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















{\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















{\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. V

    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. mcastillo356

    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. B

    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. P

    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. jisbon

    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?
  6. F

    Optimization with Newton's method

    I have a system of equations which I solved with Newtons method. Call Newtons method a function NM=f(K1,K2). K1 and K2 are input and a vector of x=x1,x2,x3,x4 is output. I have another function, SSR, the sum of square residuals. It looks like this : $$\sum...
  7. F

    A Can Newton's method work with an approximated integral

    I have a system of equations, and one of them is this : ##\int(1-U(y))Dy - H*\int(U(y)-U(y)^2)dy=0## Can Newtons method work if I approximate this integral to be ##\sum_y(1-U(y))-H\sum(U(y)-U(y)^2)=0## y is a set integers in range ##[1,1000]## I have Newtons method working for this same system...
  8. R

    Intersecting circles using Newton's Method

    Homework Statement Homework EquationsThe Attempt at a Solution My initial thought was to set the two equations equal to each other but the resulting equation is linear which gives a constant for a Newton iteration. I thought about Taylor's theorem in 2-d but I'm not so sure about that as far...
  9. C

    MHB Numerical Analysis problem Newton's method

    Can anyone help in the solution of this problem? how can i determine the zero x*??
  10. Delta31415

    Newton's method and complex roots

    1) the problem I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function (...
  11. lucasLima

    Can Newton's Method Solve This Non-Linear System Accurately?

    Homework Statement I'll try to be as clear as possible but this exercise is in portuguese and this is a free translation from me "The sum of two numbers is 3.4 .If we take each of them and summed with they square, the product of it would be 18.5856. The lowest of those number is in the interval...
  12. karush

    MHB 205.q4.2 very painful Newton's Method

    %\section{Newton's Method } %\subsubsection{205.q4.2 } %\footnote{Leeward Community College}method m , $\textsf{a. Use Newton's Method: }$0000 \begin{align} \displaystyle x_{n+1}&=x_n-\frac{f(x_n)}{f'(x_n)} \end{align} $\textsf{b. Use Newton's Method to solve: }$ \begin{align}...
  13. S

    I Newton's method for approximating solutions of functions

    In my calculus textbook, it shows that a function's solution can be approximated using an approximated function tangent to the original function based on an approximated solution, where the equation to find the approximated is L(x) = f(X0) + f'(X0)*(X-X0), where when rearranged, gives x = Xo -...
  14. P

    MHB Finding Approximate Critical Number with Newton's Method

    Use Newton's Method to approximate a critical number of the function \displaystyle f(x) = \frac{1}{2} x^8 + \frac{6}{5} x ^5+ 2 x +10 near the point x = 2 . Use x_1 = 2 as the initial approximation. Find the next two approximations, x_2 and x_3, to four decimal places each.I have been...
  15. toforfiltum

    Newton's method to approximate critical point

    Homework Statement Let ##F(x,y)=4sin(xy)+x^3+y^3## Use Newton's method to approximate the critical point that lies near ##(x,y)=(-1,-1)## Homework EquationsThe Attempt at a Solution I have a problem here because the derivative is not a square matrix. Hence, I can't find the inverse needed for...
  16. King_Silver

    Question regarding Newton's Method

    So I know how to do Newton's Method without any trouble, taking a value for n then taking it away from the function divided by the derivative of the function I get that entirely and have done roughly 22-23 examples of it over the past 3-4 days. However I have come across one question regarding...
  17. T

    Finding roots using Newton's method in MATLAB

    Homework Statement I am supposed to find the roots of the equation: 10esinx = x2 - 5x +4 in MATLAB using Newton's method with a tolerance of 10-8. There should be three roots. Homework Equations p=po - f(po)/f'(po) |p - po| < TOL The Attempt at a Solution Here is what I have for the code...
  18. Erikono

    Equations of Motion Using Newton's Method --

    Homework Statement Determine the equations of motions in terms of x and gamma. Assume small angles and that the wheel rolls without slip. The mass of the thin homogeneous large disk of radius 2R is 2m. The mass of the thin homogeneous inner disk of radius R is m. The rod of length 2R is...
  19. Erikono

    Newton's Method for Equations of Motion (Vibrations) help?

