Exam 1(b) 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 -22 and +22.
    1. What is the smallest positive floating-point number?
    2. What is the result of 228.5 + 1.704 in this number system? 
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  2. Below is the plot of g(x) = exp(0.25*x). Starting at x(0) = 8, 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. Show the convergence rate of Newton's method is (m-1)/m when applied to f(x) = (x-1)^m. 
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  4. Consider f(x) = x^4 + 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 
                 [  5.041E-01   2.143E-01   7.506E-01  ]
         A = [  6.483E-01   5.604E-01   8.562E-01  ]
             [  9.167E-01   2.797E-01   9.629E-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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