- 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?
- What is the result of 77.236 + 0.059321 in this number system?
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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.
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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
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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
- 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
[ -1.213E-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.
- What is the determinant of A?
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