- Consider a floating-point number system with base 10.
There are five digits in the fraction (mantissa) and the
exponents range between -7 and +8.
- What is the smallest positive floating-point number
in this number system?
- What is the result of 12.381 + 0.098321 in this number system?
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- Derive the formula for the method of Aitken to accelerate
the sequence x(k), k=0,1,...
Illustrate the working of Aitken acceleration with a plot.
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- Below is the plot of g(x) = 0.4x^2 + 0.2x - 1.2. Starting at
x(0) = 2.8, illustrate on the plot below how to produce
three more points defined by x(k+1) = g(x(k)), k=0,1,...
click here to see the plot
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- Consider f(x) = x^2 - 1.1 x + 0.18
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 here to see the graph
- 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 | | | | |
| 2 | | | | |
+-------+------------+------------+------------+------------+
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- Consider
[ -1.357E-01 6.112E-01 1.365E+00 ]
A = [ 8.797E-01 9.792E-01 -8.069E-01 ].
[ -2.263E-01 -1.160E+00 1.126E+00 ]
- Compute the LU decomposition of A with partial pivoting.
Use 4 decimal places with rounding, and write
all floating-point numbers in scientific format.
- What is the determinant of A?
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