Newton's Method - Matlab Function - Science and Maths Assignment Help

Assignment Task:

Consider a function f: R2 —> R2. Applying Newton's method to this function with different initial points will result in Newton converging to different zeros of f. Write a Matlab-function that produces a figure showing which initial points converge to the same zero. The Matlab-function should take four inputs: f, [xo, xi] and [yo, yi], indicating the coordinates to consider, and N, the refinement of the axes, The Matlab-function should have no output but create a plot indicating which areas in the rectangle [x0, xi] x [yo, yi] converged to the same zero under the Newton's algorithm. The results should be based on N2 points in the rectangle [xo, xi] x [yo, yi].

• Store the initial points. You should consider the combination between N points in the interval [so, xi] and N points in the interval [yo, m]. 
• Run Newton's method on the initial points and store the approximation of the zero it converged to. 
• Use these results to create a plot:

— determine the number of different zeros,

— determine which approximations of the zeros are the same,

— color boxes that converged to the same zero with the same color,

— you can use the function fill or patch to color the plot. 

Modified Newton 
Write a Matlab-function which determines the zero of a function using the modified Newton's method. The Matlab-function takes three inputs: a function, its derivative, and a starting point. The outputs are an approximation of zero, an estimate of the order of the zero, and a vector of all the iterations. The Matlab-function takes two optional inputs: a measure for the precision and the maximum number of iterations. 
• You can start out from the function newton.m on Canvas. 
• Estimate the order of the zero from the sequence of values in the iterations. 
• Can you, based on an example, say something about the order of convergence?

• Test your Matlab-function on different functions. 

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