# What is Newton's method: Definition and 96 Discussions

In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f ′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then

x

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f
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f

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{\displaystyle x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}}
is a better approximation of the root than x0. Geometrically, (x1, 0) is the intersection of the x-axis and the tangent of the graph of f at (x0, f (x0)): that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as

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=

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f
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x

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{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}}
until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.

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1. ### Using Newton's Method to Solve and Using Fsolve in Matlab

Here is my code so far and solution. When using fsolve, we obtain the result that the equation does get solved. We find the values to be 1.0000, 0.0000, and 2.0000. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the...
2. ### I Error bounds for Newton's Method

Hi, PF Sometimes it is not easy to find roots of functions. Newton gave a nice clue: the Newton's Method formula: ##x_{n+1}=x_n-\dfrac{f(x_n}{f'(x_n)}##. My concern is, now that I have understood and practiced it, comprehend what I've sketched in the summary. This is all taken from "Calculus...
3. ### Finding the ratio of the fluxion of x using Newton's method

I was asked to find the ratio of the fluxion of x to the fluxion of 1/x using Newton's synthetic method of fluxions. I do not understand how to do this.
4. ### MHB Alexander's question via email about Newton's Method

Alexander asks: Apply three iterations of Newton's Method to find an approximate solution of the equation $\displaystyle \mathrm{e}^{1.2\,x} = 1.5 + 2.5\cos^2{\left( x \right) }$ if your initial estimate is $\displaystyle x_0 = 1$. What solution do you get?
5. ### Calculating Newton's Method: Step-by-Step Guide

Since the Newton's method is as follows: $$x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}$$ $$x_{1}=x_{0}-\frac{cos(0)-1}{-sin(0)-2}$$ Is this correct? What should I proceed on from here?

35. ### Newton's Method - Cube Root Of 5

Homework Statement Use Newtons method to compute the cube root of 5. Do the first 10 iterations. x_{(0)}=1 determine the fixed points of the iteration and determine whether they are repelling/attracting. if attracting, then determine if the convergence is linear or quadratic. draw the...
36. ### MHB Felix's question at Yahoo Answers regarding Newton's method

Here is the question: I have posted a link there to this topic so the OP can see my work.
37. ### Applying Newton's Method without the need of a calculator

Hello everyone Although I do not have a specific homework question to ask, I do have a question which directly relates to a topic I do indeed have for homework. One of our topics for the semester are approximating roots using Newton's method, and as I understand the value you one obtains...
38. ### MHB An optimization problem with Newton's method

Apply Newton's method to $f(x)=(x-2)^4+(x-2)^5$ with initial guess $x_0=3$. We can observe that the sequence converges linearly with rate constant $3/4$. Now apply the iterative mathod $x_{k+1}=x_k-4f(x_k)/f'(x_k)$. This method should converge more rapidly for this problem. But how to prove that...
39. ### MHB Inverse Functions & Newton's Method: Debmnzl's Question | Yahoo! Answers

Here is the question: Here is a link to the question: A math function question please help? - Yahoo! Answers I have posted a link there to this topic so that the OP may find my response.
40. ### What is the Range of Convergence for Newton's Method on tanh(x)?

I was hoping someone could help me out with this problem. I need to find the values, that Newton's method converges for tanh(x). So far I set up the algorithm: x\tiny_{k+1} = x_{k} - \frac{tanhx}{sech^{2}x} I simplified it to: x\tiny_{k+1} = x_{k} - (\frac{1}{2}sinh(2x)) And then...
41. ### MHB Newton's method to approximate integrals?

Can we use Newton's method to approximate the value of definite integrals? (Thinking) EDIT: Ignore if the question doesn't make sense (which it probably doesn't).
42. ### [Numerical analysis] Stability and condition of Newton's method

I am confused by the concept of stability and condition. As I understand it, condition is defined by how much the output changes when the input changes. But why is it linked to the problem and not the algorithm? What if I have two algorithms that calculate the same thing but in a completely...
43. ### Particle Swarm Optimization vs. Newton's Method

I have been reading Stephen Boyd's book Convex Optimization and I have learned to form various problems like LP, QP, QCQP, SOCP or SDPs. I also learned about formulating SVM for classification problem as optimization problem. Now I am reading about Gradient Methods, Newton's method, etc...
44. ### Newton's method of estimation - using derivatives

Homework Statement Newton devised the following method for approximating a real root of the equation f(x) = 0. i.e. a real number for which f(r) = 0. We begin by guessing an approximation, say x1, to the real root r. (i) Find the equation of the line tangent to the graph of y = f(x) at the...
45. ### MATLAB What is the error in running Newton's Method in Matlab for a specific function?

I've been using this for a Newton Approximation in Matlab function x = Newton(f, fp, x, nmax, e) % f is an inline function which we apply Newton's method on % fp is an inline function that is the derivative of function f % x is the initial guess of the root % nmax is the total number...
46. ### MHB Newton's method of approximation

In an introductory calculus course I am doing I have just come across the following problem: "Given that $\sin(x)=e^{-x}$ has a solution near x=1, use Newton's method to find the solution to 4 decimal places." My question will strike you as very basic, however, I *am* a beginner and I *have*...
47. ### Comp Sci Understanding Newton's Method in C++

Homework Statement why the loop not looping ? Homework Equations fun1=x^3+4^2-x-1 fun2=3x^2+8x-1 The Attempt at a Solution #include "stdafx.h" #include<iostream> using namespace std; float fun1(float); float fun2(float); void main() { float a; cin>>a; if (fun2(a)>0)...
48. ### Physical interpretation for this? (dynamics of Newton's method)

Hello! I'm a math student, currently trying to write my diploma thesis. My field of study is complex dynamics (iteration of holomorphic/meromorphic functions, Julia sets and stuff). It's a farfetched idea, but currently I'm curious about a potential physical interpretation of the things I'm...
49. ### Newton's Method (as applied to Auto Financing)

Homework Statement A car dealer sells a new car for $18,000. He also offers to sell the same car for monthly payments of$375.00 for five years. What monthly rate is this dealer charging?Homework Equations A = [R(1 - (1 + i))^-60] / i where A = the present value, R = the monthly payment, i =...
50. ### MATLAB: Finding the 5th Root using Newton's Method

Homework Statement The solution of the nonlinear equation x^5-P=0 gives the fifth root of the number P. A numerical solution of the equation can be calculated with Newton’s method. The solution process starts by choosing a value x1 as a first estimate of the solution. Using this value, a...