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Regular version of the site
Bachelor 2021/2022

## Numerical Methods

Type: Compulsory course (Applied Mathematics)
Area of studies: Applied Mathematics
When: 3 year, 3, 4 module
Mode of studies: distance learning
Open to: students of one campus
Language: English
ECTS credits: 4

### Course Syllabus

#### Abstract

Numerical computations historically play a crucial role in natural sciences and engineering. These days however, it’s not only traditional «hard sciences»: whether you do digital humanities or biotechnology, whether you design novel materials or build artificial intelligence systems, virtually any quantitative work involves some amount of numerical computing . These days, you hardly ever implement the whole computation yourselves from scratch. We rely on libraries which package tried-and-tested, battle-hardened numerical primitives. It is vanishingly rare however that a library contains a single pre-packaged routine which does all what you need. Numerical computing involves assembling these building blocks into computational pipelines. This kind of work requires a general understanding of basic numerical methods, their strengths and weaknesses, their limitations and their failure modes. And this is exactly what this course is about. It is meant to be an introductory, foundational course in numerical analysis, with the focus on basic ideas. We will review and develop basic characteristics of numerical algorithms (convergence, approximation, stability, computational complexity and so on), and will illustrate them with several classic problems in numerical mathematics. You will also work on implementing abstract mathematical constructions into working prototypes of numerical code. Upon completion of this course, you will have an overview of the main ideas of numerical computing, and will have a solid foundation for reading up on and working with more advanced numerical needs of your specific subject area. As prerequisites for this course, we assume a basic command of college-level mathematics (linear algebra and calculus, mostly), and a basic level of programming proficiency.

#### Learning Objectives

• Basic command of modern numerical methods of applied maths
• Basic use of common packages and libraries for numeric and scientific computing

#### Expected Learning Outcomes

• Knowledge of mathematical foundations of numerical methods used in contemporary fundamental and applied research and development.
• Ability to choose an appropriate numerical method for a given problem.
• Experience working with modern software packages and libraries for numerical and scientific computing.

#### Course Contents

• Numerical mathematics.
Overview of computational problems: set-ups and approaches. Correctness, accuracy, stability, condition numbers. Roundoff errors. Machine arithmetics. Uncertainties.
• Matrix factorizations
QR factorization of a matrix. Householder reflections and Givens rotations. SVD factorization. Properties of singular vectors and singular values. Eckhart-Young theorem. Moore-Penrose pseudoinverse. Over- and under-determined systems of linear equations. Solving linear systems in a least-squares sense. Ordinary least squares.
• Eigenvalue problem.
The eigenvalue problem. Power iteration, inverse iteration. Shift-invert mode. QR algorithm.
• Systems of linear equations: direct methods.
Vector and matrix norms. Condition number of a linear system. Gaussian elimination and LU decomposition of a matrix. Pivoting. Computational complexity. Cholesky decomposition for PSD matrices. Tridiagonal systems, Thomas algorithm.
• Systems of linear equations: iterative methods.
Jacobi iteration. Seidel iteration. Convergence properties and convergence theorems. Canonic form of iterative methods. Variational approaches.
• Nonlinear equations and systems of equations.
Localization of roots. Bisection, simple iteration and its variations. Modified schemes, Newton's method. Convergence theorems. A priori and a posteriori error estimates. Multiple roots. Inverse quadratic interpolation. Hybrid schemes.
• Interpolation of functions.
Interpolation problem. Univariate interpolation. Interpolation polynomial in the Lagrange form. Runge phenomenon. Chebyshev nodes. Piecewise polynomial interpolation, splines. Hermite cubic polynomial. Construction of a spline interpolant. Boundary conditions. Monotone interpolants.
• Numerical differentiation
Finite-difference formulas. Optimal differentiation step. Construction of higher-order FD schemes. Richardson extrapolation and Neville algorithm. Algorithmic differentiation.
• Numerical integration
• Initial value problem for ODEs
Ordinary differential equations. Initial value problem. Implicit and explicit finite difference schemes. Euler rule and its modifications. Runge-Kutta methods. Linear multistep methods: convergence, approximation and stability. Zero-stability of LMMs. A-stability, stiff systems. Dahlquist barriers.
• Fredholm integral equations.
Fredholm equations of the first and second kinds. Fredholm alternative. Nystrom method of solving FIE of the 2nd kind.
• Monte Carlo methods.
Buffon's needle. MC integration in d>1. Pseudo random and quasi-random (low-discrepancy) sequences. Sampling from non-uniform distributions. Markov chain MC.

#### Assessment Elements

• In-class activity
• Exam
The final grade can be set based on the accumulated grade, without taking the exam.
• Online course
• Homework problems

#### Interim Assessment

• Interim assessment (4 module)
0.1 in-class activity + 0.2 online course + 0.5 HW + 0.2 exam

#### Recommended Core Bibliography

• Вычислительные методы для инженеров : учеб. пособие для вузов, Амосов, А. А., 2003