Proving Quadratic Convergence via Taylor Expansion

In summary, the conversation discusses a modification of Newton's method and the use of a specific equation to solve for a root. The equation is broken down into different parts and the final solution is expected to be O(En2).
  • #1
Scootertaj
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Homework Statement



The following is a modification of Newton's method:

xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn)

Homework Equations



We are supposed to use the following method:

let En = xn + p where p = root → xn = p + En

Moreover, f(xn) = f(p + En) = f(p) + f'(p)En + O(En2) = f'(p)En + O(En2) since f(p) = 0 at root

The Attempt at a Solution



Change the equation to xn+1 - xn = - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn)
Then, LHS: En+1 - En
RHS broken up into parts:
f(xn) = f(p + En) = f'(p)En + O(En2)
f(xn + f(xn)) = f(p + En + f(p + En)) = f(p + En + f'(p)En + O(En2)) = f(p) + f'(p)(En + Enf'(p) + O(En2)) + O(En2) = f'(p)(En + Enf'(p) + O(En2)) + O(En2)

So, f(xn + f(xn)) - f(xn) = [ f'(p)(En + Enf'(p) + O(En2)) + O(En2) ] - f'(p)En + O(En2)

= [ f'(p)(Enf'(p) + O(En2)) + O(En2) ] - O(En2)

Also, f(xn)2 = (f'(p)En + O(En2))2

So, En+1 - En = (f'(p)En + O(En2))2 / ( [ f'(p)(Enf'(p) + O(En2)) + O(En2) ] - O(En2) )

So, from this somehow I'm supposed to get En+1 = O(En2)
 
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  • #2
Just a bump
 

1. What is quadratic convergence?

Quadratic convergence is a type of convergence in numerical methods where the error decreases at a rate of the square of the previous error. This means that as the number of iterations increases, the error decreases at a much faster rate compared to other types of convergence.

2. What is a Taylor expansion?

A Taylor expansion is a mathematical representation of a function as an infinite sum of terms, where each term is a polynomial of increasing degree. It is used to approximate a function by considering its derivatives at a single point.

3. How is quadratic convergence proven using Taylor expansion?

In order to prove quadratic convergence using Taylor expansion, we must show that the error between the true solution and the approximation decreases at a rate of the square of the previous error. This can be done by using the Taylor series of the function and analyzing the terms of higher degree to show that they converge to zero at a faster rate compared to lower degree terms.

4. What are the advantages of proving quadratic convergence?

Proving quadratic convergence is important because it shows that a numerical method is more efficient and accurate compared to other methods with slower convergence rates. This can save time and resources in solving complex problems, especially in fields such as engineering and physics.

5. Are there any limitations to proving quadratic convergence using Taylor expansion?

Yes, there are limitations to proving quadratic convergence using Taylor expansion. This method may only work for certain types of functions and may not be applicable to all numerical methods. Additionally, the proof may become more complex for higher-order derivatives, making it difficult to apply in some cases.

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