Computational Methods and Data Visualization

Major: System Analysis
Code of subject: 6.124.00.O.062
Credits: 4.00
Department: Information Systems and Networks
Lecturer: Ph.D., Associate Professor V.A.Vysotska, Ph.D., V.A. Andrunik
Semester: 5 семестр
Mode of study: денна
Learning outcomes: As a result of studying the discipline, the applicant must be able to: 1. The student must know and understand the basic numerical methods for solving ordinary algebraic equations of systems of equations, differential and integral equations. Students will gain the necessary knowledge of the algorithmic implementation of the studied methods, methods of proving statements, areas and methods of applying the acquired knowledge in the following sections: • solving linear equations; • finding eigenvalues ??and matrix vectors; • finding singular values ??and matrix vectors; • numerical solution of nonlinear algebraic equations and their systems; • numerical solution of systems of nonlinear algebraic equations; • numerical solution of differential equations and systems (both ordinary differential equations and equations with partial derivatives); • numerical solution of systems of differential equations; • numerical solution of integral equations; • problems of function approximation; • interpolation problems of functions; • numerical integration and calculation of the derivative; • extrapolation problems; • optimization tasks; • inverse problems. 2. a trained specialist must be able to carry out a meaningful statement of the problem, followed by the transition to the use of selected numerical methods. He must also be able to design, program, test and debug programs that implement numerical methods; solve mathematical problems using mathematical packages; to make a reasonable choice of numerical method in solving a practical problem. Students must be able to solve basic classes of equations in almost numerical ways and apply the acquired knowledge to • formulation and solution of system analysis problems, • construction of algorithms for solving applied problems, • study of new information technologies, • construction of a mathematical description of applied problems, • analysis of the results of solving problems. 3. algorithmize algorithms of calculation methods for the corresponding specific tasks of system analysis. 4. apply known numerical methods and algorithms for the study of the subject area. 5. understand mathematical calculations of related and subsequent disciplines such as algorithmization and programming, machine learning, data mining, object-oriented programming, mathematical methods of operations research, etc. 6. Build your own algorithms for data analysis, including Big Data, Data Mining and Artificial Intelligence Systems. 7. perform the formulation of tasks for the design of visual data display; 8. apply methods of processing the obtained data, analyze the data according to the task; 9. to carry out graphic research of data, to carry out their visual interpretation; 10. create dynamic dashboards, interactive visualization for the web.
Required prior and related subjects: Algorithmization and programming Programming and teamwork Object-oriented programming
Summary of the subject: Fundamentals of numerical methods. Mathematical modeling. Methods of solving mathematical problems. Numerical Methods. Fundamentals of the theory of error calculation. The structure of the error of the solution of the problem. Function errors. The inverse problem of error theory. Representation of numbers in the PC. Stability. Correctness.. Methods for solving linear algebraic equations. Statement of the problem of mathematical programming. Systems of linear inequalities. The problem of linear programming. Geometric meaning. Geometric interpretation. Solving systems of linear algebraic equations. Some concepts of matrix algebra. Representation of a linear system in matrix form. Solving systems of linear equations in matrix form. Cramer's method. Gaussian method. Square roots method. Matrix method. Iterative methods. Methods for solving nonlinear equations. Numerical methods for solving problems with one variable. Separation of roots. The method of dividing a segment in half. Tangent method. Chord method. Combined method. Iteration method. Interpolation function. The problem of an approximate function. Interpolation polynomials of Newton and Lagrange. Estimation of interpolation error. Extrapolation and inverse interpolation. Interpolation of the function using splines Approximation. Methods of experimental data processing. The task of the best approximation. Uniform approximation. RMS approximation. The least squares method of approximation of a function given in a table. Construction of empirical formulas, determination of dependence parameters. Smoothing of tabular functions. Numerical integration of functions. The problem of numerical integration. Construction of quadrature formulas. Estimation of numerical integration error. Newton-Cotes quadrature formulas. Formulas of rectangles, trapezoids, Simpson. Approximate calculation of multiple integrals. Numerical differentiation. Incorrectness of the problem of numerical differentiation. Using interpolation polynomials to construct numerical differentiation formulas. Estimation of numerical differentiation error. Solving ordinary differential equations. Numerical methods for solving ordinary differential equations. Statement of the Cauchy problem. Euler's method. Runge-Kutta method. Accuracy assessment. Boundary value problems. Finite difference method. Second order equations and methods of their solution. Numerical methods for solving optimization problems. General formulation of the optimization problem. Unconditional optimization. Multicriteria optimization problems and basic approaches to their solution Analysis of statistical data. Forms and methods of presentation and preliminary statistical processing of numerical data of time sequences. Detection of time series trends by smoothing methods. Correlation analysis of time sequences. hierarchical agglomerative cluster analysis of multidimensional data. Visualization of numerical data. Visualization of geolocation (maps) data. Visualization of categorical data. Visualization of relational (network) data. Visualization of numerical and categorical data. Create dynamic, interactive visualizations in web browsers. Real-time visualization of log data.
Assessment methods and criteria: • Current control (45%): written reports on laboratory work, essay, oral examination; • Final control (55%, of exam): in written, verbally.
Recommended books: 1. Андруник В. А. Чисельні методи в комп’ютерних науках. Том 1: навчальний посібник / В. А. Андруник, В. А. Висоцька, В. В. Пасічник, Л. Б. Чирун, Л. В. Чирун. – Львів: Новий Світ – 2000, 2017. – 470 c. 2. Андруник В. А. Чисельні методи в комп’ютерних науках. Том 2: навчальний посібник / В. А. Андруник, В. А. Висоцька, В. В. Пасічник, Л. Б. Чирун, Л. В. Чирун. – Львів: Новий Світ – 2000, 2018. – 536 c. 3. Висоцька В.А., Оборська О.В. Python: алгоритмізація та програмування: навчальний посібник – Львів: Видавництво «Новий Світ – 2000», 2020. – 516 с. 4. Ришковець Ю.В., Висоцька В.А. Алгоритмізація та програмування. Частина 1: Навчальний посібник. – Львів: «Новий Світ - 200», 2018. – 337 с. 5. Ришковець Ю.В., Висоцька В.А. Алгоритмізація та програмування. Частина 2: Навчальний посібник. – Львів: «Новий Світ - 2000», 2018. – 316 с. 6. Висоцька В.А., Литвин В.В., Лозинська О.В, Дискретна математика: практикум (Збірник задач з дискретної математики: Навчальний посібник. – Львів: Новий Світ – 2000, 2019. – 575 стор. 7. Noah Iliinsky and Julie Steele. Designing Data Visualizations. - Published by O’Reilly Media, Inc., USA, 2011. – 92 p. 8. Claus O. Wilke. Fundamentals of Data Visualization. Published by O’Reilly Media, Inc., USA, 2019. – 481 p. 9. Scott Murray. Interactive Data Visualization for the Web.- Published by O’Reilly Media, Inc., USA, 2014. – 268 p. 10. Chun-houh Chen, Wolfgang Hardle, Antony Unwin. Handbook of Data Visualization. – Springer-Verlag Berlin Heidelberg, 2008. – 954p. 11. James D. Miller. Big Data Visualization. - Packt Publishing, 2017. – 299 p.