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 Incoming links: Omitted-node identities for divided differences

Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 2 / 14B / c / Solution Created 2026-09-24 Updated 2026-10-14

Write

D = f [x_{0}, \dots, x_{n}]

and let

D_{i}

be the divided difference with

x_{i}

omitted. By symmetry from part (a), reorder the full list so that its first node is

x_{i}

and its last is

x_{j}

. The recurrence from part (b) then has first omitted-node term

D_{i}

and second omitted-node term

D_{j}

, and gives

D = \frac{D _{i} - D _{j}}{x _{j} - x _{i}} .

This proves the requested identity for every pair

i \neq = j

and fixes its sign explicitly; it is one of the omitted-node identities for divided differences.

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Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 2 / 14B / d / Solution Created 2026-09-24 Updated 2026-10-14

Continue with

D, D_{i}

from part (c). Comparing omission of

x_{i}

and

x_{n}

gives

D_{i} = D_{n} + (x_{n} - x_{i}) D .

Comparing omission of

x_{0}

and

x_{n}

gives

D_{0} = D_{n} + (x_{n} - x_{0}) D

. Eliminating

D

proves

D_{i} = γ D_{n} + (1 - γ) D_{0}, γ = \frac{x _{i} - x _{0}}{x _{n} - x _{0}} .

Here

D_{n} = f [x_{0}, \dots, x_{n - 1}]

and

D_{0} = f [x_{1}, \dots, x_{n}]

, so this is exactly the required omitted-node identities for divided differences. The original nodes need not be ordered, and the formula remains valid even if

γ

lies outside

[0, 1]

.

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