A two-node collocation method already has order at least two. It has order at least three exactly when its node polynomial is orthogonal to constants on $[0,1]$:
$$
0=\int_0^1\omega(x)dx
=\frac13-\frac{c_1+c_2}{2}+c_1c_2.
$$
Equivalently,
$$
\boxed{2-3(c_1+c_2)+6c_1c_2=0}.
$$
This is the additional quadrature order condition $\sum_i b_ic_i^2=1/3$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/2/b/solution