- Suppose the basis of our floating-point numbers is ten
and all numbers have a fraction of
three decimal places long, with exponents ranging
between -9 and +9.
- What is the machine precision in this number system?
- What is the result of 72.3+2.89 in this number system?
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| /20 |
+--------+
- Consider
x dx abs(f(x))
--------------------------------------------
6.702462e-01 -1.228685e-02 1.846068e-06
6.794614e-01 -9.215139e-03 5.841073e-07
6.863727e-01 -6.911355e-03 1.848152e-07
This sequence is obtained by Newton's method.
The first column lists the successive approximations
for a root, the "dx" in the second column
is the increment added to the previous approximation
to obtain the "x" at the current line.
The last column lists the absolute value of
the function evaluated at the approximation.
- Estimate the multiplicity of the root from this sequence.
- Use the multiplicity to find a more accurate
value for the root.
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| /15 |
+--------+
- Consider the system Ax = b,
A an n-by-n matrix, det A not zero,
b an n-vector, with solution x.
The calculated solution xx satisfies A xx = b.
Let r = b - A xx. Show that
||r|| ||x - xx||
------- <= ||A|| ||A^(-1)|| -----------
||b|| ||x||
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| /15 |
+--------+
- Consider f(x) = cos(5x)
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.
- Write the values for x1, x2, f(x1), and f(x2)
in the table below:
+--------+-----------+-----------+-----------+------------+
| step | x1 | x2 | f(x1) | f(x2) |
+========+===========+===========+===========+============+
| 0 | 3.820E-1 | 6.180E-1 | | |
| 1 | | | | |
| 2 | | | | |
+--------+-----------+-----------+-----------+------------+
+--------+
| /20 |
+--------+
- Consider
[ 5.028e-01 3.046e-01 6.822e-01 ]
A = [ 7.095e-01 1.897e-01 3.028e-01 ]
[ 4.289e-01 1.934e-01 5.417e-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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| /30 |
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