https://en.wikipedia.org/wiki/Linear_difference_equation
In mathematics and in particular dynamical systems, a linear difference equation or linear recurrence relation sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree 0 or 1. Usually the context is the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t − 1, one period later as t + 1, etc.
An nth order linear difference equation is one that can be written in terms of parameters ai and b as
In the most general case the coefficients ai and b could themselves be functions of time; however, this article treats the most common case, that of constant coefficients. If the coefficients ai are polynomials in t the equation is called a linear recurrence equation with polynomial 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 initial conditions) of n of the iterates, and normally these are the n iterates that are oldest. The equation or its variable is said to be stable if from any set of initial conditions the variable's limit as time goes to infinity exists; this limit is called the steady state.
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. In econometric applications, linear difference equations are modeled with stochastic terms in the form of autoregressive (AR) models and in models such as vector autoregression (VAR) and autoregressive moving average (ARMA) models that combine AR with other features.
In mathematics and in particular dynamical systems, a linear difference equation or linear recurrence relation sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree 0 or 1. Usually the context is the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t − 1, one period later as t + 1, etc.
An nth order linear difference equation is one that can be written in terms of parameters ai and b as
In the most general case the coefficients ai and b could themselves be functions of time; however, this article treats the most common case, that of constant coefficients. If the coefficients ai are polynomials in t the equation is called a linear recurrence equation with polynomial 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 initial conditions) of n of the iterates, and normally these are the n iterates that are oldest. The equation or its variable is said to be stable if from any set of initial conditions the variable's limit as time goes to infinity exists; this limit is called the steady state.
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. In econometric applications, linear difference equations are modeled with stochastic terms in the form of autoregressive (AR) models and in models such as vector autoregression (VAR) and autoregressive moving average (ARMA) models that combine AR with other features.
Solution of homogeneous case
Characteristic equation and roots
Solving the homogeneous equation
involves first solving its characteristic equation
for its characteristic roots λi (i = 1, ..., n). These roots can be solved for algebraically if n ≤ 4, but not necessarily otherwise. If the solution is to be used numerically, all the roots of this characteristic equation can be found by numerical methods.
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.
It may be that all the roots are real or instead there may be some that are complex numbers. In the latter case, all the complex roots come in complex conjugate pairs.
Solution with distinct characteristic roots
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 ci
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 solved simultaneously for the n parameter values. If all characteristic roots are real, then all the coefficient values ci will also be real; but with non-real complex roots, in general some of these coefficients will also be non-real.
Converting complex solution to trigonometric form
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 cjλt
j and cj+1λt
j+1, the roots λj can be written as
j and cj+1λt
j+1, the roots λj can be written as
Then the two complex terms in the solution equation can be written as
where θ is the angle whose cosine is αM and whose sine is βM; 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. 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 ψ is the angle whose cosine is γ√γ2 + δ2 and whose sine is δ√γ2 + δ2.
Cyclicity
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. This can be seen in the trigonometric form of
their contribution to the solution equation, involving cos θt and sin θt.
Solution with duplicate 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
Conversion to homogeneous form
If b ≠ 0, the equation
is said to be nonhomogeneous. 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.
Stability
In the solution equation
a term with real characteristic roots converges to 0 as t
grows indefinitely large if the absolute value of the characteristic
root is less than 1. If the absolute value equals 1, the term will stay
constant as t
grows if the root is +1 but will fluctuate between two values if the
root is −1. If the absolute value of the root is greater than 1 the term
will become larger and larger over time. 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.
Thus the evolving variable x will converge to 0 if all of the characteristic roots have magnitude less than 1.
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. If any root is −1, its term will contribute permanent
fluctuations between two values. If any of the unit-magnitude roots are
complex then constant-amplitude fluctuations of x will persist.
Finally, if any characteristic root has magnitude greater than 1, then x will diverge to infinity as time goes to infinity, or will fluctuate between increasingly large positive and negative values.
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.
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.
Solution by conversion to matrix form
An alternative solution method involves converting the nth order difference equation to a first-order matrix difference equation. This is accomplished by writing w1,t = yt, w2,t = yt−1 = w1,t−1, w3,t = yt−2 = w2,t−1, and so on. Then the original single nth-order equation
can be replaced by this set of n first-order equations:
Defining the vector wi as
this can be put in matrix form as
Here A is an n × n matrix in which the first row contains a1, ..., an and all other rows have a single 1 with all other elements being 0, and b is a column vector with first element b and with the rest of its elements being 0.
This matrix equation can be solved using the methods in the article Matrix difference equation.