Introduction to numerical analysis

National Research University Higher School of Economics

About this Course

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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.

Do you have technical problems? Write to us: coursera@hse.ru

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Intermediate Level

Basic command of college-level mathematics (linear algebra and calculus). Basic level of programming proficiency: python programming language.

Approx. 18 hours to complete

English

Subtitles: English

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Intermediate Level

Basic command of college-level mathematics (linear algebra and calculus). Basic level of programming proficiency: python programming language.

Approx. 18 hours to complete

English

Subtitles: English

Instructor

Instructor rating

4/5 (18 Ratings)

Evgeni Burovski

Assistant professor

School of Applied mathematics, MIEM

12,285 Learners

1 Course

Offered by

National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more.

Syllabus - What you will learn from this course

Week

Week 1

4 hours to complete

Machine arithmetics. Systems of linear algebraic equations.

4 hours to complete

11 videos (Total 61 min), 3 readings, 2 quizzes

11 videos

About the University1m

Introduction.56s

A simple worked example.5m

Machine arithmetics. Representation of real numbers.6m

Machine epsilon. Over- and underflow.8m

A crude estimate of the machine epsilon.2m

Systems of linear equations. Cramer's rule.5m

Gaussian elimination.10m

LU decomposition: the matrix form of the Gaussian elimination.9m

When does the Gaussian elimination work?2m

LU decomposition with pivoting. Permutation matrices.8m

3 readings

About the University10m

About the course10m

Slides10m

Week

Week 2

3 hours to complete

Numerical linear algebra.

3 hours to complete

12 videos (Total 72 min), 2 readings, 1 quiz

12 videos

Introduction.1m

Sensitivity of a linear system.3m

Vector norms.5m

Matrix norms.3m

Common matrix norms.4m

Sensitivity of a linear system. Condition number.11m

Cholesky decomposition.7m

Banded matrices. Thomas algorithm.3m

Shermann-Morrison formula.8m

QR decomposition.6m

Constructing the QR decomposition: Householder reflections.13m

Constructing the QR decomposition: Givens rotations3m

2 readings

Slides10m

Week

Week 3

3 hours to complete

Non-linear algebraic equations.

3 hours to complete

11 videos (Total 71 min), 1 reading, 1 quiz

11 videos

Solving non-linear equations.6m

Localization of roots. Bisection.4m

Fixed-point iteration.4m

Aside: convergence rates and related technicalities.7m

Back to the fixed-point iteration.11m

Fine-tuning the fixed-point iteration.5m

Newton's iteration.16m

Multiple roots. Modified Newton's method.3m

Inverse quadratic interpolation.4m

Roots of polynomials.3m

Roots of polynomials: the companion matrix.3m

1 reading

Slides10m

Week

Week 4

2 hours to complete

Iterative method for linear systems.

2 hours to complete

8 videos (Total 55 min), 1 reading, 1 quiz

8 videos

Large-scale systems of linear equations.4m

Simple iteration for a linear system. Jacobi iteration.8m

Convergence criteria for simple iteration.4m

Seidel's iteration.9m

Successive over-relaxation.3m

Canonic form of two-step iterative methods for linear systems.10m

Variational approaches: minimum residual method.5m

Copy of Simple iteration for a linear system. Jacobi iteration.8m

1 reading

Slides10m

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Introduction to numerical analysis