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). Solving an equation f(x) = g(x) is the same as finding the roots of the function h(x) = f(x) – g(x). Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points, and for converging rapidly to these fixed points. The behaviour of general root-finding algorithms is studied in numerical analysis. However, for polynomials, root-finding study belongs generally to computer algebra, since algebraic properties of polynomials are fundamental for the most efficient algorithms. The efficiency of an algorithm may depend dramatically on the characteristics of the given functions. For example, many algorithms use the derivative of the input function, while others work on every continuous function. In general, numerical algorithms are not guaranteed to find all the roots of a function, so failing to find a root does not prove that there is no root. However, for polynomials, there are specific algorithms that use algebraic properties for certifying that no root is missed, and locating the roots in separate intervals (or disks for complex roots) that are small enough to ensure the convergence of numerical methods (typically Newton's method) to the unique root so located. (Wikipedia).

Graeffe's Method

Graeffe's Root-Squaring Method (also called Graeffe-Dandelin-Lobachevskiĭ or Dandelin–Lobachesky–Graeffe method) for finding roots of polynomials. The method solves for all of the roots of a polynomial by only using the coefficients and does not require derivatives nor an interation funct

From playlist Root Finding

Bisection Method

Bisection Method for finding roots of functions including simple examples and an explanation of the order. Chapters 0:00 Intro 0:14 Bisection Method 1:06 Visual Example 1:49 Difficult Functions 2:14 Order 3:16 Finding Order Example 3:38 Maximum Number of Iterations 4:28 Thanks For Watchin

From playlist Root Finding

Newton's Method

Newton's Method for finding roots of functions including finding a square root example and discussion of the order (newton's method is also known as Newton-Raphson method). Small correction around 2:26 the sign is incorrect it should be x_(n+1) = (1/2)(x_n + a/x_n). A video covering this m

From playlist Root Finding

Multiplying & Dividing Radicals Properties of Roots

I introduce finding Real Roots of Numbers and then expand into using the Property of Roots to simplify algebraic expressions that involve variables. Special care is taken to explain the differences between taking an even root versus taking an odd root. Find free review test, useful notes

From playlist Algebra 2

Root-Finding in MATLAB | Lecture 20 | Numerical Methods for Engineering

How to use the MATLAB functions root.m and fzero.m to find the roots of a polynomial and a nonlinear function. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engineers.pdf Subscribe t

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Solving l1 Regularized Least Squares via Proximal Gradient Descent

From playlist Iterative Algorithms for Least Squares and Classification Problems

Rooting a tree | Graph Theory

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Wegstein's Method

Wegstein Method for finding roots, accelerating fixed point iteration, and inducing convergence in fixed point iteration. Explained examples and discussion of order as well as how to compute q. Example code: https://github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:22 Wegstein's Me

From playlist Root Finding

Householder's Method

Householder's Method for finding roots of equations including history, derivation, examples, and fractals. Example code is available on GitHub https://github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:25 Derivation 1:58 History 2:34 Householder's Method 4:07 Householder's Method Ex

From playlist Root Finding

Newton-Raphson Algorithm in Python | Finding real and complex roots systematically

The algorithm explained: https://www.youtube.com/watch?v=qlNqPE_X4ME In this video tutorial I show you how to implement the Newton-Raphson algorithm in Python. The basic algorithm is very straightforward to implement, but there are a few tricks and things to keep in mind. In the video I s

From playlist Newton-Raphson Algorithm

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Software Development Course Day - 2 | Data Structures & Algorithms | Software Developer |Simplilearn

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Sorting Algorithms Full Course | Sorting Algorithms In Data Structures Explained | Simplilearn

This Simplilearn video is based on The Sorting Algorithms Full Course. This tutorial mainly focuses on all the major Sorting Algorithms In Data Structures Explained with detailed theory and practical examples for providing a better learning experience. This video covers the following Sort

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WHAT IS THE SQUARE ROOT ALGORITHM? How does the square root algorithm work and why | Nathan Dalaklis

I go ahead and answer the question of 'what is the square root algorithm'. Before we look at this one of many algorithms from real analysis. I'll go through a sample computation and run through the square root algorithm steps without a calculator. After looking at the square root process a

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From playlist Data Structures & Algorithms [2022 Updated]

Nexus trimester - David Gamarnik (MIT)

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Kruskals Algorithm | Kruskals Algorithm For Minimum Spanning Trees | Data Structures | Simplilearn

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Bisection Method | Lecture 13 | Numerical Methods for Engineers

Explanation of the bisection method for finding the roots of a function. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/us

From playlist Numerical Methods for Engineers

Root-finding algorithms

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