Answers to Exam 1(a) 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 17 and 333 as floating point numbers and illustrate the calculation of 17+333, using rounding. What is the calculated sum?

Answer:

       17 = +.17 10^2
      333 = +.33 10^3

denormalize 17 and round:

       17 = +.02 10^3
      333 = +.33 10^3
          ------------
            +.35 10^3

The calculated sum is 350.

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Below make the plot of g(x) = 10/(x-3). Starting at x_0 = 4, 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=-2:

Answer:

       g(x) = 10*(x-3)^(-1)
       g'(x) = -10*(x-3)^(-2)
       g'(-2) = -10/5^2 = -2/5

for x=5:
Answer:
```
       g'(5) = -10/2^2 = -5/2
```

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

Answer:

       |g'(-2)| < 1 : linear convergence
       |g'(5)| > 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^2 - 3x in the interval [0,5].

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 |   5.000 |  1.910  |  3.090  |  -2.082  |  0.2781  |
|   1  |   0.000 |   3.090 |  1.180  |  1.910  |  -2.148  |  -2.082  |
|   2  |   0.000 |   1.910 |  0.7296 |  1.180  |  -1.656  |  -2.148  |
+------+---------+---------+---------+---------+----------+----------+

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Consider the matrix

           [  4.447E-01   7.919E+08  ]
       A = [                         ]
           [  6.154E-01   9.218E-01  ]

 with its inverse

           [ -1.891E-09   1.625E+00  ]
  A^(-1) = [                         ].
           [  1.263E-09  -9.124E-10  ]

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

Answer:

       ||A||_1 = 7.919E+8 + 9.218E-1 = 7.919E+8

       ||A^(-1)||_1 = 1.625E+0 + |-9.124E-10| = 1.625E+0

       cond(A) = ||A||_1*||A^(-1)||_1 = 1.287E+9

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) = 1.287E+9 * 10^(-16)
          ||x||                        = 1.287E-7
```

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Consider the matrix

          [   2.000     8.000     0.000   ]
      A = [   9.000     4.000     4.000   ].
          [   6.000     7.000     8.000   ]

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:

                                  [   9.000     4.000     4.000   ] 2
      A  ---------------------->  [   2.000     8.000     0.000   ] 1
                                  [   6.000     7.000     8.000   ] 3


           R2 = R2 - (2/9)*R1     [   9.000     4.000     4.000   ] 2  
         ---------------------->  [   0.2222    7.111    -0.8889  ] 1
           R3 = R3 - (6/9)*R1     [   0.6667    4.333     5.333   ] 3


           R3 = R3 - 4.333/7.111*R2     [   9.000     4.000     4.000   ] 2  
         ---------------------------->  [   0.2222    7.111    -0.8889  ] 1
                                        [   0.6667    6.093     5.875   ] 3

What is the determinant of A?

Answer:

        det(A) = det(P*L*U)
               = det(P)*det(L)*det(U)
               = (-1)*1*(-9.000)*7.111*5.875
               = -376.0

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