[Solved] Numerical Analysis -Programming Assigment #1 Root Finding

1. Implement a bisection root finding method. Your function should have the signaturebisec(f,a,b,tol) and return a 1 2 list named res where f Function handle to f(x)

a, b Endpoints of initial interval [a,b] tol Tolerance res[0,0] Final approximation of the root. res[0,1] Number of iterations until convergence

• The root should have relative precision of at least tol (unless the root close to zero, in which the absolute accuracy should be tol).
• Provide a listing of the code and a source file named py

2. Implement Newtons root finding method. Your function should have the signaturenewton(f,fp,x0) and return a 1 2 list named res where f Function handle to f(x) fp Function handle to the derivative of f.

x0 Initial guess res[0,0] Final approximation of the root. res[0,1] Number of iterations until convergence

• The accuracy should be as high as possible in double precision.
• Comment how youve determined that the root is sufficiently accurate.
• Provide a listing of the code and a source file named py
• Make sure that you do not enter infinite loops!

3. Here we wish to check your implementations of Bisection and Newton methods on the functions

Use tol= 10¹² for the bisection method.

• For each function, generate a table that shows at each iteration
  • the estimate of the root,
  • the value of the function at the estimated root, the relative precision of the root.
  • Save the table in the file named csv make sure that when you save it you keep the format *.csv
• Approximate the rate of convergence of each method empirically by plotting the value in the y-axis vs. n in the x-axis (see H.W assignment 2 question 1).
• Make sure that Newtons method results in the rate p = 2. If this is not the case, decide what is the multiplicity of the root empirically (see H.W 2 Question 2).

4. Find the first 10 zeros of Bessels function J₀(x) for x 0.
   • You may use Pythons function jv found in the scipy.special library to evaluate bessel functions as needed (see https://docs.scipy.org/doc/scipy/reference/special.html)
   • The function should generate a table of the zeros accurate to 10¹².
   • plot J₀(x) with its zeros marked with symbols.
5. Implement Newtons Fractal for the complex function f(z) = z³ 1.
   • Create an initial grid of [1,1] [i,i] with 800 800 points.
   • Use Newtons method to find the root of f(z) for each of the starting points.
   • Store the number of iterations needed to approximate a given root up to 10¹⁰ in a 800 800 array. If you diverge use the maximal number of iterations. Normalize the array to values between 0 and 1. NOTICE: f(z) has 3 complex roots so you should have 3 such arrays.
   • Concatenate the three arrays into a single array IM of size 800 800 3.
   • Use the command imshow(IM) (found in the library matplotlib.pyplot) to show the resulting image.

Figure 1: Finding w.

[1] . Calculate the length of w in Figure 1. Upload your script file with the name Q6.py. Remark: The lengths 12m and 10m refer to the whole sides of the right angle triangles.