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

For the fourth-order two-stage Gauss collocation method, $b_1=b_2=1/2$, $a_{11}=a_{22}=1/4$, $a_{12}=1/4-\sqrt{3}/6$ and $a_{21}=1/4+\sqrt{3}/6$. The weights are positive. For the matrix defining algebraic stability,andHence $M=0$ 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 $\sqrt{3}/6$ attached to that entry in the converted TeX is a transcription error.