Description
- 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
- Implement Newton’s 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 you’ve 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!
- Here we wish to check your implementations of Bisection and Newton methods on the functions
| f(x) | Newton’s x0 | Bisection’s [a,b] |
| tanx − 2x | 1.4 | [1,1.4] |
| x3 − [1]x2 + 12x − 8 | 3.0 | [1,3] |
Use tol= 10−12 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 Newton’s 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).
- Find the first 10 zeros of Bessel’s function J0(x) for x ≥ 0.
- You may use Python’s 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 10−12.
- plot J0(x) with its zeros marked with symbols.
- Implement Newton’s Fractal for the complex function f(z) = z3 − 1.
- Create an initial grid of [−1,1] × [−i,i] with 800 × 800 points.
- Use Newton’s 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 10−10 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.
