Assignment 1 - Solving Nonlinear Equations; Due to December 5, 2011

• The function $f(x)=5x^2+sin(x)-e^x-10$ is given. Solve $f(x) = 0$ in [-5 5] as defined in next page by:
  1. Halving the Interval (Bisection) Method
  2. The Method of False Position (regula falsi)
  3. Newton's Method
  4. Muller's Method
  5. Fixed-point Iteration; $x=g(x)$ Method
• Which methods? See Table 4.1
  

  Table 4.1: You should use the following methods depending on the last digit of your Student ID.

  Last Digit	Methods
  0-1	1,2,3
  2-3	1,2,4
  4-5	1,2,5
  6-7	1,3,4
  8-9	1,3,5

  
  
• Hints:

  >> fplot('5*x^2+sin(x)-exp(x)-10',[-5 5])
  >> [X,FVAL]=fzero('5*x^2+sin(x)-exp(x)-10',[-5 -1])
  >> [X,FVAL]=fzero('5*x^2+sin(x)-exp(x)-10',[-1 2])
  >> [X,FVAL]=fzero('5*x^2+sin(x)-exp(x)-10',[2 5])
  

• Write one complete function.
  • You may use http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter1/mainmulfix_template.m mainmulfix_template.m as a starting template for your code.
  • You can make use of the available MATLAB codes presented in the Hands-On sessions or lectures.
  • An example calling this program;

     >> myfunction(-5, 5, tol_x, tol_y)
    

  • Tabulate the actual error $(x_n - r)$ and function values ($f(x_n)$) as given in Table 4.2.
    

    Table 4.2: Error Sequences. The number of iterations is not limited to or defined as 15. $r$ corresponds to exact value of root.

    	Method1	Method2	Method3	Method1	Method2	Method3
    n	$(x_n - r)$	$(x_n - r)$	$(x_n - r)$	$f(x_n)$	$f(x_n)$	$f(x_n)$
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    12
    13
    14
    15

    
    
  • Plot the behaviours of the errors for three methods. Analyse your plots.
  • Compare and discuss the rate of convergence (use ratios) for three methods.
  • Which method is the best and why?