Course Details

Mathematics 5 (S)

NAA019 course is part of 3 study plans

Ing. full-t. program NPC-SIS compulsory Winter Semester 1st year 4 credits

Ing. full-t. program NPA-SIS compulsory Winter Semester 1st year 4 credits

Ing. combi. program NKC-SIS compulsory Winter Semester 1st year 4 credits

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

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Learning outcomes

The aim of the course is to give students the basic knowledge from numerical mathematics, to be able to use it in technical applications.

Prerequisites

Basic courses of mathematics for bachelor students, MATLAB programming (as in the recommended course at MAT FCE).

Corequisites

Not required.

Planned educational activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.

Forms and criteria of assessment

Evaluation of work on specified tasks, periodical tests, examination in the form of final written test.

Objective

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 instruction, the form of instruction, and the form of compensation of the absences

Vymezení kontrolované výuky a způsob jejího provádění stanoví každoročně aktualizovaná vyhláška garanta předmětu.

Lecture

2 hours/week, 13 weeks, elective

Syllabus of lectures

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.

Practice

1 hours/week, 13 weeks, compulsory

Syllabus of practice

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.