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#### 100% online

Start instantly and learn at your own schedule.

#### Intermediate Level

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

#### Approx. 21 hours to complete

Suggested: 7 weeks of study, 5-6 hours/week...

#### English

Subtitles: English

#### 100% online

Start instantly and learn at your own schedule.

#### Intermediate Level

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

#### Approx. 21 hours to complete

Suggested: 7 weeks of study, 5-6 hours/week...

#### English

Subtitles: English

### Syllabus - What you will learn from this course

Week
1
5 hours to complete

## Machine arithmetics. Systems of linear algebraic equations.

...
10 videos (Total 60 min), 1 reading, 2 quizzes
10 videos
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
Week
2
3 hours to complete

## Numerical linear algebra.

...
12 videos (Total 72 min), 1 quiz
12 videos
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
Week
3
3 hours to complete

## Non-linear algebraic equations.

...
11 videos (Total 71 min), 1 quiz
11 videos
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
Roots of polynomials.3m
Roots of polynomials: the companion matrix.3m
Week
4
3 hours to complete

## Iterative method for linear systems.

...
7 videos (Total 46 min), 1 quiz
7 videos
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
Week
5
3 hours to complete

## Interpolation and approximation. Modeling of data.

...
9 videos (Total 65 min), 1 quiz
9 videos
Linear least squares problem.5m
Ordinary least squares: the normal equations.3m
Ordinary least squares: QR decomposition of the design matrix.7m
Global polynomial interpolation.8m
Lagrange interpolating polynomial.6m
Quantifying interpolation errors. Runge phenomenon.7m
Chebyshev nodes.8m
Interpolation of the Runge function.7m
Week
6
3 hours to complete

## Numerical calculus: derivatives and integrals.

...
13 videos (Total 80 min), 1 quiz
13 videos
Numerical derivatives: finite differences.7m
Truncation and roundoff errors: an interplay.9m
Higher order schemes.4m
Richardson extrapolation.7m
Simple geometric quadratures: Trapezoids, Simpson's rule and all that.7m
Error bounds for quadratures. Romberg extrapolation.4m
Integrals with singularities.6m
A check of convergence.4m
Week
7
3 hours to complete

## Initial value problem for ordinary differential equations.

...
7 videos (Total 64 min), 1 quiz
7 videos
Approximation and convergence.10m
Truncation error or Euler-like schemes.6m
Runge-Kutta methods.10m
Asymptotic stability of ODEs. Stiffness.12m
Linear Multistep methods.7m
Zero-stability of linear multistep methods.7m

## Instructor

### Evgeni Burovski

Assistant professor
School of Applied mathematics, MIEM

## About 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. Learn more on www.hse.ru...