Ceng 375 Numerical Computing
Midterm
Nov 12, 2009 14.40-16.30
Good Luck!

Each question is 25 pts.

1. The following function is given
   
   $$f(x)=3*sin(x)- e^x/4-1$$
   
   
   This nonlinear equation ($f(x)=0$) is solved by using four methods, namely Bisection, Regula Falsi, Newton's, Muller's methods. See the following MATLAB commands;

   >> f = inline ( ' 2*sin( x) - exp ( x)/4 -1');
   >> df = inline ( ' 2*cos( x) - exp ( x)/4');
   >> fplot(f,[-7 -3]); grid on;
   >> format short e
   >> bisect(f,-7,-5,fzero(f,[-7 -5]),1e-5);
   >> regula(f,-7,-5,fzero(f,[-7 -5]),1e-5,eps,20);
   >> newton(f,df,-7,fzero(f,[-7 -5]),1e-5,eps,20);
   >> muller(f,-7,-6,-5,fzero(f,[-7 -5]),1e-5,eps,20);
   

   Plot of the function is given at the following figure;

   
   
   Then, the following tables are obtained.
   

   Table: Obtained root values at each iteration for all of four methods.

   iteration	Bisection	Regula	Newton	Muller
   1	-6.0000e+00	-5.5672e+00	-5.4650e+00	-5.7134e+00
   2	5.5000e+00	-5.7373e+00	-5.8008e+00	-5.7604e+00
   3	-5.7500e+00	-5.7575e+00	-5.7596e+00	-5.7591e+00
   4	-5.8750e+00	-5.7590e+00	-5.7591e+00	-5.7591e+00
   5	-5.8125e+00	-5.7591e+00	-5.7591e+00	-
   6	-5.7812e+00	-5.7591e+00	-	-

   
   
   

   Table: Obtained function values at each iteration for all of four methods.

   iteration	Bisection	Regula	Newton	Muller
   1	-4.4179e-01	3.1174e-01	4.5882e-01	7.8084e-02
   2	4.1006e-01	3.7524e-02	-7.3051e-02	-2.2184e-03
   3	1.5762e-02	2.8928e-03	-8.1042e-04	4.1882e-06
   4	-2.0681e-01	2.0926e-04	-1.0968e-07	-4.0674e-11
   5	-9.3753e-02	1.5061e-05	-2.2204e-15	-
   6	-3.8525e-02	1.0836e-06	-	-

   
   

   i

   Analyze these tables. Is the convergence sustained for the each methods? For the sustained ones; at which iteration and why?

   ii

   If the exact value is given as $-5.7591e+00$, fill the following table for two methods. Choose two methods and use scientific notation with five significant figures.;
    
   

   iteration	$Error_1$	$Error_2$	$Error~Ratio_1$	$Error~Ratio_2$
   1
   2
   3
   4
   5
   6

    
   

   iii

   What can you say about the speed of convergences for each method?
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   iv

   Which method is the best one? Why?
   
   
   
   
   
   
   
   
   
   

2. Consider the same function with the previous question:

   v

   Find the root in the interval of $[-5,-3]$ with Newton's method. Hint: Newton's method uses the algorithm:
   

   $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

   
   

   vi

   Estimate the error at the last iteration in your answer to part i. Hint: To estimate the error, compute one more iteration.
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   vii

   Approximately how many iterations of the bisection method would have been required to achieve the error value of $1e-5$?
   

3. Consider this pair of equations:
   
   $$x^2+4y^2=4$$
   
   
   
   
   $$y-x^2+2x=0.5$$
   
   
   Plot of the system is given at the following figure;
   

   
   
   
   Solve this system by iteration. Start with something like $y=???$ and proceed only two iterations.
4. Consider the linear system;
   
   $$\begin{array}{r} x_1-2x_2+4x_3=6\\ 8x_1-3x_2+2x_3=2\\ -1x_1+10x_2+2x_3=4\\ \end{array}$$
   
   

   viii

   Solve this system by Gaussian elimination with pivoting. How many row interchanges are needed?

   ix

   What is the value of determinant?

   x

   Obtain the $LU$ decomposition of the system.

   xi

   Repeat without any row interchanges (only for the first item). Do you get the same results? Why?