Newton-Raphson & Bisection Methods to Find x2- 2 Root

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In summary, the conversation discusses using the Newton raphson method and bisection method to find the root of x2-2 with a given starting point and accuracy. The speaker was able to find the root using the Newton raphson method, but is unsure of how to estimate the number of iterations needed for the bisection method. They suggest using a formula involving the interval and accuracy.
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Homework Statement



Using the Newton raphson method with x0=6, find the root of x2- 2 to 3dp.

Then estimate the number of iterations the bisection method that would be required to achieve the same accuracy.

Homework Equations





The Attempt at a Solution



I have done the first part with the Newton raphson method and have found that at x5 and x6 the answer of the root of the function is 1.414

However I am not sure how to estimate the number of iterations the bisection method will use, my guess is you use the following formula:

make the interval [1,2] (b=2, a=1) and k= accuracy
then

n must be greater than or equal to: log(b-a)+klog10[tex]/[/tex] log2
 
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Assuming k is the number of decimal digits in the precision (i.e. k = 6 for a precision of 10-6) and assuming you just forgot to type a parenthesis around the two log terms before diving with log 2, I get same result. If that is any help at this time :smile:
 

FAQ: Newton-Raphson & Bisection Methods to Find x2- 2 Root

1. What is the Newton-Raphson method?

The Newton-Raphson method is an iterative algorithm for finding the roots of a given function. It starts with an initial guess for the root and then uses the slope of the function at that point to refine the guess in each iteration until a satisfactory root is found.

2. How does the Newton-Raphson method work?

The method uses the derivative of the function to find the slope at a given point. The next guess for the root is then calculated by subtracting the function value at the current guess from the previous guess, divided by the slope at the current guess. This process is repeated until the root is found.

3. What is the Bisection method?

The Bisection method is another iterative algorithm for finding the roots of a given function. Unlike the Newton-Raphson method, it does not require the derivative of the function. It works by repeatedly bisecting an interval in which the root is known to lie and choosing the subinterval in which the function changes sign to continue the search.

4. How does the Bisection method work?

The method starts with an initial interval in which the root is known to lie and then divides it in half. If the function changes sign between the two subintervals, the one in which the sign changes is chosen for the next iteration. This process is repeated until the root is found within a given tolerance.

5. Which method is better for finding the root of x2-2?

Both methods can be used to find the root of x2-2, but the Newton-Raphson method is usually faster and more efficient since it converges to the root in fewer iterations. However, the Bisection method is more robust and can handle functions with multiple roots or when the initial guess is far from the root.

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