-
Consider the polynomial p(x) = x^2 + 2x - 3.
- 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 x/sqrt(x+1) is
2 3 4 35 5 63 6 231 7 429 8
x - 1/2 x + 3/8 x - 5/16 x + --- x - --- x + ---- x - ---- x
128 256 1024 2048
6435 9 10
+ ----- x + O(x )
32768
Use this Maclaurin expansion to construct a Padéapproximation
for x/sqrt(x+1) as a quotient of two quadrics.
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- Give the continued-fraction representation for x/(2 x^2 - 5).
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- Use central differences and Richardson extrapolation to
compute a sixth order approximation for the derivative
of arctan(x) at x = 0.4.
- 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
a
/ ( -a ) ( a )
| f(x) dx = w f(----) + w f(---) for a > 0.
/ 1 ( 2 ) 2 ( 2 )
-a
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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