Monolayer

Template:Redirect-distinguish In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms.

The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation.

An example of a recurrence relation is the logistic map:

$x_{n+1}=r{x}_n(1-x_n),$

with a given constant r; given the initial term x₀ each subsequent term is determined by this relation.

Some simply defined recurrence relations can have very complex (chaotic) behaviours, and they are a part of the field of mathematics known as nonlinear analysis.

Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.

Fibonacci numbers

The Fibonacci numbers are the archetype of a linear, homogeneous recurrence relation with constant coefficients (see below). They are defined using the linear recurrence relation

$F_n=F_{n-1}+F_{n-2}$

with seed values:

$F_0=0$

$F_1=1$

Explicitly, recurrence yields the equations:

$F_2=F_1+F_0$

$F_3=F_2+F_1$

$F_4=F_3+F_2$

etc.

We obtain the sequence of Fibonacci numbers which begins:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

It can be solved by methods described below yielding the closed-form expression which involve powers of the two roots of the characteristic polynomial t² = t + 1; the generating function of the sequence is the rational function

$\frac{t}{1-t-t^2}.$

Structure

Linear homogeneous recurrence relations with constant coefficients

An order d linear homogeneous recurrence relation with constant coefficients is an equation of the form

$a_n=c_1{a}_{n-1}+c_2{a}_{n-2}+\cdots +c_d{a}_{n-d},$

where the d coefficients c_i (for all i) are constants.

More precisely, this is an infinite list of simultaneous linear equations, one for each n>d−1. A sequence which satisfies a relation of this form is called a linear recurrence sequence or LRS. There are d degrees of freedom for LRS, i.e., the initial values $a_0,\dots ,a_{d-1}$ can be taken to be any values but then the linear recurrence determines the sequence uniquely.

The same coefficients yield the characteristic polynomial (also "auxiliary polynomial")

$p(t)=t^d-c_1{t}^{d-1}-c_2{t}^{d-2}-\cdots -c_d$

whose d roots play a crucial role in finding and understanding the sequences satisfying the recurrence. If the roots r₁, r₂, ... are all distinct, then the solution to the recurrence takes the form

$a_n=k_1{r}_1^n+k_2{r}_2^n+\cdots +k_d{r}_d^n,$

where the coefficients k_i are determined in order to fit the initial conditions of the recurrence. When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as (x−r)³, with the same root r occurring three times, then the solution would take the form

$a_n=k_1{r}^n+k_{2}n{r}^n+k_3{n}^2{r}^n.$

Rational generating function

Linear recursive sequences are precisely the sequences whose generating function is a rational function: the denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.

The simplest cases are periodic sequences, $a_n=a_{n-d},n\ge d$, which have sequence $a_0,a_1,\dots ,a_{d-1},a_0,\dots$ and generating function a sum of geometric series:

$\frac{a_0+a_1{x}^1+\cdots +a_{d-1}{x}^{d-1}}{1-x^d}=(a_0+a_1{x}^1+\cdots +a_{d-1}{x}^{d-1})+(a_0+a_1{x}^1+\cdots +a_{d-1}{x}^{d-1})x^d+(a_0+a_1{x}^1+\cdots +a_{d-1}{x}^{d-1})x^{2d}+\cdots .$

More generally, given the recurrence relation:

$a_n=c_1{a}_{n-1}+c_2{a}_{n-2}+\cdots +c_d{a}_{n-d}$

with generating function

$a_0+a_1{x}^1+a_2{x}^2+\cdots ,$

the series is annihilated at a_d and above by the polynomial:

$1-c_1{x}^1-c_2{x}^2-\cdots -c_d{x}^d.$

That is, multiplying the generating function by the polynomial yields

$b_n=a_n-c_1{a}_{n-1}-c_2{a}_{n-2}-\cdots -c_d{a}_{n-d}$

as the coefficient on $x^n$, which vanishes (by the recurrence relation) for n ≥ d. Thus

$(a_0+a_1{x}^1+a_2{x}^2+\cdots )(1-c_1{x}^1-c_2{x}^2-\cdots -c_d{x}^d)=(b_0+b_1{x}^1+b_2{x}^2+\cdots +b_{d-1}{x}^{d-1})$

so dividing yields

$a_0+a_1{x}^1+a_2{x}^2+\cdots =\frac{b_0+b_1{x}^1+b_2{x}^2+\cdots +b_{d-1}{x}^{d-1}}{1-c_1{x}^1-c_2{x}^2-\cdots -c_d{x}^d},$

expressing the generating function as a rational function.

