Lecture

Error Analysis and Stability in Numerical Methods

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In course

MATH-251(d): Numerical analysis

This course offers an introduction to numerical methods for the solution of mathematical problems as: solution of systems of linear and non-linear equations, functions approximation, integration and differentiation and solution of differential equations. ...

Description

This lecture covers error analysis, stability, and adaptive time stepping in numerical methods. Topics include truncation error, convergence order, absolute stability, equilibrium points, linear systems, and error control. The instructor explains the concepts using examples and discusses methods' stability conditions.

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Instructor

Pablo Antolin Sanchez

Teaching & PhD PhD Students Philipp Christoph Weder, Davor Dobrota Past EPFL PhD Students as codirector Ondine Chanon Courses Analysis III MATH-202(c) The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to their use in solving multidisciplinary problems in scientific engineering. Analysis III (for IC) MATH-203(d) The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to thei

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Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (32)

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Congestion pricing

Congestion pricing or congestion charges is a system of surcharging users of public goods that are subject to congestion through excess demand, such as through higher peak charges for use of bus services, electricity, metros, railways, telephones, and road pricing to reduce traffic congestion; airlines and shipping companies may be charged higher fees for slots at airports and through canals at busy times. Advocates claim this pricing strategy regulates demand, making it possible to manage congestion without increasing supply.

Round-off error

In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error.

Road pricing

Road pricing (also road user charges) are direct charges levied for the use of roads, including road tolls, distance or time-based fees, congestion charges and charges designed to discourage the use of certain classes of vehicle, fuel sources or more polluting vehicles. These charges may be used primarily for revenue generation, usually for road infrastructure financing, or as a transportation demand management tool to reduce peak hour travel and the associated traffic congestion or other social and environmental negative externalities associated with road travel such as air pollution, greenhouse gas emissions, visual intrusion, noise pollution and road traffic collisions.

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

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