9600-1004/01 – Programming of Numerical Methods (PNM)

Gurantor departmentIT4InnovationsCredits6
Subject guarantorprof. Ing. Tomáš Kozubek, Ph.D.Subject version guarantorprof. Ing. Tomáš Kozubek, Ph.D.
Study levelundergraduate or graduate
Study languageCzech
Year of introduction2016/2017Year of cancellation
Intended for the facultiesUSPIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
KOZ75 prof. Ing. Tomáš Kozubek, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2

Subject aims expressed by acquired skills and competences

Upon the successful completion of the course students will be able to actively use new terms in the field of numerical methods, which are essential for understanding modern computational methods.

Teaching methods

Lectures
Tutorials
Project work

Summary

One of the important methods for solving many technical problems (for example full text search, signal analysis, optimal control, or solving differential equations) are deeper results of linear algebra and numerical mathematics. The aim of the course is to introduce students to these results. Upon the successful completion of the course, students will be able to use an appropriate numerical method for a given class of problems and choose it on the basis of acquired theoretical knowledge. It includes analysis of errors, stability, and computational demands. Within the tutorials, students will implement these methods and conduct numerical experiments on selected model problems in MATLAB.

Compulsory literature:

1. Yousef Saad, Iterative Methods for Sparse Linear Systems, Second Edition, Apr 30, 2003, 2. Gene H. Golub and Charles F. Van Loan, Matrix Computations (Johns Hopkins Studies in the Mathematical Sciences), Dec 27, 2012

Recommended literature:

1. Tomáš Kozubek, Tomáš Brzobohatý, Václav Hapla, Marta Jarošová, Alexandros Markopoulos – Lineární algebra s Matlabem, http://mi21.vsb.cz/modul/linearni-algebra-s-matlabem

Way of continuous check of knowledge in the course of semester

E-learning

Další požadavky na studenta

No other requirements.

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

1. Introduction • Motivational Examples (full text search, computation of string/membrane deflection using mesh methods, signal and image analysis) • Fundamentals of Linear Algebra (vector space, basis, linear representation, matrix, scalar multiplication, orthogonality, norm) • Correctness, Stability, Types of Errors • Numerical Approximation on a Computer • Insight into the Analysis of Computational Demands and Complexity • Storage Formats for Dense and Sparse Matrices (CSR, CSC, …) 2. Direct Solvers for Systems of Linear Equations • Summary of Systems Types and Their Solvability • Gaussian Elimination • Inverse Matrix • LU Decomposition • Cholesky and LDLT Decomposition • Stabilization with Partial and Complete Pivoting 3. Orthogonal and Spectral Problems • Gram-Schmidt process, Its Versions (classical, modified, iterative) • Householder Transformation, Givens Transformation • QR Decomposition • Eigenvalue and Spectral Decomposition • Eigenvalue Estimations • Dominant Eigenvalue Computation (power method, Lanczos method, spectral shift and reduction) • Calculation of Spectral Decomposition using QR algorithm • Singular Value Decomposition (SVD) • Generalized Inversion 4. Iterative Solvers for Systems of Linear Equations • Linear Methods (Jacobi, Gauss-Seidel, and Successive Over-relaxation (SOR) methods) • Gradient Methods (method of steepest descent, Krylov methods) • Preconditioning 5. Numerical Methods for Solving Non-linear Equations • Root Separation • Bisection Method • Simple Iteration Method • Newton’s Method 6. Interpolation and Approximation problems • Polynomial Interpolation • Lagrange Polynomial Interpolation • Newton Polynomial Interpolation • Linear and Cubic Spline • Method of Least Squares • Orthogonal Systems of Functions 7. Numerical Differentiation and Integration

Conditions for subject completion

Full-time form (validity from: 2016/2017 Winter semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
        Credit Credit 40  20
        Examination Examination 60  20
Mandatory attendence parzicipation:

Show history

Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2018/2019 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 1 Compulsory study plan
2017/2018 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 1 Compulsory study plan
2016/2017 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 1 Compulsory study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner