2002 Paper 12 Question 9

2002 Paper 12 Question 9
2002 Paper 12 Question 9
Numerical Analysis II
(a) In the Chebyshev characterisation theorem, the best L∞ approximation to
f (x) over a closed finite interval by a polynomial pn−1 (x) of degree n − 1 has
the property that the maximum error |e(x)| is attained at M distinct points
ξj such that e(ξj ) = −e(ξj−1 ). What is M ?
[2 marks]
(b) Let x = m × 2k represent a √
normalised number in a floating-point
implementation. When computing x show how the domain of the problem
can be reduced to x ∈ [1, 4). Find√
the coefficients a, b which minimise ||e(x)||∞
over [1, 4] where e(x) = ax + b − x.
[8 marks]
(c) Taking full account of symmetry, describe the form of the best polynomial
approximation p(x) to x4 over [−1, 1] by a polynomial of lower degree. Sketch
x4 and p(x), showing the extreme values of |e(x)| where e(x) = x4 − p(x).
Hence compute the coefficients of p(x).
[10 marks]
1
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