An attractive idea is to combine the reliability of the bisection method with the speed of Newton's method. Such a combination is implemented by running the bisection method until we have a narrow interval, and then switch to Newton's method for speed.
Write a script that implements this idea. Start with the bisection method with an interval [a,b] and switch to Newton's method when the current interval in the bisection method is a fraction ‘s’ of the initial interval (when the interval has length s(b−a)). Potential divergence of Newton's method is still an issue, so if the approximate root jumps out of the narrowed interval (where the solution is known to lie), one can switch back to the bisection method. The value of ‘s’ can be set by the user.
Refer to the assignments for the Bisection and Newton’s method for details on those methods. You may utilize any work done on those assignments.
Find the solution of f(x)=cos(2x)+sin(√x)=0 over 2<x<5 using the combined method as follows:
Define a maximum number of iterations (103) allowed and an error tolerance ϵ (10-6) and fraction of the original rang s=0.1.
Define an initial range of ‘a’ and ‘b’, and convergence ϵ. Define the fraction ‘s’.
If the current range (b-a) is greater than s(b-a) (the original ‘b’ and ‘a’, then use the bisection method to determine new range (new ‘a’ and ‘b’)
If the current range is less than or equal to s(b-a), then use Newton’s method to solve for the root. (a good initial guess is the average of the current range)
If the current guess for Newton’s method goes outside the current range, then use bisection to get a new range.
Once the root has been found (refer to the Newton’s method), output messages as per the Newton’s method.
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I can do the task. Have a lot of experience in designing algorithms for numerical methods in MATLAB. Familiar with both Newton's and bisection methods
hi I have 12+ years experience in MATLAB and an expert in numerical methods. I can complete your task perfectly. I assure quality work in quick time. Let me know if you have any clarifications. regards
I have done a very similar project to this on Matlab and will be able to finish this within 2 days. I will also be able to provide comments and any other required technical help.
Hello I am a professional mathematician having 9 years of teaching and project work experience in MATLAB. You have come up with a nice idea which I can surely automate using MATLAB.