The denominator is $x^{d}p\left(x^{-1}\right),$ a transform of the auxiliary polynomial (equivalently, reversing the order of coefficients); one could also use any multiple of this, but this normalization is chosen both because of the simple relation to the auxiliary polynomial, and so that $b_0=a_0$.

Relationship to difference equations narrowly defined

Given an ordered sequence ${\left\{a_n\right\}}_{n=1}^{\mathrm{\infty }}$ of real numbers: the first difference $\mathrm{\Delta }(a_n)$ is defined as

$\mathrm{\Delta }(a_n)=a_{n+1}-a_n$.

The second difference ${\mathrm{\Delta }}^2(a_n)$ is defined as

${\mathrm{\Delta }}^2(a_n)=\mathrm{\Delta }(a_{n+1})-\mathrm{\Delta }(a_n)$,

which can be simplified to

${\mathrm{\Delta }}^2(a_n)=a_{n+2}-2a_{n+1}+a_n$.

More generally: the k^th difference of the sequence a_n is written as ${\mathrm{\Delta }}^k(a_n)$ is defined recursively as

${\mathrm{\Delta }}^k(a_n)={\mathrm{\Delta }}^{k-1}(a_{n+1})-{\mathrm{\Delta }}^{k-1}(a_n)=\sum _{t=0}^k(\genfrac{}{}{0ex}{}{k}{t})(-1{)}^t{a}_{n+k-t}$.

(The sequence and its differences are related by a binomial transform.) The more restrictive definition of difference equation is an equation composed of a_n and its k^th differences. (A widely used broader definition treats "difference equation" as synonymous with "recurrence relation". See for example rational difference equation and matrix difference equation.)

Linear recurrence relations are difference equations, and conversely; since this is a simple and common form of recurrence, some authors use the two terms interchangeably. For example, the difference equation

$3{\mathrm{\Delta }}^2(a_n)+2\mathrm{\Delta }(a_n)+7a_n=0$

is equivalent to the recurrence relation

$3a_{n+2}=4a_{n+1}-8a_n$

Thus one can solve many recurrence relations by rephrasing them as difference equations, and then solving the difference equation, analogously to how one solves ordinary differential equations. However, the Ackermann numbers are an example of a recurrence relation that do not map to a difference equation, much less points on the solution to a differential equation.

See time scale calculus for a unification of the theory of difference equations with that of differential equations.

Summation equations relate to difference equations as integral equations relate to differential equations.

From sequences to grids

Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). Multi-variable or n-dimensional recurrence relations are about n-dimensional grids. Functions defined on n-grids can also be studied with partial difference equations.

Solving

General methods

For order 1, the recurrence

$a_n=r{a}_{n-1}$

has the solution a_n = rⁿ with a₀ = 1 and the most general solution is a_n = krⁿ with a₀ = k. The characteristic polynomial equated to zero (the characteristic equation) is simply t − r = 0.

Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that a_n = rⁿ is a solution for the recurrence exactly when t = r is a root of the characteristic polynomial. This can be approached directly or using generating functions (formal power series) or matrices.

Consider, for example, a recurrence relation of the form

$a_n=A{a}_{n-1}+B{a}_{n-2}.$

When does it have a solution of the same general form as a_n = rⁿ? Substituting this guess (ansatz) in the recurrence relation, we find that

$r^n=A{r}^{n-1}+B{r}^{n-2}$

must be true for all n > 1.

Dividing through by rⁿ⁻², we get that all these equations reduce to the same thing:

$r^2=Ar+B,$

$r^2-Ar-B=0,$

which is the characteristic equation of the recurrence relation. Solve for r to obtain the two roots λ₁, λ₂: these roots are known as the characteristic roots or eigenvalues of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution

$a_n=C{\lambda }_1^n+D{\lambda }_2^n$

while if they are identical (when A² + 4B = 0), we have

$a_n=C{\lambda }^n+Dn{\lambda }^n$

This is the most general solution; the two constants C and D can be chosen based on two given initial conditions a₀ and a₁ to produce a specific solution.

