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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-80/3/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 80 / 3 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

For the fourth-order two-stage Gauss collocation method,

b_{1} = b_{2} = 1/2

a_{11} = a_{22} = 1/4

a_{12} = 1/4 - 3 /6

and

a_{21} = 1/4 + 3 /6

. The weights are positive. For the matrix defining algebraic stability,

m_{11} = m_{22} = 2 (1/2) (1/4) - (1/2)^{2} = 0,

(1)

and

m_{12} = m_{21} = \frac{1}{2} (\frac{1}{4} - \frac{3}{6}) + \frac{1}{2} (\frac{1}{4} + \frac{3}{6}) - \frac{1}{4} = 0.

(2)

Hence $M = 0$ is positive semidefinite, so the method is algebraically stable. Its two-stage contractivity estimate follows from part (b). The PDF has

a_{22} = 1/4

; the extra

3 /6

attached to that entry in the converted TeX is a transcription error.

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