Course Details

Mathematics 5 (S)

Academic Year 2025/26

NAA019 course is part of 3 study plans

​ NPA-SIS Winter Semester 1st year

​ NPC-SIS Winter Semester 1st year

​ NKC-SIS Winter Semester 1st year

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Language of instruction

Czech, English

Credits

4 credits

Semester

winter

Forms and criteria of assessment

course-unit credit and examination

Offered to foreign students

To offer to students of all faculties

Course on BUT site

https://www.vut.cz/en/students/courses/detail/289555

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.