- Compute the Newton form of the polynomial interpolating
x^3 + 2 x through the points 0, 1, and 2.
Show how to evaluate the Newton form most efficiently
at x = 1.5. Write all numbers with four
significant decimal places, using scientific format.
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- Consider the operator D defined by Df(x) = f'(x),
and the operator E defined by Ef(x) = f(x+h)
for any h > 0. Show that E = e^(hD).
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- The Maclaurin expansion of cosh(x) is
1 2 1 4 1 6 ( 8 )
1 + --- x + ---- x + ----- x + O( x ).
2 24 720 ( )
Use this Maclaurin expansion to construct a
Padé approximation
for cosh(x) as a quotient of two quadrics.
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- Consider
Pi
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| sin(x) dx.
/
0
- Apply the composite trapezoidal rule to approximate
this integral, using 1,2, and 4 subintervals of [0,Pi].
Give all symbolic formulas before numerical evaluation.
Write the numbers with six decimal places in scientific format.
- Apply Romberg integration to improve the approximations
from above. Write all numbers with six decimal
places in scientific format.
- How many decimal places are correct in the last number
computed by Romberg integration? Justify your answer.
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- Consider the quadrature formula
b
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| f(x) dx ~ w f(x1) + w f(x2).
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a
Formulate the conditions on w, x1, and x2
so that the degree of the polynomials integrated exactly
by this quadrature formula is as high as possible.
Do not solve this system of equations.
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