Solving Nonlinear Equations

• We have given the following function;
  
  $$f(x)=3x + sin(x) - e^x$$
  
  

• To obtain the true value for the root $r$, which is needed to compute the actual error. MATLAB is used as:

  >> solve('3*x + sin(x) - exp(x)')
  ans=
  .36042170296032440136932951583028
  

• Use the function used in the previous item, and write a MATLAB program for Muller's method:
  An algorithm for Muller's method :
  
• Use the function used in the previous item, and write a MATLAB program for Fixed-point Iteration; $x=g(x)$ Method:
  Iteration algorithm with the form $x=g(x)$
  
• Tabulate the actual error values as the following; (See Table 1. The number of iterations is not limited to or defined as 15.)
  

  Table 1: The Error Sequences

  n	Muller $(x_n - r)$	Fixed-point $(x_n - r)$	Muller $f(x_n)$	Fixed-point $f(x_n)$
  1
  2
  3
  4
  5
  6
  7
  8
  9
  10
  12
  13
  14
  15

  
  
• Plot the behaviours of the errors (use ratios) for both cases. Compare and discuss the rate of convergence.
• A pair of equations:
  

  $x^2 + y^2 = 4$

  $e^x + y = 1$

  
  Solve this system by expanding both functions as a Taylor series (begin with $x_0 = 1, y_0 = - 1.7$) and by Iteration (begin with $x=1$)
• Tabulate the actual error values as the following; (See Table 2. The number of iterations is not limited to or defined as 15.)
  

  Table 2: 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