-
- Consider the points (0,0), (1,1), and (2,3).
Use these points to compute the table of divided differences
and give the Newton form of the interpolating polynomial through
these points.
- Give an argument why the divided differences are independent of
the order of the points, i.e.: f[x_0,x_1,x_2] = f[x_1,x_0,x_2].
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| /20 |
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- Consider the quotient
3 2
1 - - x
4
q := --------
1 2
1 - - x
4
Compute a continued fraction representation of q.
Count the number of arithmetical operations it takes to
evaluate q, before and after the conversion.
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| /20 |
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- Why does Romberg integration for
the integral of f(x) over [a,b]
use only even powers of 2? Explain.
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| /15 |
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- Consider the values in the table below:
+-----------------------------------------------------------------------------------------------+
| x | 0.00000 | .125000 | .250000 | .375000 | .500000 | .625000 | .750000 | .875000 | 1.00000 |
+-----------------------------------------------------------------------------------------------+
|f(x) | 0.00000 | .382683 | .707107 | .923880 | 1.00000 | .923880 | .707107 | .382683 | 0.00000 |
+-----------------------------------------------------------------------------------------------+
Perform all your calculations with six significant decimal places.
Write all numbers in scientific notation.
- To approximate f'(0.0), compute
forward differences Delta f(0.0,h),
for h=.5,.25,.125.
- Apply extrapolation using the values for Delta f(0.0,h).
- How accurate is your final approximation for f'(0.0)?
Justify your answer.
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| /30 |
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-
- Consider the quadrature rule
b
/
| f(x) dx = w0 f(a) + w1 f((a+b)/2) + w2 f(b).
/
a
Set up the system of equations in the weights w0, w1,
and w2 to be satisfied for the highest possible algebraic
degree of accuracy.
Do NOT solve this system.
- What is the algebraic degree of accuracy attained by this rule?
Justify your answer.
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| /15 |
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