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- Consider f(x) = ln(x).
Give the interpolating polynomial in the form of Lagrange,
using three points: (1,ln(1)), (2,ln(2)), and (3,ln(3)).
- Give one advantage and one disadvantage of the method of Lagrange.
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| /20 |
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- Set up the linear system for the natural spline for ln(x)
over [1,2], using 4 intervals of equal length.
Do NOT solve the system.
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| /20 |
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- Show that the central difference operator delta f(x,h) applied
to a function f with step size h >0 at x to approximate
f'(x) has an error of order O(h^2).
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| /15 |
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- Consider the values in the table below:
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| x | 1.00000 | 1.12500 | 1.25000 | 1.37500 | 1.50000 | 1.62500 | 1.75000 | 1.87500 | 2.00000 |
+----------------------------------------------------------------------------------------------+
|f(x)| .841471 | .902268 | .948985 | .980893 | .997495 | .998531 | .983986 | .954086 | .909297 |
+----------------------------------------------------------------------------------------------+
Perform all your calculations with six significant decimal places.
- Apply the composite Trapezoidal rule to compute
T(h), for h = 0.5,0.25,0.125.
- Use the three values for T(h) from above in Romberg integration.
- How accurate is your final approximation for
the integral of f(x) over [1,2]?
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| /30 |
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- Consider the quadrature rule
+h
/
| f(x) dx = f(x0) + f(x1) + f(x2) + f(x3), for h > 0.
/
-h
- Set up the system of equations in the unknown
nodes x0, x1, x2, and x3 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?
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| /15 |
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