Linear recurrence with constant coefficients information
Further information: Constant-recursive sequence
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients[1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) 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. A linear recurrence denotes 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.
The solution of such an equation is a function of t, 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.
^Chiang, Alpha (1984). Fundamental Methods of Mathematical Economics (Third ed.). New York: McGraw-Hill. ISBN 0-07-010813-7.
^Baumol, William (1970). Economic Dynamics (Third ed.). New York: Macmillan. ISBN 0-02-306660-1.
and 26 Related for: Linear recurrence with constant coefficients information
combinatorics, linear algebra, and dynamical systems), a linearrecurrencewithconstantcoefficients: ch. 17 : ch. 10 (also known as a linearrecurrence relation...
theorist Édouard Lucas. Like every sequence defined by a linearrecurrencewithconstantcoefficients, the Fibonacci numbers have a closed-form expression...
{\text{otherwise}},\end{cases}}} with initial term a 0 = 0. {\displaystyle a_{0}=0.} A linearrecurrencewithconstantcoefficients is a recurrence relation of the form...
sequence satisfying a linearrecurrencewithconstantcoefficients. This theorem states that, if such a sequence has zeros, then with finitely many exceptions...
the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by...
of polynomials satisfies a linearrecurrencewithconstantcoefficients; these coefficients are identical to the coefficients of the fraction denominator...
the discussion of linearity.) If the ai are constants (independent of x and y) then the PDE is called linearwithconstantcoefficients. If f is zero everywhere...
zero), the coefficients of all series involved in second linearly independent solutions can be calculated straightforwardly from tandem recurrence relations...
undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations...
Method of undetermined coefficientsRecurrence relation Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications...
_{k},\;k=0,1,\ldots } is a sequence of functions that satisfy the linearrecurrence relation ϕ k + 1 ( x ) = α k ( x ) ϕ k ( x ) + β k ( x ) ϕ k − 1 (...
applied to this differential equation with step size h yields a linearrecurrence relation with characteristic polynomial π ( z ; h λ ) = ( 1 − h λ β s ) z...
[0,2^{w}-1]} . The Mersenne Twister algorithm is based on a matrix linearrecurrence over a finite binary field F 2 {\displaystyle {\textbf {F}}_{2}} ...
satisfies a linearrecurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since...
for some purposes such as linearrecurrence relations (see below). C ( p ) {\displaystyle C(p)} is defined from the coefficients of p ( x ) {\displaystyle...
other terms, which are assumed to be known, are usually called constants, coefficients or parameters. An example of an equation involving x and y as unknowns...
coefficients, there is a negative binomial distribution in which the multiset coefficients occur. Multiset coefficients should not be confused with the...
constantcoefficient ordinary differential equation: d u d x = c u + x 2 . {\displaystyle {\frac {du}{dx}}=cu+x^{2}.} Homogeneous second-order linear...
sampling Linear classifier Linear discriminant analysis Linear least squares Linear model Linear prediction Linear probability model Linear regression...
satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial...
that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e. The quality...
the coefficients of their Taylor series at any point satisfy a linearrecurrence relation with polynomial coefficients, and that this recurrence relation...
software packages KPP and Autochem can be used. Consider the linearconstantcoefficient inhomogeneous system where y , f ∈ R n {\displaystyle \mathbf...
Q ) {\displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation x n = P ⋅ x n − 1 − Q ⋅ x n − 2 {\displaystyle...
Global Information