Numerical Methods (FSI-2NU)

Academic year 2020/2021
Supervisor: doc. Ing. Jiří Šremr, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The aim of the course is to familiarize students with essential methods applied for solving numerical problems and provide them with an ability to solve such problems individually by hand and especially on computer. Students ought to realize that only the knowledge of substantial features of particular numerical methods enables them to choose a suitable method and an appropriate software product.
Learning outcomes and competences:
Students will be made familiar with a basic collection of numerical methods. They will make sense of errors in mathematical modelling, learn to find zeros of nonlinear equation and to solve systems of linear equations. They will master the basics of approximation including the least squares method, manage to use quadrature formulas and obtain an initial insight into the unconstrained minimization.
Prerequisites:
Numerical linear algebra, approximation of functions, numerical differentiation and integration, differential and integral calculus, basic Matlab programming.
Course contents:
Students will be made familiar with a basic collection of numerical methods. They will make sense of errors in mathematical modelling, learn to find zeros of nonlinear equation and to solve systems of linear equations. They will master the basics of approximation including the least squares method, manage to use quadrature formulas and obtain an initial insight into the unconstrained minimization.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in seminars. Students have to pass two check tests successfully and to work out semester assignment solved by means of MATLAB. A student can receive up to 20 points for both tests and up to 10 points for a semester assignment, in total up to 30 points. A necessary condition for course credit acquirement is a gain of at least 15 points, including at least 10 points in both check tests. Students, who reach course-unit credits, thus obtain from 15 to 30 points, which will be included in the final course classification.
FORM OF THE EXAMINATIONS: The exam has a practical and a theoretical part. In the practical part students solve several numerical examples by hand using a scientific calculator. In the theoretical part they answer several questions to basic notions in order to check up how they understand the subject. Students will obtain 0--70 points as a result of the exam.
FINAL ASSESSMENT: The final point course classicifation is the sum of points obtained from both the seminars (15--30) and the exam (0--70).
FINAL COURSE CLASSIFICATION: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.
Controlled participation in lessons:
Attendance at seminars is checked. Lessons are planned according to the week schedules. Absence may be replaced by the agreement with the teacher.
Type of course unit:
    Lecture  13 × 1 hrs. optionally                  
    Computer-assisted exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture Two-hour lessons take place every other week.
Week 1-2. Introduction to computing: Error analysis. Computer arithmetic. Conditioning of problems, stability of algorithms.
Solving linear systems: Gaussian elimination. LU decomposition. Pivoting.
Week 3-4. Solving linear systems: Effect of roundoff errors. Conditioning. Iterative methods (Jacobi, Gauss-Seidel, SOR method).
Week 5-6. Interpolation: Lagrange, Newton and Hermite interpolation polynomial. Piecewise linear and piecewise cubic Hermite interpolation. Cubic interpolating spline. Least squares method.
Week 7-8. Numerical differentiation: Basic formulas. Richardson extrapolation.
Numerical integration: Basic quadrature rules (midpoint, trapezoidal and Simpson's rule). Gaussian quadrature. Composite quadrature. Adaptive quadrature.
Week 9-10. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
Solving nonlinear systems: Newton's method, fixed point iteration.
Week 11-12. Minimization of a function of one variable: golden ratio, quadratic interpolation.
Minimization methods for multivariable functions: Nelder-Mead method, steepest descent and Newton's method.
Week 13. Teacher's reserve.
    Computer-assisted exercise Seminars are organized in biweekly cycles, alternatively in a classical classroom and in a computer lab. The seminar schedule corresponds to the subject of the corresponding lecture.
Literature - fundamental:
1. HEATH, Michael T. Scientific computing: an introduction survey. 2nd ed. Boston: McGraw-Hill, 2002, 563 s. ISBN 0-07-239910-4.
2. MOLER, Cleve B. Numerical computing with MATLAB. Philadelphia: SIAM, 2004, xi, 336 s. : il. ISBN 0-89871-560-1.
3. DAHLQUIST, Germund a Ake BJÖRCK. Numerical Methods. New Jersey: Englewood Cliffs, 1974, 573 s.
4. MATHEWS, John H. a Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River: Pearson Prentice Hall, 2004, ix, 680 s. : il. ISBN 0-13-191178-3.
Literature - recommended:
1. ČERMÁK, Libor a Rudolf HLAVIČKA. Numerické metody. Brno: Akademické nakladatelství CERM, 2016.
2. MOLER, Cleve B. Numerical computing with MATLAB. Philadelphia: SIAM, 2004, xi, 336 s. : il. ISBN 0-89871-560-1.
3. MATHEWS, John H. a Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River: Pearson Prentice Hall, 2004, ix, 680 s. : il. ISBN 0-13-191178-3.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-ZSI-P full-time study STI Fundamentals of Mechanical Engineering -- Cr,Ex 4 Compulsory 1 1 S
B-FIN-P full-time study --- no specialisation -- Cr,Ex 4 Compulsory 1 1 S