Lecture

Numerical Derivatives of Order 1: Finite Differences Progressive

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture introduces the concept of numerical derivatives of order 1 using progressive finite difference formulas. The instructor explains the calculation of derivatives based on differences between function values at different points. The lecture covers the application of these formulas in numerical analysis, specifically focusing on the development of Taylor series. Various examples are provided to illustrate the computation of derivatives and the importance of understanding the relationship between function values. Additionally, the lecture discusses the implications of different choices of parameters on the accuracy of derivative calculations.

In MOOC

Numerical Analysis for Engineers

This course is based on the book "Introduction à l'analyse numérique", J. Rappaz M. Picasso, Ed. PPUR. Basic tools are described in the first 5 chapters. The last two chapters deal with the numerical solution of differential equations. More specifically, we will study the following chapters of the book: Chapter 1: interpolation, how to approximate a function by a polynomial? Chapter 2: how to approximate derivatives with finite difference formulas? Chapter 3: How can integrals be approximated by quadrature

Official source

Related concepts (17)

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Numerical differentiation

In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function. The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). Choosing a small number h, h represents a small change in x, and it can be either positive or negative.

Finite difference

A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.

Finite difference method

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.

Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

Related lectures (64)

Numerical Differentiation: Part 1

Covers numerical differentiation, forward differences, Taylor's expansion, Big O notation, and error minimization.

Numerical Derivatives of Order 1 MOOC: Numerical Analysis for Engineers

Covers the concept of numerical derivatives of order 1 and the importance of choosing a small fixed value for h.

Numerical Differentiation: Backward and Central Differences

Explores backward and central differences for numerical differentiation, analyzing their properties and error analysis.

Numerical Methods: Boundary Value Problems

Explores numerical methods for boundary value problems, including heat diffusion and fluid flow, using finite difference methods.