Hands-on-Solving Nonlinear Equations with MATLAB II

1. We have given the following function;
   
   $$f(x)=3x + sin(x) - e^x$$
   
   
   1. To obtain the true value for the root $r$, which is needed to compute the actual error. MATLAB is used as:

      
   2. Comparing Muller's and Fixed-point Iteration methods. (Download http://siber.cankaya.edu.tr/NumericalComputations/mfiles/chapter1/muller.mmuller.m, http://siber.cankaya.edu.tr/NumericalComputations/mfiles/chapter1/fixedpoint.mfixedpoint.m)
      save with the name muller.m. Then;

      
      
      save with the name fixedpoint.m. Then;

      
   3. Plot the behaviours of the errors (may use ratios) for both cases. Compare and discuss the rate of convergence.
      Solution:

      
      
      save with the name main.m. Then;

      >> main
      

      For the rate of convergence: Muller's method converges much faster than fixed-point iteration.
2. The following MATLAB command plots the function
   $f_1(x)=x^2-3x+2$

   
   
   and the following finds the roots; (What are 1 -3 2?)

   >> roots([1 -3 2])
   ans =
        2
        1
   

   These are distinct real roots. Apply same procedure for the following functions
   
   $$f_2(x)=x^2-10x+25$$
   
   
   
   
   $$f_3(x)=x^2-17x+72.5$$
   
   
   comment the outputs. You should observe, repeated real roots, and complex roots.
3. A pair of equations:
   

   $x^2 + y^2 = 4$

   $e^x + y = 1$

   1. Write a MATLAB program to solve this system by
      1. expanding both functions as a Taylor series expansion (begin with $x_0 = 1, y_0 = - 1.7$). See lecture notes.
      2. and by Iteration (begin with $x=1$). See lecture notes.
   2. Tabulate the actual error values as the following; (See Table 1. Number of iterations is not limited to or defined as 15.)
      

      Table 1: The Error Sequences

      n	Expansion $f(x_n)$	Iteration $f(x_n)$
      1
      2
      3
      4
      5
      6
      7
      8
      9
      10
      12
      13
      14
      15