Course Details

# Mathematics 5 (S)

Academic Year 2023/24

NAA019 course is part of 3 study plans

NPC-SIS Winter Semester 1st year

NKC-SIS Winter Semester 1st year

NPA-SIS Winter 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.

Course Guarantor

Institute

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.

Knowledge

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.

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.

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.

Prerequisites

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

Language of instruction

Czech, English

Credits

4 credits

Semester

winter

Forms and criteria of assessment

course-unit credit and examination

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

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

Offered to foreign students

To offer to students of all faculties

Course on BUT site

Lecture

13 weeks, 2 hours/week, elective

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 weeks, 1 hours/week, compulsory

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.