Course detail

Mathematics 5 (S)

FAST-CA001Acad. year: 2022/2023

Errors in numeric calculations, solvig transcendental equations in one and several unknowns using iteration methods. Interpolation and approximation of function. Numerical differentiation and integration and their application to solving boundary value problems for ordinary differential equations. Applications given by the specialization.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

The outputs of this course are the skills and the knowledge which enable the graduates understanding of basic numerical problems and of the ideas on which the procedures for their solutions are based. In their future practice they will be able to recognize the applicability of numerical methods for the solution of technical problems and use the existing universal programming systems for the solution of basic types of numerical problems and their future improvements effectively.

Prerequisites

Basic notions of the theory of functions of one real variable (derivative, limit, continuos functions, elementary functions). Calculating integrals of functions of one variable, knowing about their basic applications.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Errors in numerical calculations, approximation of the solutions of one equation in one real variable by bisection and by iteration
2. Approximation of the solutions of one equation in one real variable by iteration, the Newton method and its modifications
3. Norms of matrices and vectors, calculations of the inverse matrices
4. Solutions of systems of linear equations with speciál matrice and the condition numer of a matrix
5. Solutions of systems of linear equations by iteration
6. Solutions of systems of non—linear equations
7. Lagrange interpolation by polynomials and cubic splines, Hermite interpolation by polynomials and Hermite cubic splines
8. The discrete least squares Metod, numerical differentiation
9. Classical formulation of the boundary—value problem for the ODE of second order and its approximation by the finite diference method
10. Numerical integration. Variational formulation of the boundary—value problem for the ODE of second order
11. Discertization of the variational boundary—value problem for the ODE of second order by the finite element method
12. Classical and variational formulations of the boundary—value problem for the ODE of order four
13. Discertization of the variational boundary—value problem for the ODE of order four by the finite element method



Work placements

Not applicable.

Aims

The students should understand the basic principles of numeric calculation, the factors that influence numeric calculation. They should be able to solve selected basic problems in numerical mathematics, understand the principle of iteration methods for solving the equation f(x)=0 and systems of linear algebraic equations mastering the calculation algorithms. They should learn how to get the basics of interpolation and approximation of functions to solve practical problems. They should be acquainted with the principles of numerical differentiation to be able to numerically solve boundary problems for ordinary differential equations. They should be able to numerically solve definite integrals.

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-P-C-SI Master's

    branch S , 1 year of study, winter semester, compulsory
    branch S , 1 year of study, winter semester, compulsory
    branch S , 1 year of study, winter semester, compulsory

  • Programme N-K-C-SI Master's

    branch S , 1 year of study, winter semester, compulsory
    branch S , 1 year of study, winter semester, compulsory
    branch S , 1 year of study, winter semester, compulsory

  • Programme N-P-E-SI Master's

    branch S , 1 year of study, winter semester, compulsory
    branch S , 1 year of study, winter semester, compulsory
    branch S , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

1. Errors in numerical computations. Contractive mappings, application to solution of nonlinear algebraic equations: simple iterative method, Newton method, method of secants. 2. Direct methods for solution of systems of linear algebraic equations, namely multiplicative decompositions: LU decomposition, Choleski decomposition, idea of QR decomposition. 3. Iterative and relaxation methods for solution of systems of linear algebraic equations, namely Jacobi and Gauss-Seidel methods including relaxation. 4. Conjugate gradient method, namely for systems of linear algebraic equations. Newton method for nonlinear systems. 5. Conditionality of systems of equations. Least squares method: idea, discrete case. 6. Lagrange interpolating polynomial, namely Newton form. Hermite interpolating polynomial. 7. Cubic splines: idea for Lagrange splines, calculations for Hermite splines. 8. Numerical differentiation. Finite difference method, application to boundary value problems for ordinary differential equations of order 2. 9. Numerical integration: rectangular, trapezoidal and Simpson rule, including approximation error estimate. Idea of more-dimensional numerical integration. 10. Finite element method, application to boundary value problems for ordinary differential equations of order 2. 11. Time-dependent problems: Euler explicit and implicit method, Crank-Nicolson method and Runge-Kutta methods, application to initial value problems for ordinary differential equations of order 1. 12. Continuation and completion of preceding themes, comments to engineering applications. 13. Finite element method for partial differential equations, example of equation of heat transfer.

Exercise

13 hod., compulsory

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

1.-2. Introduction to MATLAB: MATLAB environment, MATLAB online, assignment to variables, double dot, operations with number and vectors, plot, comments, MATLAB help, cycle for-end and condition if-else-end. Setting individual semester work. 3.-4. Repetition of methods for solution of 1 nonlinear equation: function graph and root estimate, script for 1 specific example and method of bisection, generalization for an arbitrary functions and initial inputs (for, if, plot, anonymous function). 5.-7. Implementation of iterative methods for solution of systems of linear algebraic equations: matrix operations (*, .*, +, inv, det, size and similar), vector norm, creation of solver with a lower triangular matrix, consequently creation of script for Gauss-Seidel method in matrix notation, creation of a function including check of inputs (diagonal dominance, etc.). 8.-9. Approximation of functions: least squares method in matrix form, usage of prepared Gauss-Seidel iteration for solution of a normal equation, Lagrange interpolation – form of a polynomial and setting coefficients, possible relation to numerical integration following composed rectangular rule. 11.-12. Ordinary differential equations: explicit and implicit Euler method for order 1, finite difference method for order 2, utilization of prepared solver of systems of linear algebraic equations, comparison with finite element method. 13. Evaluation of semester work.