In the case of complex eigenvalues (which also gives rise to complex values for the solution parameters C and D), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form. In this case we can write the eigenvalues as ${\lambda }_1,{\lambda }_2=\alpha \pm \beta i.$ Then it can be shown that

$a_n=C{\lambda }_1^n+D{\lambda }_2^n$

can be rewritten asPrimarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

$a_n=2M^n(E\cos (\theta n)+F\sin (\theta n))=2G{M}^n\cos (\theta n-\delta ),$

where

$\begin{array}{lcl}M=\sqrt{{\alpha }^2+{\beta }^2}& \cos (\theta )=\frac{\alpha }{M}& \sin (\theta )=\frac{\beta }{M}\\ C,D=E\mp Fi& & \\ G=\sqrt{E^2+F^2}& \cos (\delta )=\frac{E}{G}& \sin (\delta )=\frac{F}{G}\end{array}$

Here E and F (or equivalently, G and δ) are real constants which depend on the initial conditions. Using

${\lambda }_1+{\lambda }_2=2\alpha =A,$

${\lambda }_1\cdot {\lambda }_2={\alpha }^2+{\beta }^2=-B,$

one may simplify the solution given above as

$a_n=(-B{)}^{\frac{n}{2}}(E\cos (\theta n)+F\sin (\theta n)),$

where a₁ and a₂ are the initial conditions and

$\begin{array}{rl}E& =\frac{-A{a}_1+a_2}{B}\\ F& =-i\frac{A^2{a}_1-A{a}_2+2a_{1}B}{B\sqrt{A^2+4B}}\\ \theta & =a\cos \left(\frac{A}{2\sqrt{-B}}\right)\end{array}$

In this way there is no need to solve for λ₁ and λ₂.

In all cases—real distinct eigenvalues, real duplicated eigenvalues, and complex conjugate eigenvalues—the equation is stable (that is, the variable a converges to a fixed value (specifically, zero)); if and only if both eigenvalues are smaller than one in absolute value. In this second-order case, this condition on the eigenvalues can be shown to be equivalent to |A| < 1 − B < 2, which is equivalent to |B| < 1 and |A| < 1 − B.

The equation in the above example was homogeneous, in that there was no constant term. If one starts with the non-homogeneous recurrence

$b_n=A{b}_{n-1}+B{b}_{n-2}+K$

with constant term K, this can be converted into homogeneous form as follows: The steady state is found by setting b_n = b_n−1 = b_n−2 = b* to obtain

$b^{*}=\frac{K}{1-A-B}.$

Then the non-homogeneous recurrence can be rewritten in homogeneous form as

$[b_n-b^{*}]=A[b_{n-1}-b^{*}]+B[b_{n-2}-b^{*}],$

which can be solved as above.

The stability condition stated above in terms of eigenvalues for the second-order case remains valid for the general n^th-order case: the equation is stable if and only if all eigenvalues of the characteristic equation are less than one in absolute value.

Solving via linear algebra

A linearly recursive sequence y of order n

$y_{n+k}-c_{n-1}{y}_{n-1+k}-c_{n-2}{y}_{n-2+k}+\cdots -c_0{y}_k=0$

is identical to

$y_n=c_{n-1}{y}_{n-1}+c_{n-2}{y}_{n-2}+\cdots +c_0{y}_0.$

Expanded with n-1 identities of kind $y_{n-k}=y_{n-k},$ this n-th order equation is translated into a system of n first order linear equations,

${\overrightarrow{y}}_n=\left[\begin{array}{c}{y}_n\\ y_{n-1}\\ ⋮\\ y_1\end{array}\right]=\left[\begin{array}{cccc}-c_{n-1}& -c_{n-2}& \cdots & -c_0\\ 1& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\end{array}\right]\left[\begin{array}{c}{y}_{n-1}\\ y_{n-2}\\ ⋮\\ y_0\end{array}\right]=C{\overrightarrow{y}}_{n-1}=C^n{\overrightarrow{y}}_0.$

Observe that the vector ${\overrightarrow{y}}_n$ can be computed by n applications of the companion matrix, C, to the initial state vector, $y_0$. Thereby, n-th entry of the sought sequence y, is the top component of ${\overrightarrow{y}}_n,y_n={\overrightarrow{y}}_n[n]$.

