Calculus of Finite Differences

a branch of mathematics that studies functions of a discretely (discontinuously) varying argument, in contrast to differential calculus and integral calculus, where the argument is assumed to vary continuously. The expressions

Δy_k ≡ Δf(x_k) = f(x_k+1)− f(x_k)

first order differences

Δ²y_k ≡ Δ²f(x_k) = Δf(x_k+1)− Δf(x_k)

=f(x_k+2) − 2f(x_k+1) + f(x_k+2)

second order differences

. . . . . . . . . . . . . . . . . . . .

Δⁿy_k ≡ Δⁿf(x_k) = Δ^n-1f(x_k+1) − Δ^n-1f(x_k)

are the “forward” finite differences for a sequence of values y₁ = f(x₁), y₂ = f(x₂, . . . ,y_k = f(x_k), . . . , of the function f(x) corresponding to the sequence x₀, . . . ,x_k, . . . of values of the argument; here x_k = x₀ + kh, with h constant and k integral.

Likewise, the “backward” finite differences Δⁿy_k are defined by the equalities

Δⁿy_k = ∇ⁿy_k+1

In connection with interpolation, frequent use is made of central differences δⁿy, which are computed for odd n at the points x = x_i + 1/2h and for even n at the points x = x_i, by the formulas

δf(x + 1/2h) ≡ δy_i+1/2 = f(x_i+1) − f(x_i)

δf(x_i) ≡ δ²y_i ≡ δy_i+1/2 − δy_i-1/2

. . . . . . . . . . . . . . . . . . . .

δ^2m-1f(x_i + 1/2h) ≡ δ^2m-1y_i+1/2 = δ^2m-2y_i+1 − δ^2m

δ^2mf(x_i)≡δ^2m-1y_i+1/2 −δ^2m-1y_i-1/2

They are supplemented with the arithmetic means

for m = 1, 2, . . . and the mean

for m = 0. The central differences δⁿy are related to the finite differences Δⁿy by the equations

δ^2my_i = ∆^2my_i-m

δ^2m+1y_i+1/2 = ∆^2m+1y_i-m

If the values of the argument do not form an arithmetic progression, that is, if x_k + x_k is not constant, then instead of finite differences we use divided differences, defined successively by the formulas

The connection between finite differences and derivatives is given by the formula, where. There exists a perfect analogy between the role of finite differences in the theory of functions of a discrete argument and the role of derivatives in the theory of functions of a continuous argument. Finite differences are at the core of a number of branches of numerical analysis, such as interpolation of functions, numerical differentiation and integration, and numerical methods for solving differential equations.

For example, in order to obtain an approximate solution of (an ordinary or partial) differential equation, we often replace the derivatives by the corresponding differences divided by the powers of the differences of the arguments and solve the resulting difference equation.

An important branch of the calculus of finite differences is devoted to the solution of difference equations of the form

(1)F[x, Δf(x), . . . , Δⁿf(x)] = 0

a problem that in many ways is similar to the solution of nth-order differential equations. Equation (1) is usually written in the form

Φ[x, f(x), f(x₁), . . . , f(x_n)] = 0

with differences expressed by the corresponding values of the function. A particularly simple case is that of a linear homogeneous equation with constant coefficients:

f(x + n) + a₁f(x + n − 1) + . . . + a_nf(x) = 0

where a₁, . . . , a_n are constants. In order to solve this equation, we must find the roots λ₁, λ₂, . . . , λ_n of its characteristic equation

λⁿ + aλ^n-1 + . . . + a_n = 0

The solution of the equation is then represented in the form

f(x) = C₁λ₁^x +C₂λ₂^x + . . . + C_nλ_n^x

where C₁, C₂, . . . , C_n are arbitrary constants (it is assumed here that no two of the numbers λ₁, λ₂, . . . , λ_n are equal).