Answers to Exam 1(c) Wed 20 Feb 2002

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.

What is the machine precision in this number system?
Answer:
```
       B^(-p) = 10^(-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?

Answer:

      387 = +.39 10^3
       25 = +.25 10^2

denormalize 25 and round:

       25 = +.03 10^3
      387 = +.39 10^3
          ------------
            +.42 10^3

The calculated sum is 420.

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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,...

Answer: Follow the link for the cobweb picture.

Compute the convergence rate of this fixed-point iteration

for x=-3:

Answer:

       g(x) = 21*(x-4)^(-1)
       g'(x) = -21*(x-4)^(-2)
       g'(-3) = -21/7^2 = -21/49 = -3/7

for x=7:
Answer:
```
       g'(7) = -21/3^2 = -21/9 = -7/3
```

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

Answer:

       |g'(-3)| < 1 : linear convergence
       |g'(7)| > 1 : divergence

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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):

Answer:

+------+---------+---------+---------+---------+----------+----------+
| step |    a    |    b    |   x_1   |   x_2   |  f(x_1)  |  f(x_2)  |
+------+---------+---------+---------+---------+----------+----------+
|   0  |  0.000  |  7.000  |  2.674  |  4.326  | 1.260E+2 | 1.498E+3 |
|   1  |  0.000  |  4.326  |  1.653  |  2.674  |  5.729   | 1.260E+2 |
|   2  |  0.000  |  2.674  |  1.021  |  1.653  | -2.974   |  5.729   |
+------+---------+---------+---------+---------+----------+----------+

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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 ]

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

Answer:

       ||A||_1 = 6.449E+10 + 6.602E-1 = 6.449E+10

       ||A^(-1)||_1 = 1.223E+0 + |-1.558E-11| = 1.223E+0

       cond(A) = ||A||_1*||A^(-1)||_1 = 7.887E+10

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?
Answer:
```
       ||x - xx||
       ----------  <= cond(A)*10^(-16) = 7.887E+10 * 10^(-16)
          ||x||                        = 7.887E-6
```

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Consider the matrix

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

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.

Answer:

                                   [   7.000    1.000    3.000  ] 2
       A  ---------------------->  [   5.000    3.000    6.000  ] 1
                                   [   4.000    1.000    5.000  ] 3


            R2 = R2 - (5/7)*R1     [   7.000    1.000    3.000  ] 2
          ---------------------->  [   0.7143   2.286    3.857  ] 1
            R3 = R3 - (4/7)*R1     [   0.5714   0.4286   3.286  ] 3


            R3 = R3 - 0.4286/2.286*R2     [   7.000    1.000    3.000  ] 2
          ----------------------------->  [   0.7143   2.286    3.857  ] 1
                                          [   0.5714   0.1875   2.563  ] 3

What is the determinant of A?

Answer:

        det(A) = det(P*L*U)
               = det(P)*det(L)*det(U)
               = (-1)*1*(-7.000)*2.286*2.563
               = -41.01

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