Exam 1(b) Wed 20 Feb 2002

1. Consider the representation of floating-point numbers with base 10 and 2 digits in the fraction part. The values for the exponents are between -10 and +10.

   1. What is the machine precision in this number system?

   2. Represent the numbers 13 and 577 as floating point numbers and illustrate the calculation of 13+577, using rounding. What is the calculated sum?

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2. Below make the plot of g(x) = 4/(x+3). Starting at x_0 = -7, illustrate on the plot below how to produce four more points defined by x(k+1) = g(x(k)), k=0,1,...

   Compute the convergence rate of this fixed-point iteration

   1. for x=1:

   2. for x=-4:

   What conclusions can you make from the rates you computed above?

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3. Apply 3 steps of the golden section search method to find the minimum of the function f(x) = x^3 - 2x in the interval [0,2]. Write the values for a,b, x_1, x_2, f(x_1), and f(x_2) in the table (4 decimal places):

   +------+---------+---------+---------+---------+----------+----------+
   | step |    a    |    b    |   x_1   |   x_2   |  f(x_1)  |  f(x_2)  |
   +------+---------+---------+---------+---------+----------+----------+
   |   0  |   0.000 |   2.000 |         |         |          |          |
   |   1  |         |         |         |         |          |          |
   |   2  |         |         |         |         |          |          |
   +------+---------+---------+---------+---------+----------+----------+
   

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4. Consider the matrix

              [  8.537E-01   4.966E-01  ]
          A = [                         ]
              [  5.936E-01   8.998E+08  ]
   
    with its inverse
   
              [  1.171E+00   -6.465E-10  ]
     A^(-1) = [                          ].
              [ -7.728E-10    1.111E-09  ]
   

   1. Compute the condition number of A using the norm ||.||_1.

   2. Suppose we wish to solve the system A x = b, using the matrix A from above. Assuming a relative error of 10^(-16) on the coefficients of the matrix, what is the bound on the relative error of the solution?

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5. Consider the matrix

             [  6.000     0.000     9.000  ]
         A = [  1.000     4.000     4.000  ].
             [  2.000     7.000     4.000  ]
   

   1. Compute the LU decomposition of A with partial pivoting. Calculate with four decimal places, using rounding: write the answer of every step rounded to four decimal places, and use the rounded number in the calculations of the next step.

   2. What is the determinant of A?

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