 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-71/4/2/a/solution

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

Use the Cox-de Boor recurrence. For unit knots the order-two hat is

L (u) = u

[0, 1]

L (u) = 2 - u

[1, 2]

, and zero elsewhere. The order-three quadratic cardinal B-spline is

q (u) = \frac{u}{2} L (u) + \frac{3 - u}{2} L (u - 1) = ⎩ ⎨ ⎧ \frac{1}{2} u^{2}, \frac{3}{4} - (u - \frac{3}{2})^{2}, \frac{1}{2} (3 - u)^{2}, 0, 0 \leq u \leq 1, 1 \leq u \leq 2, 2 \leq u \leq 3, otherwise .

Thus

N_{j} (t) = q (t - j)

. At

x_{i} = i + 3/2

, the only possible arguments in its support are

3/2

and

1/2, 5/2

at neighboring indices. Their values give

N_{j} (x_{i}) = ⎩ ⎨ ⎧ \frac{3}{4}, \frac{1}{8}, 0, j = i, ∣ j - i ∣ = 1, ∣ j - i ∣ \geq 2.

Indices outside

1, \dots, n

are absent, so the first and last rows have just one neighboring nonzero value.

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