Answers to Exam 1(c) Fri 7 Oct 2005

Consider a floating-point number system with base 10. There are five digits in the fraction (mantissa) and the exponents range between -9 and +8.

What is the smallest positive floating-point number in this number system?
```
    +.10000E-9 = 10^(-10)
```

What is the result of 77.236 + 0.059321 in this number system?

    77.236 = +.77236E+2
    0.059321 = +.59321E-1 = +.00059321E+2 (denormalize)

    +.77236E+2
    +.00059321E+2
   ---------------
    +.77295321E+2 = +.77295E+2 = 77.295

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Derive the formula for Newton's method for one equation f(x) = 0.
Illustrate the working of Newton's method with a plot.

  Taylor expansion of f(x+dx) = f(x) + dx*f'(x) + O((dx)^2).
  We discard the higher order term O((dx)^2).
  We look for the update dx to x so that f(x+dx) = 0.
  Therefore, dx must satisfy 0 = f(x) + dx*f'(x) or dx = -f(x)/f'(x).

  So we obtain x(k+1) = x(k) - f(x(k))/f'(x(k)).

  To illustrate Newton's method, we draw the tangent line at x(k)
  and mark x(k+1) where the tangent meets the horizontal x-axis.
  See the solution handout for the plot.

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Below is the plot of g(x) = (1.6 x)^(1/2). Starting at x(0) = 0.3, illustrate on the plot below how to produce three more points defined by x(k+1) = g(x(k)), k=0,1,...

click to see the plot
See the solution handout for the answer.

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Consider f(x) = x^2 - 1.4 x + 0.24 over the interval [0,1].

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.
click to see the graph
See the solution handout for the answer.

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  | -1.489E-1  | -2.433E-1  |
|   1   |  6.180E-1  |  7.639E-1  | -2.433E-1  | -2.459E-1  |
|   2   |  7.639E-1  |  8.541E-1  | -2.459E-1  | -2.263E-1  |
+-------+------------+------------+------------+------------+

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Consider

             [ -1.213E-01  -3.148E-01   6.446E-01  ]
         A = [ -1.111E+00  -5.082E-01   2.092E-01  ].
             [ -2.098E-01  -2.774E+00   4.738E-01  ]

Compute the LU decomposition of A with partial pivoting. Use 4 decimal places with rounding, and write all floating-point numbers in scientific format.

           [ -1.111E+00  -5.082E-01   2.092E-01  ] 2
  A -----> [ -1.213E-01  -3.148E-01   6.446E-01  ] 1
           [ -2.098E-01  -2.774E+00   4.738E-01  ] 3

            .1213      [                                     ]
 R2 := R2 - ------ R1  [ -1.111E+00  -5.082E-01   2.092E-01  ] 2
            1.111      [                                     ]
 --------------------> [  1.092E-01  -2.593E-01   6.218E-01  ] 1
            .2098      [                                     ]
 R3 := R3 - ------ R1  [  1.888E-01  -2.678E+00   4.343E-01  ] 3
            1.111      [                                     ]

                       [ -1.111E+00  -5.082E-01   2.092E-01  ] 2
 --------------------> [  1.888E-01  -2.678E+00   4.343E-01  ] 3
                       [  1.092E-01  -2.593E-01   6.218E-01  ] 1

            .2593      [                                     ]
 R3 := R3 - ------ R2  [ -1.111E+00  -5.082E-01   2.092E-01  ] 2
            2.678      [                                     ]
 --------------------> [  1.888E-01  -2.678E+00   4.343E-01  ] 3
                       [                                     ]
                       [  1.092E-01   9.683E-02   5.797E-01  ] 1
                       [                                     ]

     [ 0 1 0 ]         [     1           0           0       ]
 P = [ 0 0 1 ]     L = [  1.888E-01      1           0       ]
     [ 1 0 0 ]         [  1.092E-01   9.683E-02      1       ]

                       [ -1.111E+00  -5.082E-01   2.092E-01  ]
                   U = [     0       -2.678E+00   4.343E-01  ]
                       [     0           0        5.797E-01  ]

What is the determinant of A?

  det(A) = det(P*L*U)
         = det(P)*det(L)*det(U)
         = (+1)*(+1)*(-1.111E+00)*(-2.678E+00)*(5.797E-01)
         = +1.725E+00

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