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Consider the polynomial p(x) = x^2 - x + 1.
- Construct the Newton form of p(x) by divided differences,
using the points (x_i,p(x_i)), with x_i = i, for i=0,1,2,3.
-
Explain why the last element f_0123 in the table of divided
differences you constructed above is (or should have been) zero.
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- The Maclaurin expansion of sqrt(x+1) is
2 3 4 5 21 6 33 7
1 + 1/2 x - 1/8 x + 1/16 x - 5/128 x + 7/256 x - ---- x + ---- x
1024 2048
429 8 715 9 10
- ----- x + ----- x + O(x )
32768 65536
Use this Maclaurin expansion to construct a Padé approximation
for sqrt(x+1) as a quotient of two quadrics.
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- Give the continued-fraction representation for x/(x^2 - 7).
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- Use central differences and Richardson extrapolation to
compute a sixth order approximation for the derivative
of arccos(x) at x = 0.7.
- Start with h=0.1 and divide h by 2 (i.e.: r = 1/2)
to build the first column of the extrapolation table.
Show how you applied the formulas (for partial credit).
Write your answers with ten decimal places.
- How many decimal places are correct in your final answer
from Richardson extrapolation? Explain.
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- Consider the quadrature rule
2a
/
| f(x) dx = w f(-a) + w f(a) for a > 0.
/ 1 2
-2a
Determine the weights w_1 and w_2 so that the
rule has the highest possible algebraic degree of precision.
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- How would you compute a least squares approximation
using Chebyshev polynomials?
For example, suppose we have five samples (x_i,y_i), i=1,2,...,5.
We wish to fit the data with a second degree polynomial
p(x) = a_0 T_0(x) + a_1 T_1(x) + a_2 T_2(x).
Describe how you would find the coefficients a_0, a_1, and a_2.
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