1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Computer science and econometrics - pierwszego stopnia z tyt. licencjata
5. Introduction to Numerical Methods

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Introduction to Numerical Methods - opis przedmiotu

Cel przedmiotu

The aim of the course Introduction to Numerical Methods is to familiarize students with the basic methods of approximate solution of mathematical problems, with particular emphasis on the choice of method for the problem to be solved. The choice of method depends on the formulation of the problem, the complexity of the method and the accuracy of the calculations. The given tasks and problems are illustrated with a large number of examples. After completing this course, the student should be prepared to independently use numerical methods and programming tools (e.g., Python) to solve applied mathematics problems arising in science, technology or engineering.

Wymagania wstępne

Zapisz zmiany

Students should pass the subjects: mathematical analysis 1 and 2, linear algebra 1 and 2.

Zakres tematyczny

Lecture

Computer arithmetic

1. Floating-point arithmetic (1 hour).
2. Absolute and relative errors. Real and machine numbers (1 hr).
3. Loss of significant digits (1 hr).
4. Stability and instability of algorithms. Conditioning (1 hr).

Solving nonlinear equations

1. Bisection method (1 hr).
2. Newton's method (2 hrs).
3. The secant method (1 hr).
4. Iterative methods (1 hr).
5. Calculation of polynomial roots (1 hr).

Solving systems of linear equations

1. Norms and error analysis (2 hrs).
2. LU factorization (2 hrs).
3. Gauss elimination (2 hrs).
4. Iterative methods (2 hrs).
5. Fastest gradient and conjugate gradient methods (2 hrs).

Interpolation and approximation of functions

1. Polynomial interpolation (4 hrs).
2. Chebyshev polynomials (2 hrs).

Numerical integration

1. Basic methods of calculating integrals (Simpson's method, trapezoidal formula, Newton-Cotes method) (4 hrs).

Exercises

Computer arithmetic

1. Floating-point systems - system conversion, machine numbers, absolute and relative errors (1 hour).
2. Stability and instability of algorithms. Conditioning (1 hr).

Solving nonlinear equations

1. Bisection, Newton's and secant methods - application of relevant formulas and theorems of convergence of methods (4 hrs.).

Solving systems of linear equations

1. Matrix norms, condition number and error analysis (1 hr).
2. Colloquium (1 hr).
3. LU factorization, Gauss elimination - application of relevant formulas and theorems of convergence of methods (2 hrs).
4. Jacobi, Gauss-Seidel, JOR and SOR iterative method - application of relevant formulas and theorems on convergence of methods (2 hrs).

Interpolation and approximation of functions

1. Polynomial interpolation methods - application of appropriate formulas and theorems on convergence of methods (2 hrs).
2. Colloquium (1 hr).

Laboratory

Computer arithmetic

1. Introduction to a selected mathematical package/programming language. (2 hrs).
2. Implementation of simple algorithms - study of stability and instability of solutions (2 hrs).

Solving nonlinear equations

1. Bisection, Newton's and secant methods - implementation of algorithms, numerical solution of problems, interpretation of results, learning about the functions available in the math package (4 hrs.).

Solving systems of linear equations

1. LU distributions, Gauss elimination, Jacobi method, Gauss-Seidel method, JOR and SOR - implementation of algorithms, numerical problem solving, interpretation of results, learning about functions available in the math package (5 hrs).

Interpolation and approximation of functions

1. Method of least squares, Newton's interpolation method, Lagrange and splines method - implementation of algorithms, numerical problem solving, interpretation of results, learning about functions available in the mathematics package (2 hrs).

Metody kształcenia

Lecture available in electronic version; during exercises and laboratories students solve computational problems analytically and using a given programming language  (e.g., Python).

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

The course grade consists of the grade from exercises and laboratory (40%) and the grade from the written exam (60%). The condition for taking the exam is a positive grade from exercises. The condition for passing the course is a positive grade from exercises, laboratories and the exam.

Literatura podstawowa

1. A. Björck, G.Dahlquist, Metody numeryczne, PWN, Warszawa, 1987.
2. D. Kincaid, W.Cheney, Analiza numeryczna, WNT, Warszawa, 2006.
3. J. Stoer, R .Bulirsch, Wstęp do analizy numerycznej, PWN, Warszawa, 1987.
4. R. L. Burden, J. D. Faires, Numerical analysis, Prindle, Weber & Schmidt, Boston, Massachusetts, 1981.
5. Z. Fortuna, B. Macukow, J. Wasowski, Metody numeryczne, WNT, Warszawa, 1993.

Literatura uzupełniająca

1. A. Quarteroni, R. Sacco, F. Saleri, Numerical mathematics, Springer, 2002.
2. A. Quarteroni, F. Saleri, Scientific Computing with Matlab and Octave, Springer, 2006.
3. P. Deuflhard, A. Hohmann Numerical analysis in modern scientific computing. An introduction, Springer, 2003.
4. A. Ralston, Wstep do analizy numerycznej, PWN, Warszawa 1983.
5. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer, 1993.

Uwagi

Zmodyfikowane przez dr Ewa Synówka (ostatnia modyfikacja: 07-02-2024 18:45)