Finite differences.

The finite difference of the first order for y₀ is the value: D y₀ = y₁–y₀ (the first finite difference). Analogously: D y₁ = y₂–y₁, …, D y_k = y_k+1–y_k.

The second finite differences:

D ²y₀ = D y₁– D y₀

…

D ²y_k = D y_k+1– D y_k

The finite difference of the k-th order: D ^ky_i = D ^k–1y_i+1– D ^k–1y_i.

Let us consider the term:

D ²y₀ = D y₁– D y₀ = y₂–y₁–(y₁–y₀) = y₂–2y₁+y₀

D ³y₀ = D ²y₁– D ²y₀ = y₃–2y₂+y₁–(y₂–2y₁+y₀) = y₃–3y₂+3y₁–y₀

…

D ^my₀ = (–1)^kC y_m–k

D ^my_i = (–1)^kC y_m+i–k

The table of differences:

Divided differences.

The first divided difference:

f(x₀, x₁) = [x₀, x₁] = y_{0, 1} = =

y_{1, 2} =

The divided difference of the second order:

y_{0, 1, 2} = ; y_{1, 2, 3} =

Let us consider the table of the function:

y_{0, 1} = =

y_{0, 1, 2} =

=

y_{0, 1, …, m} =

Properties:

1. D(φ + ψ) = Dφ + Dψ: (φ +ψ)_{0, 1} = φ _{0, 1}+ψ _{0, 1}

2. D(cy) = cDy: (cy)_{0, 1} = cy_{0, 1}

3.: y_{0, 1, …, k} = y_{3, 0, 1, …, k}

Connection between fin. and div. dif. (h = const)