Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 67 / 5 / a

Solution

Let

C_{k} (u)

be the order-

k

[0, k]

. The Cox-de Boor recurrence on unit-spaced knots reads

C_{k} (u) = \frac{u}{k - 1} C_{k - 1} (u) + \frac{k - u}{k - 1} C_{k - 1} (u - 1), k \geq 2.

Start with

C_{1} = 1_{[0, 1)}

. The order-two hat satisfies

C_{2} (1) = 1

and vanishes at its support endpoints. Substituting these values into the recurrence gives

C_{3} (1) = C_{3} (2) = 1/2

and then

C_{4} (1) = \frac{1}{6}, C_{4} (2) = \frac{2}{3}, C_{4} (3) = \frac{1}{6} .

The continuous cubic spline is zero at

0, 4

and outside its support. Since

N_{j} (t) = C_{4} (t - j)

and

x_{i} = i + 2

N_{j} (x_{i}) = ⎩ ⎨ ⎧ 2/3, 1/6, 0, j = i, j = i - 1 or j = i + 1, ∣ i - j ∣ \geq 2.

Only indices

1 \leq j \leq n

are retained. In particular no nonexistent spline is added to an endpoint row.

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