## Linear difference equation |

In **linear difference equation**^{[1]}^{:ch. 17}^{[2]}^{:ch. 10} or ** linear recurrence relation** equates to a polynomial that is linear in the various iterates of a *t*, one period earlier denoted as *t*–1, one period later as *t*+1, etc.

An *n*-th order linear difference equation is one that can be written in terms of *a*_{i} and *b* as

or equivalently as

The equation is called *homogeneous* if and *inhomogeneous* if . Since the longest time lag between iterates appearing in the equation is *n*, this is an *n*-th order equation, where *n* could be any positive integer. When the longest lag is specified numerically so *n* does not appear notationally as the longest time lag, *n* is occasionally used instead of *t* to index iterates.

In the most general case the coefficients *a*_{i} and *b* could themselves be functions of time; however, this article treats the most common case, that of constant coefficients.

The * solution* of such an equation is a function of time, and not of any iterate values, giving the value of the iterate at any time. To find the solution it is necessary to know the specific values (known as

Difference equations are used in a variety of contexts, such as in

- solution of homogeneous case
- stability
- solution by conversion to matrix form
- see also
- references

Solving the homogeneous equation involves first solving its

for its characteristic roots *i* = 1, ..., *n*. These roots can be solved for *n* ≤ 4, but

It may be that all the roots are

If no characteristic roots share the same value, the solution of the homogeneous linear difference equation can be written in terms of the characteristic roots as

where the coefficients *c*_{i} can be found by invoking the initial conditions. Specifically, for each time period for which an iterate value is known, this value and its corresponding value of *t* can be substituted into the solution equation to obtain a linear equation in the *n* as-yet-unknown parameters; *n* such equations, one for each initial condition, can be *n* parameter values. If all characteristic roots are real, then all the coefficient values *c*_{i} will also be real; but with non-real complex roots, in general some of these coefficients will also be non-real.

If there are complex roots, they come in pairs and so do the complex terms in the solution equation. If two of these complex terms are and the roots can be written as

where the non-subscript *i* is the *M* is the

where is the angle whose cosine is and whose sine is the last equality here made use of

Now the process of finding the coefficients and guarantees that they are also complex conjugates, which can be written as Using this in the last equation gives this expression for the two complex terms in the solution equation:

which can also be written as

where is the angle whose cosine is and whose sine is

Depending on the initial conditions, even with all roots real the iterates can experience a transitory tendency to go above and below the steady state value. But true cyclicality involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate characteristic roots. This can be seen in the trigonometric form of their contribution to the solution equation, involving and

In the second-order case, if the two roots are identical (), they can both be denoted as and a solution may be of the form

If *b* ≠ 0, the equation

is said to be non-homogeneous. To solve this equation it is convenient to convert it to homogeneous form, with no constant term. This is done by first finding the equation's *steady state value*—a value *y* * such that, if *n* successive iterates all had this value, so would all future values. This value is found by setting all values of *y* equal to *y* * in the difference equation, and solving, thus obtaining

assuming the denominator is not 0. If it is zero, the steady state does not exist.

Given the steady state, the difference equation can be rewritten in terms of deviations of the iterates from the steady state, as

which has no constant term, and which can be written more succinctly as

where *x* equals *y*–*y* *. This is the homogeneous form.

If there is no steady state, the difference equation

can be combined with its equivalent form

to obtain (by solving both for *b*)

in which like terms can be combined to give a homogeneous equation of one order higher than the original.

Other Languages

català: Successió recurrent lineal

Deutsch: Lineare Differenzengleichung

français: Suite récurrente linéaire

日本語: 線型回帰数列