Course Details

Numerical methods

Academic Year 2024/25

NAA027 course is part of 1 study plan

​ NPC-GK Summer Semester 1st year

Introduction to numerical mathematics, namely interpolation and approximations of functions, numerical differentiation and quadrature, analysis of algebraic and differential equations and their systems.

Credits

2 credits

Language of instruction

Czech

Semester

summer

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Forms and criteria of assessment

course-unit credit

Entry Knowledge

Basic knowledge of mathematical analysisi at the level of bachelor courses, ability to study mathematical textbooks.

Aims

To understand fundamentals of numerical methods for the interpolation and approximation of functions and for the solution of algebraic and differential equations, reqiured in the technical practice.
Following the aim of the course, students will be able to apply numerical approaches to standard engineering problems.

Basic Literature

DALÍK J.: Numerické metody. CERM Brno 1997. (cs)
VALA J.: Numerická matematika. FAST VUT v Brně 2021. (cs)

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/281855

Exercise

13 weeks, 2 hours/week, compulsory

Syllabus

  • 1. Errors in numerical calculations. Linear spaces and operators, fixed point theorems. Iterative methods for the analysis of nonlinear algebraic and selected further equations.
  • 2. Iterative and coupled methods for the analysis of linear algebraic equations, relaxation methods, method of conjugated gradients.
  • 3. Multiplicative decomposition of matrices. Numerical evaluation of eigenvalues and eigenvectors of matrices and of inverse matrices, algorithms for special matrices.
  • 4. Condition numbers of systems of linear equations. Least squares method, pseudoinverse matrices.
  • 5. Generalizations of methods from 3. and 4. to the analysis of systems of nonlinear equations.
  • 6. Lagrange and Hermite interpolation of functions of 1 variable, namely polynoms and splines.
  • 7. Approximation of functions of 1 variable using the least squares methos: linear and nonlinear approach.
  • 8. Approximation of function of more variables.
  • 9. Numerical differentiation. Finite difference method for the analysis of selected initial and boundary problems for ordinary differential equations.
  • 10. Numerical quadrature. Finite element method for the analysis of selected initial and boundary problems for ordinary differential equations.
  • 11. Time-dependent problems. Time discretization: Euler methods, Cranka-Nicholson method, Runge-Kutta methods, Newmark method.
  • 12. Generalization of 9. and 10. for pro partial differential equations of evolution, e.g. heat transfer equations, fluid flow equations and equations of dynamics of building structures.
  • 13. Sensitivity and inverse problems. Identification of uncertain material parameters from known measurement results. Selected engineering application, due to other courses.

An iniciative own study of theoretical backgroung is assumed, without any supporting lectures.