Lecture

Root Finding Methods: Secant and Newton's Methods

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 discusses various numerical methods for finding roots of functions, focusing on the secant method and Newton's method. The instructor begins by introducing the secant method, explaining its iterative process and the importance of selecting a suitable initial interval. The lecture emphasizes the significance of maintaining a consistent slope throughout the iterations, which is a key feature of the secant method. Following this, the instructor transitions to Newton's method, detailing its reliance on the derivative of the function to improve convergence rates. The lecture highlights the conditions under which these methods are effective, including the necessity for a good initial approximation. The instructor also compares the convergence rates of the secant method and Newton's method, illustrating the advantages of Newton's method in terms of speed and efficiency. Throughout the lecture, practical examples and visual aids are used to enhance understanding, making the concepts accessible to students.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis, Differential anaysis

Related concepts (32)

Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is which gives rise to the sequence of iterated function applications which is hoped to converge to a point . If is continuous, then one can prove that the obtained is a fixed point of , i.e., More generally, the function can be defined on any metric space with values in that same space.

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

Root-finding algorithms

In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).

Halley's method

In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. It is named after its inventor Edmond Halley. The algorithm is second in the class of Householder's methods, after Newton's method. Like the latter, it iteratively produces a sequence of approximations to the root; their rate of convergence to the root is cubic. Multidimensional versions of this method exist.

Muller's method

Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0. It was first presented by David E. Muller in 1956. Muller's method is based on the secant method, which constructs at every iteration a line through two points on the graph of f. Instead, Muller's method uses three points, constructs the parabola through these three points, and takes the intersection of the x-axis with the parabola to be the next approximation.

Related lectures (64)

Nonlinear Equations: Methods and Applications

Covers methods for solving nonlinear equations, including bisection and Newton-Raphson methods, with a focus on convergence and error criteria.

Advanced Analysis II: Derivatives and Notations

Covers derivatives, notations, composition of functions, and multiple derivatives in advanced analysis.

Nonlinear Equations: Fixed Point Method Convergence

Covers the convergence of fixed point methods for nonlinear equations, including global and local convergence theorems and the order of convergence.

Computational Geomechanics: Unconfined Flow Analysis

Explores unconfined flow analysis in geomechanics, emphasizing iterative solution methods and boundary condition considerations.

Numerical Methods: Iterative Techniques