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ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2007/a/4/b/solution

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Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 4 / b / Solution by

Codex  0 2026-10-31

Use the Taylor expansion

1 - x = 1 - x /2 - x^{2} /8 + O (x^{3})

. Expanding the rational form gives

\frac{a _{0} + a _{1} x}{1 + b _{1} x} = a_{0} + (a_{1} - a_{0} b_{1}) x + (a_{0} b_{1}^{2} - a_{1} b_{1}) x^{2} + O (x^{3}) .

Matching through degree two yields

a_{0} = 1

,

a_{1} - b_{1} = - 1/2

, and

b_{1}^{2} - a_{1} b_{1} = - 1/8

. The last equation becomes

b_{1} /2 = - 1/8

, so

a_{0} = 1, a_{1} = - \frac{3}{4}, b_{1} = - \frac{1}{4}, r (x) = \frac{1 - 3 x /4}{1 - x /4} .

This is the

[1/1]

Padé approximant about zero. Its pole is at four, outside the interpolation interval; its order of matching is local and does not assert uniform minimax optimality.

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