Ceng 375 Numerical Computing
Final
Jan 15, 2009 13.00-15.00
Good Luck!

1. (10pts) Choose only two questions.

   i

   What are the advantages and disadvantages of numerical analysis?

   ii

   Describe truncation and round-off errors. Give example.

   iii

   Describe the concept of ill-conditionness. Give an example.

   iv

   What information can be obtained from the determinant of a matrix?

   v

   Why do we need pivoting while solving sets of equations by elimination methods? Can we skip pivoting and under which circumstances?

   vi

   What does singularity mean for a matrix? Make a comparison of singular and nonsingular matrices.

   vii

   What information can be obtained from the condition number of a matrix?

   viii

   What are the differences between the interpolation and curve fitting?

2. (20pts) Choose only two questions.

   ix

   For the given data points; we suggest the relation $y(x)=\alpha e^{\beta x}$.
   

   1. First, construct the normal equations.
   2. Then, describe the remaining steps.

   x

   Solve the following linear system by either by Jacobi or Gauss-Seidel iterations;
   

   $$\begin{array}{r} 4x-y+z=7\\ -2x+y+5z=15\\ 4x-8y+z=-21\\ \end{array}$$

   
   

   1. Start by $P_0=(1,2,2)$. Iterate only two steps.
   2. Compare Jacobi or Gauss-Seidel methods.

   xi

   Consider the function:
   

   $$f(x) = cos(x)- x$$

   
   

   1. Show that this function has a simple root in the interval $0 < x < 1$
   2. Estimate this root using two iterations of the Secant Method. The secant algorithm is
      
      $$x_{n+1}=x_n-f(x_n)\frac{(x_{n-1}-x_n)}{f(x_{n-1})-f(x_n)}$$
      
      

   3. Estimate the error in your answer to part ii.

3. (20pts) Consider the matrix
   
   $$A=\left[ \begin{array}{rrr} 3 &-1 &2\\ 1 & 1 &3\\ -3 & 0 &5\\ \end{array} \right]$$
   
   

   xii

   Use the Gaussian elimination method to triangularize this matrix and from that gets its determinant.

   xiii

   Get the inverse of the matrix through Gauss-Jordan method.

4. (20pts)

   xiv

   Find the Fourier coefficients for $f(x)=x^2-1$ if it is periodic and one period extends from $x=-1$ to $x=2$. Do not evaluate the integrals.

   xv

   Write the Fourier series expansion for this function until $3^{rd}$ term.

5. (20pts) Write the expression to economize the the Maclaurin series for $e^{2x}$ with the precision 0.008 by using Chebyshev polynomials.
   Hint: The two-term recursion formula
   
   $$\begin{array}{l} T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)\\ T_0(x)=1\\ T_1(x)=x\\ \end{array}$$
   
   

6. (20pts) Consider the function $f(x)=x^3$. Following table within the five digit accuracy is given.

   $x_i$	$f_i$
   0.00000	0.00000
   0.20000	0.00800
   0.40000	0.06400
   0.60000	0.21600
   0.80000	0.51200
   1.00000	1.00000
   1.20000	1.72800

   xvi

   Approximate $\int_0^{1.2} f(x)$dx using the Trapezoidal Rule and a step size of $h = 0.2$.

   xvii

   Approximate $\int_0^{1.2} f(x)$dx using the Trapezoidal Rule and a step size of $h = 0.4$.

   xviii

   Estimate the error in your answers;

   1. Find the exact value of the integral simply by integrating the given function. Then, find the errors for parts i and ii.
   2. Also use the following global error formula to find the errors for parts i and ii.
      
      $$Global~error=(-1/12)h^3nf''(\xi)$$
      
      

   3. Analyze and compare these error values.