Recurrence relation

In mathematics is a difference equation ( DzGl ) (also known as recursion ) is a sequence defined recursively. That is, each follower is a function of the foregoing sequence elements:

For natural numbers. A special form of the linear difference equations.

Origin and application

In the engineering science the difference equation is an arithmetic rule for calculating an output sequence, respectively, of an output signal. For the purpose of time series analysis can be a difference equation also define more general than equation can be calculated with the values ​​of a time series that are related recursively.

After the discretization of a continuous signal can be calculated from the sequence no longer derivation, one has to use the difference quotient here. A difference equation is the discrete-time counterpart of the differential equation and finds its application mainly in digital signal processing (eg, in connection with the design of filters ) but also in complicated numerical simulations such as weather or climate prediction.

Application examples from the time series analysis are the eradication rate of annuity loan ( deterministic context ) or the stock of unemployed ( stochastic context ).

Related difference equation and Z-transform

The Z-transform assumes the same position for time-discrete signals ( sequences ) such as the Laplace transform of the continuous time domain.

From the z- image area to the discrete time

There are many possibilities: is through tables, Taylor series expansion, nichtabbrechende Division inter alia Another aid

The transfer function

Expressions can easily be translated with the help of a transfer function of the variables and a shock ( in the discrete ) into a difference equation.

For example, considering the function

Then obtained as the relationship between and 1

Multiplying this equation from to

Is obtained using the shifting theorem

By comparing coefficients in the example considered

A solution of this difference equation can be determined by the geometric series, to the denominator of is reduced to the form by expanding:

Resulting in


This is an application of the method of generating functions.

Note: The discrete delta shock or the Kronecker Delta always has the value 0, except in the case where it takes the value 1.

Difference equations in economics

In economic theory difference equations are mainly used to analyze the development of economic variables over time. Especially in the growth theory and business cycle theory temporal sequences are very much displayed in the form of difference equations.

It is based on the assumption that, for example, the gross domestic product developed on a specific path towards a long-run equilibrium in which all capacities are utilized. Depending on the solution of the difference equation results in the development path as asymptotic path or a wielding the course ( approximately cosine curves). However, it is inevitable that the mathematical modeling (eg gross domestic product), some simplifying assumptions have to be made ( eg via the formation of camps, consumption as a share of GDP or investment increase by profit expectations ).

  • Application of the 1st order difference equation

Another classic example is the Spinnwebtheorem (also cobweb theorem ). The development of prices and quantities followed by recursive functions or, in mathematical terms, the general differential equations of first order.

  • Application of differential equation of 2nd order

The multiplier - accelerator model to explain why economic growth is not monotonic runs, but typically follows a business cycle. The model can be developed from the growth model of Harrod and Domar out one particular variant comes from Paul A. Samuelson (1939) and John Richard Hicks (1950).