    Homework Statement Determine the equations of motions in terms of x and gamma. Assume small angles and that the wheel rolls without slip. The mass of the thin homogeneous large disk of radius 2R is 2m. The mass of the thin homogeneous inner disk of radius R is m. The rod of length 2R is...
  20. P

    Questions on Newton's Method: Investigating Convergence & Initial Guesses

    I have three questions regarding Newton's method. https://en.m.wikipedia.org/wiki/Newton-Raphson#Failure_of_the_method_to_converge_to_the_root According to this wikipedia article, "if the first derivative is not well behaved in the neighborhood of a particular root, the method may overshoot, and...
  21. P

    Optimization using Newton's method gradient hessian

    Hello, This is my first post here. So I hope I'm posting in the right place, sorry if not. http://homes.soic.indiana.edu/classes/spring2012/csci/b553-hauserk/Newtons_method.pdf I am trying to solve the following numerical optimization function using Netwon's Method: So, if I have the gradient...
  22. kostoglotov

    Newton's Method generalized to 3 dimensions

    Homework Statement Prelim: my question is about a very specific part of a question whereby the student is asked to derive the final formula for the general solution in two vars, but I will post the entire question for clarify. Newton's Method for approximating the roots of an equation f(x)=0...
  23. David Carroll

    How to calculate efficiency of Newton's Method

    Greetings. I was wondering if anyone knew of a way to calculate the efficiency of Newton's Method for a given function: I have an equation f(x) and I'm trying to find a value of x = x0 such that f(x0) = 0. So I start with a guess x0 and then use that to find a second (usually closer) guess...
  24. S

    Newton's Method utilizing Mathcad

    Homework Statement I am writing a simple program in Mathcad for Newton's Method. The function is x^3-5*x^2+3*x+4. The root is between 0 and 3, and we want a tolerance of 0.00001. My code is the exact same as a program I made in Matlab that works, but since I am new to Mathcad, I am getting an...
  25. Maged Saeed

    Initial guess in Newton's method

    How to find the Initial guess for Newton method with high accuracy ?? Is there a way rather than using mean value theorem [which is used to test whether there is a solution on a closed interval to the equation] ?
  26. 5

    How Does Including the Quadratic Term Affect the Newton Iteration Formula?

    Homework Statement The Newton iteration formula is based on a Taylor series expansion of the function f(x) around an estimate of the root xn, truncated after the linear term. You are asked to derive a more accurate iteration scheme as follows: Start from the Taylor series expansion of f(x)...
  27. A

    Why Does Initial Approximation Affect Convergence Speed in Newton's Method?

    Homework Statement Use Newton's method with x1 = 1 to find the root of the equation x3 - x = 1 to correct six decimal places. Do the question again with x1 = 0.6 Do the question again with x1 = 0.57 You probably need to do it in an excel sheet. With each try, it takes longer to...
  28. D

    Successive Substitution vs Newton's Method

    Homework Statement Name disadvantages of the Successive Method vs Newtons for solving nonlinear equations? Homework Equations The Attempt at a Solution I went all through the textbook and this is all I could find on the successive method disadvantages but these are not compared to...
  29. Z

    Sign confusion when taking gradient (Newton's Method)

    I'm watching a lecture on Newton's method with n-dimensions but I am kind of hung up on why the professor did not use the negative sign while taking the first gradient? Is there a rule that explains this or something that I'm forgetting? The rest makes sense but highlighted in red is the part I...
  30. Z

    Sign confusion when taking gradient (Newton's Method)

    I'm watching a lecture on Newton's method with n-dimensions but I am kind of hung up on why the professor did not use the negative sign while taking the first gradient? Is there a rule that explains this or something that I'm forgetting? The rest makes sense but highlighted in red is the part I...
  31. MarkFL

    MHB しおり's question at Yahoo Answers regarding Newton's method

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  32. S

    Solving Matlab Newton's Method Homework

    Homework Statement So, I've written a program that carries out Newton's method. The root of the equation which I am trying to find is approximately 13.1. This is fine and my program returns the correct value when my initial guess is around this value (up to about x = 50 as my initial guess)...
  33. twoski

    MATLAB Mastering Newton's Method and Graphing in Matlab: A Step-by-Step Tutorial

    Hey guys, I have to make some graphs for an assignment and a friend told me it would be easiest with matlab. I need to make an x(k+1) vs. x(k) graph for Newton's method as well as the Chord method. I also need to make a few charts of the discrete logistic equation. Plotting functions isn't...
  34. H

    [Wolfram Mathematica] Using Newton's method to solve non-linear system

    Hi. This is not actually not part of the homework; but it's something I'd like to do. I have to solve the following system using Newton-Raphson's method: $$\begin{matrix} \frac{X}{\mu }+Y=1 \\ X=\left( \lambda -\left( K-1 \right)X \right)Y \\ \end{matrix}$$ Surfing the...
  35. twoski

    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. MarkFL

    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. M

    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. I

    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. MarkFL

    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. M

    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. P

    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. S

    [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. N

    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. B

    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. R

    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. DeusAbscondus

    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. S

    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. U

    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. C

    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. P

    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...