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Introduction to numerical analysis

Approx. 18 hours to complete

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Course Summary

This course teaches students how to use numerical methods to solve a range of mathematical problems. It covers topics such as interpolation, numerical integration, and linear systems.

Key Learning Points

Learn how to solve mathematical problems using numerical methods
Gain skills in interpolation, numerical integration, and linear systems
Get hands-on experience with programming assignments

Learning Outcomes

Ability to solve mathematical problems using numerical methods
Skills in interpolation, numerical integration, and linear systems
Experience with programming assignments

Prerequisites or good to have knowledge before taking this course

Basic knowledge of calculus and linear algebra
Familiarity with programming concepts

Course Difficulty Level

Intermediate

Course Format

Online
Self-paced

Similar Courses

Numerical Methods for Engineers
Applied Linear Algebra

Related Education Paths

Notable People in This Field

Cleve Moler
Gilbert Strang

Related Books

Description

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

Outline

Machine arithmetics. Systems of linear algebraic equations.
About the University
Introduction.
A simple worked example.
Machine arithmetics. Representation of real numbers.
Machine epsilon. Over- and underflow.
A crude estimate of the machine epsilon.
Systems of linear equations. Cramer's rule.
Gaussian elimination.
LU decomposition: the matrix form of the Gaussian elimination.
When does the Gaussian elimination work?
LU decomposition with pivoting. Permutation matrices.
About University
Rules on the academic integrity in the course
About the course
Slides

Numerical linear algebra.
Introduction.
Sensitivity of a linear system.
Vector norms.
Matrix norms.
Common matrix norms.
Sensitivity of a linear system. Condition number.
Cholesky decomposition.
Banded matrices. Thomas algorithm.
Shermann-Morrison formula.
QR decomposition.
Constructing the QR decomposition: Householder reflections.
Constructing the QR decomposition: Givens rotations
Slides
Slides

Non-linear algebraic equations.
Solving non-linear equations.
Localization of roots. Bisection.
Fixed-point iteration.
Aside: convergence rates and related technicalities.
Back to the fixed-point iteration.
Fine-tuning the fixed-point iteration.
Newton's iteration.
Multiple roots. Modified Newton's method.
Inverse quadratic interpolation.
Roots of polynomials.
Roots of polynomials: the companion matrix.
Slides

Iterative method for linear systems.
Large-scale systems of linear equations.
Simple iteration for a linear system. Jacobi iteration.
Convergence criteria for simple iteration.
Seidel's iteration.
Successive over-relaxation.
Canonic form of two-step iterative methods for linear systems.
Variational approaches: minimum residual method.
Copy of Simple iteration for a linear system. Jacobi iteration.
Slides

Interpolation and approximation. Modeling of data.
Interpolation and approximation. Modelling of data.
Linear least squares problem.
Ordinary least squares: the normal equations.
Ordinary least squares: QR decomposition of the design matrix.
Global polynomial interpolation.
Lagrange interpolating polynomial.
Quantifying interpolation errors. Runge phenomenon.
Chebyshev nodes.
Interpolation of the Runge function.
Slides

Numerical calculus: derivatives and integrals.
Numerical derivatives.
Numerical derivatives: finite differences.
Truncation and roundoff errors: an interplay.
Higher order schemes.
Richardson extrapolation.
Integration: numeric quadratures.
Convergence rates of simple quadratures.
Simple geometric quadratures: Trapezoids, Simpson's rule and all that.
Error bounds for quadratures. Romberg extrapolation.
Integrals with singularities.
A check of convergence.
Recap: Newton-Cotes vs Gaussian quadratures.
Gaussian quadratures.
Slides

Initial value problem for ordinary differential equations.
Initial value problem for an ODE. Discretization.
Approximation and convergence.
Truncation error or Euler-like schemes.
Runge-Kutta methods.
Asymptotic stability of ODEs. Stiffness.
Linear Multistep methods.
Zero-stability of linear multistep methods.
Slides

Summary of User Reviews

Key Aspect Users Liked About This Course

The course content is well-structured and easy to understand, even for beginners in the field.

Pros from User Reviews

The course covers a wide range of topics related to numerical analysis.
The instructors are knowledgeable and responsive to student questions.
The course offers practical examples and interactive exercises to reinforce learning.
The course is well-organized and easy to follow.

Cons from User Reviews

Some users found the course content to be too basic.
Some users felt that the course was too theoretical and lacked practical applications.
Some users found the course to be too time-consuming.
Some users found the quizzes and assignments to be too difficult.

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