Difference equation

In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms.

For example:

<math>x_{n+1} = r x_n (1 - x_n) \,<math>

Some simply defined recurrence relations can have very complex (chaotic) behaviours and are sometimes studied by physicists and mathematicians in a field of mathematics known as nonlinear analysis.

Linear homogeneous recurrence relations

Recurrence relations, particular linear recurrence relations, can be solved in order to obtain a non-recursive function of n.

The term linear simply means the coefficients of the recursively defined variables (e.g. x_n) is a constant, not another variable and hence do not depend on x. Linear recurrence relations must have some initial conditions, as the first number in the sequence can not depend on other numbers in the sequence and must be set to some value.

Solving linear recurrence relations

Solutions to recurrence relations are found by systematic means, often by using generating functions (formal power series) or by noticing the fact that rⁿ is a solution for particular values of r.

For recurrence relations in the form:

<math>a_{n}=Aa_{n-1}+Ba_{n-2} \,<math>

we have the solution rⁿ:

<math>r^{n}=Ar^{n-1}+Br^{n-2} \,<math>

Dividing through by <math>r^{n-2}<math> we get:

<math>r^2=Ar+B \,<math>

<math>r^2-Ar-B=0 \,<math>

This is known as the characteristic equation of the recurrence relation. Solve for r to obtain the two roots <math>\lambda_1, \lambda_2 <math>, and if these roots are distinct, we have the solution

<math>a_n = C\lambda_1^n+D\lambda_2^n \,<math>

while if they are identical (when A²+4B=0), we have

<math>a_n = C\lambda^n+Dn\lambda^n \,<math>

where C and D are constants.

Additionally, if the equation is of the form <math>a_{n}=Aa_{n-1}+B<math> you can substitute 2 for n and get <math>r^2=Ar+B<math> as above. The constants C and D can be found from the "side conditions" that are often given as <math>a_0=a<math>, <math>a_1=b<math>.

Different solutions are obtained depending on the nature of the roots of the characteristic equation.

If the recurrence is inhomogeneous, a particular solution can be found by the method of undetermined coefficients and the solution is the sum of the solution of the homogeneous and the particular solutions.

Interestingly, the method for solving linear differential equations is similar to the method above — the "intelligent guess" for linear differential equations is e^x.

This is not a coincidence. If you consider the Taylor series of the solution to a linear differential equation:

<math>

\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n} <math>

you see that the coefficients of the series are given by the n-th derivative of f(x) evaluated at the point a. The differential equation provides a linear difference equation relating these coefficients.

This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation.

Example: Fibonacci numbers

The Fibonacci numbers are defined using a linear recurrence relation:

<math>F_{n} = F_{n-1}+F_{n-2} \,<math>

<math>F_{0} = 0 \,<math>

<math>F_{1} = 1 \,<math>

and has solution (letting <math>\Phi = {1+\sqrt{5} \over 2}<math> be the golden ratio)

<math>F_n = {\Phi^n \over \sqrt{5}} - {(1-\Phi)^n \over \sqrt{5}}<math>

The initial conditions are:

<math>F_{0} = 0 \,<math>

<math>F_{1} = 1 \,<math>

Therefore, the sequence of Fibonacci numbers is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ...

Related topics

• recursion
• Master theorem
• Circle points segments proof