Exam 1(c) 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 387 and 25 as floating point numbers and illustrate the calculation of 387+25, using rounding. What is the calculated sum?

                                                           +--------+
                                                           |    /15 |
                                                           +--------+
   

2. Below make the plot of g(x) = 21/(x-4). Starting at x_0 = 13, 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=-3:

   2. for x=7:

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

                                                           +--------+
                                                           |    /20 |
                                                           +--------+
   

3. Apply 3 steps of the golden section search method to find the minimum of the function f(x) = x^5 - 4x in the interval [0,7]. 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 |   7.000 |         |         |          |          |
   |   1  |         |         |         |         |          |          |
   |   2  |         |         |         |         |          |          |
   +------+---------+---------+---------+---------+----------+----------+
   

                                                           +--------+
                                                           |    /15 |
                                                           +--------+
   

4. Consider the matrix

              [ 8.216E-01    8.180E-01 ]
          A = [                        ]
              [ 6.449E+10    6.602E-01 ]
   
   with its inverse
   
              [ -1.251E-11   1.550E-11 ]
     A^(-1) = [                        ].
              [  1.223E+00  -1.558E-11 ]
   
   

   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?

                                                           +--------+
                                                           |    /20 |
                                                           +--------+
   

5. Consider the matrix

               [ 5.000  3.000  6.000 ]
           A = [ 7.000  1.000  3.000 ].
               [ 4.000  1.000  5.000 ]
   

   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?

                                                           +--------+
                                                           |    /30 |
                                                           +--------+