Computational Mathematics

General data

Course ID:	1000-113aMOBa
Erasmus code / ISCED:	11.102 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Computational Mathematics
Name in Polish:	Matematyka obliczeniowa (potok I)
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	Polish
Type of course:	obligatory courses
Short description:	In the course we present several methods of finding usually approximate solutions to basic mathematical problems when finding exact solutions is either impossible or impractical. The course includes: elements of error analysis, polynomial and spline interpolation, elements of approximation, quadrature rules, Gauss quadratures, numerical solutions of systems of linear equations, real roots of the equations, numerical eigenproblem.
Full description:	1/ Elements of fl rounding error analysis 2/Interpolation 2.1 Polynomial interpolation - specifying Lagrange interpolation problem - existence and uniqueness of a solution - finite differences algorithm - approximation error estimates 2.2 Spline approximation - definition of spline spaces - linear splines - cubic splines 3/Approximation 3.1 approximation in Hilbert spaces - existence and uniqueness - algorithms including the Gram-Schmidt process - orthogonal polynomials - properties and application to the polynomial approximation problem -Chebyshev polynomials and their properties - Linear Least Square problem as a special case of an approximation problem 3.2 Uniform approximation - (optional if time permits) 4/Numerical integration 4.1 Interpolation quadratures -Gauss quadratures 4.2 Quadrature rules: trapezoidal and Simpson rules 5/Numerical methods of solving system of algebraic equations - LU decompositions with partial pivoting - QR orthogonal decomposition: Housholder method - condition of the matrices and their influence on rounding error analysis of LU decomposition - an application of QR factorization of M x N matrix to Linear Least Square problems 6/ Roots of a nonlinear equation - bisection method - Newton method - Secant method - Banach iteration method - order of convergence 7/Numerical Eigenproblem (optional if time permits) - Power Method - Inverse Power Method
Bibliography:	David Kincaid and Ward Cheney, Numerical analysis. Mathematics of scientific computing. 2nd ed., Brooks/Cole Publishing Co., Pacific Grove, CA, 1996.

This course is not currently offered.

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