Spring08EEE-2443/26/2008

Midterm

  • pen book/note
  • 1 hour and 15 minutes (4:00-5:15 PM).
  • Calculator is allowed.
  • 5 problems, 20 points each.
  • Show all intermediate results for partial credit.
  • All answers must have at least 3 significant figures.

No loose papers will be accepted.

!!! GOOD LUCK !!!

  1. The orders of the matrices [A], [A1], [A2] are 33 and [A] = [A1] [A2]. Determine [A]-1 if :

Answer:

  1. The Newton-Raphson technique is used to find the root of y = sin(x). We all know that the roots are xR = n where n = 0, ±1, ±2,±3, …

a)If the initial guess is x0 = 0.5, which root will be obtained by this technique, why?

b)If the initial guess is x0 = 3, which root will be obtained by this technique, why?

c)If the initial guess is x0 = , which root will be obtained by this technique, why?

d)Will the Newton-Raphson technique work for this function: y = sin2(x)? Why or why not?

(Note: You do not need to do any calculations for the problem. Select an answer and explain the reason for your choice)

Answer:

a)xR = 0

b)xR = 

c)The technique will fail or it will yield a root that is unpredictable because the slope at x0 =  is infinite, which will project the next point to an extremely large value.

d)It will work but converge very slowly and as it approaches the root the derivative f’(x) also approaches zero, which will cause 0/0. Special procedures are needed to take care of this problem.

Acceptable answer: No, it will not work for this function because theroots are multiple roots.

  1. Use the bisection method to determine the root of y = x2 – e-x between x = 0 and 1 (perform two steps of the calculation only).

Answer:

Step / f(x) = / x2 - e-x / a
1 / XL / 0 / XR / 0.5 / XU / 1
f(XL) / -1 / f(XR) / -0.3565307
f(XL)*f(XR) / 0.35653066
2 / XL / 0.5 / XR / 0.75 / XU / 1
f(XL) / -0.3565307 / f(XR) / 0.09013345
f(XL)*f(XR) / -0.0321353
XR / 0.625
  1. Use the Euler’s method (h = 0.1) to numerically determine the solution of the following ODEat x = 2.1 and x = 2.2.
  1. Numerically determine the solution of the following ODE (h = 0.5):.

a)Use the Euler’s method to estimate the solutionat x = 0.5.

b)Use the result from part a) to perform two (2) additional iteration steps with the Huen’s method to improve the accuracy of the solutionat x = 0.5.