Iterative Methods: Faster Convergence Analysis

  • Thread starter Firepanda
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In summary: Basically in my notes I have a page titled 'Rate of convergence' which is all basically just a proof using the taylor expansion to show that Newtons method is a quadratic convergence.
  • #1
Firepanda
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http://img142.imageshack.us/img142/6899/asdaps7.jpg

I can't see how to do this at all, I can see how the methods come about easily enough, and of course find the root if needed and then show which converged faster. But I have nothing in my notes to hint me on how I can find which converges the fastest without working anything out..

Any ideas?

Thanks
 
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  • #2
Well, it says "this question is based on theory only". Okay, what theory do you know?
 
  • #3
HallsofIvy said:
Well, it says "this question is based on theory only". Okay, what theory do you know?

I know how to use the x0 close to the root to find an xn as an approximation the the root, also some notes on errors, but nothing on rate of convergence to the root.
 
  • #4
Firepanda said:
http://img142.imageshack.us/img142/6899/asdaps7.jpg

I can't see how to do this at all, I can see how the methods come about easily enough, and of course find the root if needed and then show which converged faster. But I have nothing in my notes to hint me on how I can find which converges the fastest without working anything out..

Any ideas?

Thanks

You said you don't have anything in your notes about this. Is there something in your text (assuming you're using a textbook) that discusses convergence rates?
 
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  • #5
Mark44 said:
You said you don't have anything in your notes about this. Is there something in your text (assuming you're using a textbook) that discusses convergence rates?

Basically in my notes I have a page titled 'Rate of convergence' which is all basically just a proof using the taylor expansion to show that Newtons method is a quadratic convergence.

I've just recently done more looking into it and everywhere says this under all google searches so I assume this is the only bit of theory I need to know?

In that case I'll take a wild stab in the dark and say the 1st equation has a faster rate as it seems liek the x^3 term would have a bigger impact on the next term than simply diving by x.
 
  • #6
Anyone? I really can't get this and it seems like I should know it fundamentally to understand the method better.
 
  • #7
Look-up "Fixed point iteration". In particular, what can said about the convergence rate realtive to the magnitude of the derivative of the iteration formula near the root. Compare the magnitude of the derivatives of these two functions over the domain.
 

FAQ: Iterative Methods: Faster Convergence Analysis

1. What are iterative methods and why are they important in scientific research?

Iterative methods are a class of mathematical algorithms used to solve problems by repeatedly refining an initial estimate until a desired level of accuracy is achieved. They are important in scientific research because they offer a systematic and efficient approach to solving complex problems that cannot be solved analytically.

2. How do iterative methods achieve faster convergence compared to other methods?

Iterative methods achieve faster convergence by using previous approximations to refine the solution in each iteration. This allows for a more targeted approach, as the algorithm can focus on areas where the solution is still inaccurate, rather than starting from scratch each time.

3. What factors affect the convergence rate of iterative methods?

The convergence rate of iterative methods can be affected by several factors, including the initial guess, the choice of iterative method, the problem's characteristics (such as linearity or nonlinearity), and the desired level of accuracy. Additionally, the convergence rate can be impacted by the presence of any errors or noise in the system.

4. How can one determine the convergence rate of an iterative method?

The convergence rate of an iterative method can be determined by analyzing the behavior of the error function over multiple iterations. If the error decreases at a consistent rate, the method is said to have linear convergence. If the error decreases at an accelerating rate, the method has superlinear convergence. Nonlinear convergence occurs when the error decreases at a decreasing rate.

5. What are the advantages and limitations of using iterative methods?

The main advantage of iterative methods is their ability to solve complex problems efficiently. They also allow for flexibility in choosing the level of accuracy and can be applied to a wide range of problems. However, iterative methods may not always converge to a solution, and the convergence rate can be affected by various factors. Additionally, the choice of iterative method and initial guess can greatly impact the performance of the algorithm.

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