Difference equations are used in a variety of contexts, such as in economics to model the evolution through time of variables such as gross domestic product, the inflation rate, the exchange rate, etc.
They are used in modeling such time series because values of these variables are only measured at discrete intervals.
A linear recurrence with constant coefficients is an equation of the following form, written in terms of parameters a1, ..., an and b:
is called the order of the recurrence and denotes the longest time lag between iterates.
whose roots play a crucial role in finding and understanding the sequences satisfying the recurrence.
To solve this equation it is convenient to convert it to homogeneous form, with no constant term.
in which like terms can be combined to give a homogeneous equation of one order higher than the original.
The roots of the characteristic polynomial play a crucial role in finding and understanding the sequences satisfying the recurrence.
Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that
This can be approached directly or using generating functions (formal power series) or matrices.
), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form.
However, for use in a theoretical context it may be that the only information required about the roots is whether any of them are greater than or equal to 1 in absolute value.
If all the characteristic roots are distinct, the solution of the homogeneous linear recurrence
where θ is the angle whose cosine is α/M and whose sine is β/M; the last equality here made use of de Moivre's formula.
Now the process of finding the coefficients cj and cj+1 guarantees that they are also complex conjugates, which can be written as γ ± δi.
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 cyclicity involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate characteristic roots.
In the second-order case, if the two roots are identical (λ1 = λ2), they can both be denoted as λ and a solution may be of the form
In the homogeneous case yi is a para-permanent of a lower triangular matrix [6] The recurrence
The method for solving linear differential equations is similar to the method above—the "intelligent guess" (ansatz) for linear differential equations with constant coefficients is
is a complex number that is determined by substituting the guess into the differential equation.
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.
The rule of thumb (for equations in which the polynomial multiplying the first term is non-zero at zero) is that:
The z-transforms are a class of integral transforms that lead to more convenient algebraic manipulations and more straightforward solutions.
There are cases in which obtaining a direct solution would be all but impossible, yet solving the problem via a thoughtfully chosen integral transform is straightforward.
A pair of terms with complex conjugate characteristic roots will converge to 0 with dampening fluctuations if the absolute value of the modulus M of the roots is less than 1; if the modulus equals 1 then constant amplitude fluctuations in the combined terms will persist; and if the modulus is greater than 1, the combined terms will show fluctuations of ever-increasing magnitude.
If the largest root has absolute value 1, neither convergence to 0 nor divergence to infinity will occur.
If all roots with magnitude 1 are real and positive, x will converge to the sum of their constant terms ci; unlike in the stable case, this converged value depends on the initial conditions; different starting points lead to different points in the long run.
A theorem of Issai Schur states that all roots have magnitude less than 1 (the stable case) if and only if a particular string of determinants are all positive.
[2]: 247 If a non-homogeneous linear difference equation has been converted to homogeneous form which has been analyzed as above, then the stability and cyclicality properties of the original non-homogeneous equation will be the same as those of the derived homogeneous form, with convergence in the stable case being to the steady-state value y* instead of to 0.