difference equation (redirected from Recurrence relation)

difference equation

Equation involving differences between successive values of a function of a discrete variable (i.e., one defined for a sequence of values that differ by the same amount, usually 1). A function of such a variable is a rule for assigning values in sequence to it. For example, f(x + 1) = xf(x) is a difference equation. Methods developed for solving such equations have much in common with methods for solving linear differential equations, which difference equations are often used to approximate.

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difference equation - A relation between consecutive elements of a sequence. The first difference is D u(n) = u(n+1) - u(n) where u(n) is the nth element of sequence u. The second difference is D2 u(n) = D (D u(n)) = (u(n+2) - u(n+1)) - (u(n+1) - u(n)) = u(n+2) - 2u(n+1) + u(n) And so on. A recurrence relation such as u(n+2) + a u(n+1) + b u(n) = 0 can be converted to a difference equation (in this case, a second order linear difference equation): D2 u(n) + p D u(n) + q u(n) = 0 and vice versa. a, b, p, q are constants.