Exam 1(a) Wed 23 Feb 2005

  1. Suppose the basis of our floating-point numbers is ten and all numbers have a fraction (mantissa) of 4 decimal places long, with exponents ranging between -11 and +11.
    1. What is the smallest positive floating-point number?
    2. What is the result of 407.1 + 2.657 in this number system? 
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  2. Below is the plot of g(x) = 2 sin(x). Starting at x(0) = 3, illustrate on the plot below how to produce three more points defined by x(k+1) = g(x(k)), k=0,1,... 

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  3. Consider the nonlinear equation f(x) = 0. Show that the order of Newton's method is at least one, provided the derivative f' is nonzero at the root. 
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  4. Consider f(x) = x^2 + exp(-x) over the interval [0,1].
    1. Starting with [0,1], apply two steps of the golden section search method, and indicate on the graph below where you do the function evaluations.
      In addition, mark the new intervals as [a1,b1], [a2,b2], and [a3,b3] on the x-axis.

    2. Write the values for x1, x2, f(x1), and f(x2) (4 decimal places, scientific notation): 
         +--------+-----------+-----------+-----------+------------+
   |  step  |    x1     |     x2    |   f(x1)   |   f(x2)    |
   +========+===========+===========+===========+============+
   |    0   |  3.820E-1 |  6.180E-1 |           |            |
   |    1   |           |           |           |            |
   |    2   |           |           |           |            |
   +--------+-----------+-----------+-----------+------------+

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  5. Consider 
                 [  1.904E-01   8.174E-01   9.799E-01  ]
         A = [  6.207E-01   8.729E-01   4.415E-01  ]
             [  4.511E-01   3.919E-01   8.609E-01  ]
    1. Compute the LU decomposition of A with partial pivoting. Use 4 decimal places with rounding, and write all floating-point numbers in scientific format.
    2. What is the determinant of A? 
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