Task 1 (50%): Design a Matab scriipt to solve f(x) =0 for x using the following numerical methods:
· Fixed point solution
· Newton Method
Your Matlab scriipt (One scriipt file for both approaches) must perform the following:
1. Plotting the function f(x) and determine the value x which satisfy f(x) =0 graphically (from the plot).
2. Selecting an initial value for the numerical solution
3. Implementing the fixed-point algorithm and the Newton method for solving f(x)=0 using a for loop which calculates the value of x at each iteration.
4. Computing the absolute error at each iteration,
5. Plotting the value of x versus iteration number,
6. Plotting the error versus the iteration number
7. Repeating the above steps for a different initial value.
Task 2 (50%): Write a report with the following structure
1. Introduction to the problems: show the mathematical formulation of the engineering problem.
2. Numerical Solution: present the steps of the numerical solution and explain how you selected the function g(x) and the initial value x0 and show the Matlab Code you used.
3. Results: include tables of the values of x at each iteration, the plots you generated from the Matlab simulations and comment on them explaining what they illustrate.
4. Discussion: discuss the convergence of the numerical algorithm: explain the effect of the initial value and the choice of the function g(x) on the convergence of both approaches (Fixed point and Newton’s algorithms) and show the number of iterations required to obtain a convergent solution.
A conclusion: summary of what you have done and the results you obtained and a comparison between the two methods and their convergence.
