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Numerical Methods
Major: Software Engineering
Code of subject: 6.121.00.O.014
Credits: 5.00
Department: Software
Lecturer: Assoc. Prof. Melnyk Nataliia
Semester: 2 семестр
Mode of study: денна
Завдання: The study of an educational discipline involves the formation of competencies in students of education:
general competences:
• Ability to learn and master modern knowledge.
professional competences:
• Ability to apply and develop fundamental and interdisciplinary knowledge to successfully solve software engineering tasks.
• Ability to algorithmic and logical thinking.
• Ability to demonstrate understanding of the scientific and mathematical principles underlying information technology.
Learning outcomes: Learning outcomes Learning and teaching methods Methods of assessing the level of achievement of learning outcomes
PR05. Know and apply relevant mathematical concepts, methods of domain, system and object-oriented analysis and mathematical modeling for software development. Lectures and practical classes: information-receptive method; reproductive method. Independent work: reproductive method. Current and examination control.
Knowledge assessment methods: selective oral survey in practical classes; protection of laboratory works; test papers in practical classes; tests Exam - written component, test control, oral component.
PR13. Know and apply methods of developing algorithms, designing software and data and knowledge structures. Lectures and practical classes: information-receptive method; reproductive method. Independent work: reproductive method. Current and examination control.
Knowledge assessment methods: selective oral survey in practical classes; protection of laboratory works; test papers in practical classes; tests Exam - written component, test control, oral component.
PR27. Use the theory and methods of selected sections of mathematics and physics to build computational algorithms that are implemented in software systems of various purposes. Lectures and practical classes: information-receptive method; reproductive method. Independent work: reproductive method. Current and examination control.
Knowledge assessment methods: selective oral survey in practical classes; protection of laboratory works; test papers in practical classes; tests Exam - written component, test control, oral component.
PR28. Be able to mathematically formulate technical problems used in software engineering. Lectures and practical classes: information-receptive method; reproductive method. Independent work: reproductive method. Current and examination control.
Knowledge assessment methods: selective oral survey in practical classes; protection of laboratory works; test papers in practical classes; tests Exam - written component, test control, oral component.
Required prior and related subjects: • Linear algebra and analytic geometry
• Mathematics Analysis
• Basic programming
• Object-Oriented Programming
Summary of the subject: Mathematical modeling and computational experiment. Elements of the theory of errors. Solving nonlinear equations. Direct methods for solving systems of linear algebraic equations. Iterative methods for solving systems of linear algebraic equations. Solving systems of nonlinear equations. Interpolation functions. Approximation functions. Numerical differentiation. Numerical integration. Numerical methods of function optimization. Numerical methods for solving the Cauchy problem for ordinary differential equations. Numerical solution for ordinary differential equations.
Опис: 1. Introduction. Mathematical modeling and computational experiment.
2. Basic concepts of error theory. Approximate calculations and elements of error theory. Sources and classification of errors.
3. Classification of nonlinear equations. Approximate solution of nonlinear equations. Root separation methods. The method of dividing a segment in half.
4. Solving nonlinear equations. Chord method. Newton's (tangent) method. Method of simple iteration (successive approximations).
5. Direct methods of solving SLAR. The inverse matrix method. Kramer's method. Gauss method. Gaussian method with selection of the main element. Application of the Gaussian method for finding determinants. LU-distribution method. The square root method. The method of running.
6. Iterative methods of solving SLAR. Simple iteration method (Jacobi) Convergence of the iterative process. Criteria for terminating the iterative process. Seidel's method.
7. Solving systems of nonlinear equations. Simple iteration method. Newton's method.
8. Approximation of functions. Interpolation of tabular functions. Lagrange interpolation polynomial. Newton's interpolating polynomial. Inverse interpolation.
9. Approximation of functions. Formulation of the function approximation problem. The method of least squares for approximation of functions. Partial cases of approximation polynomials.
10. Numerical integration. Formulation of the problem of numerical integration. Method of rectangles. Trapezium method. Simpson's method. Selection of the integration step.
11. Numerical methods of optimization of functions. One-dimensional optimization problem. Algorithm for separation of unimodality segments. Numerical methods of one-dimensional optimization. The method of dividing a segment in half. Golden ratio method. Fibonacci method. Comparison of optimization methods.
12. Numerical differentiation. Formulation of the problem of numerical differentiation. Numerical differentiation formulas. Errors of numerical differentiation formulas. Approximate differentiation based on Newton interpolation. Error in determining the derivative.
13. Solving Cauchy problems by numerical methods. Differential equations. Cauchy's problem. Euler's method. The Runge-Kutta method. Geometric interpretation of determining the solution of the Cauchy problem by the Runge-Kutta method of the 4th order. Multi-step forecasting and correction methods.
14. Approximate solution of differential equations with partial derivatives. Classification of mathematical physics equations. Classical examples of equations. Boundary conditions. Basic concepts of the finite difference method. Grid method for solving elliptic type differential equations.
15. Finite element method of solving differential equations with partial derivatives. The difference between the finite difference method and the finite element method. The main types of elements in the finite element method. Stages of the method of finite elements. Solving the Fredholm equations.
Assessment methods and criteria: 1. The defense of laboratory works includes a demonstration of the work of the program in accordance with the individual version, preparation of a written report and answers to questions on the topic of the work.
2. Selective oral examination in practical classes consists in solving problems at the blackboard and answering questions from the formed lists for each topic.
3. Control measures include testing at the National Emergency Service and written problem solving.
4. The examination work consists of a written component (test) and an oral component (individual survey).
Критерії оцінювання результатів навчання: Current control - 40 points
laboratory classes - 30 points
practical classes - 10 points
Examination control - 60 points
written component - 55 points
oral component - 5 points
Total for the discipline - 100 points
Порядок та критерії виставляння балів та оцінок: 1. Each laboratory work is evaluated for 3 points. If the work is defended late, the grade is reduced by 1 point for each week of delay (starting with the third week of delay, the grade for laboratory work is 0 points).
Protection of laboratory work is carried out according to the schedule:
No. 1 second, third weeks of study;
No. 2 fourth, fifth week of study;
No. 3, the sixth week of study;
No. 4 of the seventh, eighth week of study;
No. 5 ninth week of study;
No. 6 tenth week of study;
No. 7 – the eleventh week of study;
No. 8 – the twelfth week of study;
No. 9 – the thirteenth week of study;
No. 10 fourteenth, fifteenth week of study.
2. The points of the current control are calculated before the beginning of the session.
3. A student who completed less than 50% of the work of the current control is considered uncertified. For him, repeated study of the discipline is proposed.
4. A student who completed more than 50% of the work of the current control, but not all 100%, is considered ineligible for the exam and has the opportunity to complete and defend the work and pass the exam at the commission.
Recommended books: 1. Мельник Н.Б. Чисельні методи: Конспект лекцій. – Львів: Видавництво Львівської політехніки, 2019. – 116 с.
2. Гавриш В.І., Мельник Н.Б. Чисельні методи. Лабораторний практикум: навчальний посібник. – Львів, Видавництво Львівської політехніки, 2018. – 136 с.
3. Коссак О., Тумашова О., Коссак О. Методи наближених обчислень: Навч. посібн. – Львів: Бак, 2003. – 168 с.
4. Фельдман Л.П., Петренко А.І., Дмитрієва О.А. Чисельні методи в інформатиці. – К.: Вид. група BHV, 2006. – 480 с.
5. Шахно С.М., Дудикевич А.Т., Левицька С.М. Практикум з чисельних методів: навч. посібник. – Львів: ЛНУ імені Івана Франка, 2013. – 432 с.
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