-
- Consider the points (0,0), (1,1), and (2,3).
Use the method of Neville to find the value of the interpolating
polynomial at the point 1.5.
- Prove the formula of Neville:
if p[i,..,j-1](x(k)) = f(k), for k=i,..,j-1, and
p[i+1,..,j](x(k)) = f(k), for k=i+1,..,j,
then
(x(k)-x(j))*p[i,..,j-1](x(k)) - (x(k)-x(i))*p[i+1,..,j](x(k))
p[i,..,j](x(k)) = -------------------------------------------------------------
x(i) - x(j)
for k=i,..,j.
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| /20 |
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- Consider the Maclaurin expansion of f(x) = (1-x^2)^(1/2):
1 2 1 4 1 6 5 8 7 10 / 12\
1 - - x - - x - -- x - --- x - --- x + O\x /
2 8 16 128 256
Set up the system of linear equations which must be solved
to construct a Pade approximation of f as a quotient of
two quadrics.
Do NOT solve the system.
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| /20 |
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- Derive the first order extrapolation formula to compute
f'(x0) using forward differences Delta f(x0,h)
for any h > 0, reducing h by the factor r, 0 < r < 1.
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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'(0.5), compute
central differences delta f(0.5,h),
for h=.5,.25,.125.
- Apply extrapolation using the values for delta f(0.5,h).
- How accurate is your final approximation for f'(0.5)?
Justify your answer.
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| /30 |
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-
- Consider the quadrature rule
+h
/
| f(x) dx = w0 f(x0) + w1 f(x1), for h > 0.
/
-h
Set up the system of equations in the weights w0, w1
and nodes x0, x1 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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