Çankaya University Mcs 331 Numerical Methods Final Examination Dec 31, 2014 09.30 - 11.30 Good Luck!

NAME-SURNAME:

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DEPARTMENT:

DURATION: 120 minutes

$\diamond$ Answer all the questions.

$\diamond$ Write the solutions explicitly and clearly.

Use the numerical terminology.

$\diamond$ You are allowed to use Formulae Sheet.

$\diamond$ Calculator is allowed.

$\diamond$ You are not allowed to use any other

electronic equipment in the exam.

$\diamond$ I declare hereby that I fulfilled the requirements

for the attendance according to the University

regulations and I accept that my examination

will not be valid otherwise.

Question	Grade	Out of
1		15
2		10
3		15
4		20
5		20
6		20
7		20
TOTAL		120

1. Answer the following questions, choose only 4 of them.

   i

   What are the advantages and disadvantages of numerical analysis?

   ii

   Describe the general working of a bracketing method. What are the assumptions for this family of methods?

   iii

   Describe truncation and round-off errors. Give example.

   iv

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

   v

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

   vi

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

   vii

   What are the differences between the interpolation and curve fitting?

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

   viii

   Get the inverse of the matrix through either Gaussian elimination or Gauss-Jordan method.

   ix

   Check your result: $AA^{-1}=I$

3. For the given data points;
   
   $$\begin{array}{rr} x & y \\ \hline 1 & 1.06 \\ 2 & 1.12 \\ 3 & 1.34 \\ 5 & 1.78 \\ \end{array}$$
   
   

   x

   construct the divided-difference table.

   xi

   interpolate for $x=4$.

   xii

   extrapolate for $x=5.5$.

4. A material is tested for cyclic fatigue failure whereby a stress (S), in MPa, is applied to the material and the number of cycles (N) needed to cause failure is measured. The results are in the table below.
   

   X	Y
   Cycles (N)	Stress (S)
   1	1100
   10	1000
   100	925
   1000	800
   10000	625
   100000	550
   1000000	420

   When a log-log plot of stress versus cycles is generated, the data trend shows a linear relationship; $Y'=aX'+b$. Use least-squares method to determine a best-fit equation for this data. Hints:

   xiii

   Start by taking logarithms of the data.

   xiv

   Construct the normal equations.

   xv

   Find the values $a$ and $b$.

   xvi

   Determine the best-fit equation for $Y(X)$.

   
   
5. Write the expression to economize the the Maclaurin series for $e^{2x}$ with the precision 0.08 by using Chebyshev polynomials. Do not perform the calculation.
6. Use the Fourier series to approximate the square wave function (see Figure).
   
   
   
   
7. If the velocity distribution of a fluid flowing through a pipe is known (see Figure), the flow rate $Q$ (that is, the volume of water passing through the pipe per unit time) can be computed by $Q=\int \upsilon dA$, where $\upsilon$ is the velocity and $A$ is the pipe's cross-sectional area. For a circular pipe, $A = \pi r^2$ and $dA = 2\pi r dr$. Therefore, $Q=\int_0^r \upsilon (2\pi r) dr$, where $r$ is the radial distance measured outward from the center of the pipe.
   
   
   
   
   

   $r_i$	$q_i$
   0.0000	0.0000
   0.5000
   3.0000

   xvii

   First fill the table within the four digit accuracy.

   xviii

   Approximate the integral with a step size of $h = 0.5$.

   xix

   Approximate the integral with a step size of $h = 1.0$.

   xx

   Estimate the error in your answers;

   • Exact value of the integral is 44.7419. Find the errors for parts ii and iii.
   • Also use the global error formula to find the errors for parts ii and iii.
   • Analyze and compare these error values.