Gurantor department | IT4Innovations | Credits | 6 |

Subject guarantor | prof. Ing. Tomáš Kozubek, Ph.D. | Subject version guarantor | prof. Ing. Tomáš Kozubek, Ph.D. |

Study level | undergraduate or graduate | ||

Study language | Czech | ||

Year of introduction | 2016/2017 | Year of cancellation | |

Intended for the faculties | USP | Intended for study types | Follow-up Master |

Instruction secured by | |||
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Login | Name | Tuitor | Teacher giving lectures |

KOZ75 | prof. Ing. Tomáš Kozubek, Ph.D. |

Extent of instruction for forms of study | ||
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Form of study | Way of compl. | Extent |

Full-time | Credit and Examination | 2+2 |

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.

Lectures

Tutorials

Project work

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.

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

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

No other requirements.

Subject has no prerequisities.

Subject has no co-requisities.

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

Task name | Type of task | Max. 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 |

Show history

Academic year | Programme | Field of study | Spec. | Form | Study language | Tut. centre | Year | W | S | Type 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 |

Block name | Academic year | Form of study | Study language | Year | W | S | Type of block | Block owner |
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