numerical algorithms, propagation of the errors. Stability of the algorthims. Ill-posed methods.
2) Systems of linear algebraic equations, direct and iterative metods.
The Gauss elimination method, pivoting. LU decomposition.
Systems with special matrices: The Choleski theorem and the Choleski method, tridiagonal systems.
Iterative methods: the Jacobi method, the Gauss-Seidel method. The problem of the convergence of the iteration methods.
3) Eigenvalues and eigenvectors of matrices. The Jacobi-method. The Householder transformation and the QR algorithm.
Iterative methods: the power method, convergence.
4) Singular value decomposition and its applications. Linear regression.
5) Interpolation: divided difference tables, polynomial interpolation, cubic splines.
6) The solution of nonlinear equations in 1D: bisection, Newton's and secant method, fixed-point iteration.

Literature

MÍKA, Stanislav. Numerické metody algebry. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1982. 169 s. info
HUMLÍČEK, J. Základní metody numerické matematiky. 1. vyd. Praha: Státní pedagogické nakladatelství, 1981. 171 s. info
CELÝ, Jan. Programové moduly pro fyzikální výpočty. 1. vyd. Brno: Rektorát UJEP, 1985. 99 s. info
PRESS, William H. Numerical recipes in C :the art of scientific computing. 2nd ed. Cambridge: Cambridge University Press, 1992. xxvi, 994. ISBN 0-521-43108-5. info
MARČUK, Gurij Ivanovič. Metody numerické matematiky. 1. vyd. Praha: Academia, 1987. 528 s. info
CELÝ, Jan. Řešení fyzikálních úloh na mikropočítačích. 1. vyd. Brno: Rektorát Masarykovy university, 1990. 108 s. ISBN 80-210-0126-7. info
PANG, Tao. An introduction to computational physics. 2nd ed. Cambridge: Cambridge University Press, 2006. xv, 385 s. ISBN 0-521-82569-5. info

Teaching methods