- Consider a floating-point number system with base 10.
There are five digits in the fraction (mantissa) and the
exponents range between -8 and +9.
- What is the smallest positive floating-point number
in this number system?
- What is the result of 42.873 + 0.079872 in this number system?
+--------+
| /20 |
+--------+
- Derive the formula for the secant method for one equation f(x) = 0.
Illustrate the working of the secant method with a plot.
+--------+
| /15 |
+--------+
- Below is the plot of g(x) = -0.3x^2 + 0.8 x + 2.5.
Starting at x(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 here to see the plot
+--------+
| /15 |
+--------+
- Consider f(x) = x^2 - 0.5 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 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 | | | | |
+-------+------------+------------+------------+------------+
+--------+
| /20 |
+--------+
- Consider
[ 9.229E-02 -1.324E+00 1.976E+00 ]
A = [ -6.501E-01 1.201E+00 -3.308E-01 ].
[ 2.245E+00 -1.265E+00 -1.277E+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?
+--------+
| /30 |
+--------+