Eigendecomposition, ${\overrightarrow{y}}_n={\overrightarrow{C}}^n{\overrightarrow{y}}_0=c_1{\lambda }_1^n{\overrightarrow{e}}_1+c_2{\lambda }_2^n{\overrightarrow{e}}_2+\cdots +c_n{\lambda }_n^n{\overrightarrow{e}}_n$ into eigenvalues, ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$, and eigenvectors, ${\overrightarrow{e}}_1,{\overrightarrow{e}}_2,\dots ,{\overrightarrow{e}}_n$, is used to compute ${\overrightarrow{y}}_n.$ Thanks to the crucial fact that system C time-shifts every eigenvector, e, by simply scaling its components λ times,

$C{\overrightarrow{e}}_i={\lambda }_i{\overrightarrow{e}}_i=C\left[\begin{array}{c}{e}_{i,n}\\ e_{i,n-1}\\ ⋮\\ e_{i,1}\end{array}\right]=\left[\begin{array}{c}{\lambda }_i{e}_{i,n}\\ {\lambda }_i{e}_{i,n-1}\\ ⋮\\ {\lambda }_i{e}_{i,1}\end{array}\right]$

that is, time-shifted version of eigenvector,e, has components λ times larger, the eighenvector components are powers of λ, ${\overrightarrow{e}}_i={\left[\begin{array}{ccccc}{\lambda }_i^{n-1}& \cdots & {\lambda }_i^2& {\lambda }_i& 1\end{array}\right]}^T,$ and, thus, recurrent linear homogenous equation solution is a combination of exponential functions, ${\overrightarrow{y}}_n=\sum _1^n{c}_i{\lambda }_i^n{\overrightarrow{e}}_i$. The components $c_i$ can be determined out of initial conditions:

${\overrightarrow{y}}_0=\left[\begin{array}{c}{y}_0\\ y_{-1}\\ ⋮\\ y_{-n+1}\end{array}\right]=\sum _{i=1}^n{c}_i{\lambda }_i^0{\overrightarrow{e}}_i=\left[\begin{array}{cccc}{\overrightarrow{e}}_1& {\overrightarrow{e}}_2& \cdots & {\overrightarrow{e}}_n\end{array}\right]\left[\begin{array}{c}{c}_1\\ c_2\\ \cdots \\ c_n\end{array}\right]=E\left[\begin{array}{c}{c}_1\\ c_2\\ \cdots \\ c_n\end{array}\right]$

Solving for coefficients,

$\left[\begin{array}{c}{c}_1\\ c_2\\ \cdots \\ c_n\end{array}\right]=E^{-1}{\overrightarrow{y}}_0={\left[\begin{array}{cccc}{\lambda }_1^{n-1}& {\lambda }_2^{n-1}& \cdots & {\lambda }_n^{n-1}\\ ⋮& ⋮& \ddots & ⋮\\ {\lambda }_1& {\lambda }_2& \cdots & {\lambda }_n\\ 1& 1& \cdots & 1\end{array}\right]}^{-1}\left[\begin{array}{c}{y}_0\\ y_{-1}\\ ⋮\\ y_{-n+1}\end{array}\right].$

This also works with arbitrary boundary conditions $\underset{\text{n}}{\underbrace{y_a,y_b,\dots }}$, not necessary the initial ones,

$\left[\begin{array}{c}{y}_a\\ y_b\\ ⋮\end{array}\right]=\left[\begin{array}{c}{\overrightarrow{y}}_a[n]\\ {\overrightarrow{y}}_b[n]\\ ⋮\end{array}\right]=\left[\begin{array}{c}\sum _{i=1}^n{c}_i{\lambda }_i^a{\overrightarrow{e}}_i[n]\\ \sum _{i=1}^n{c}_i{\lambda }_i^b{\overrightarrow{e}}_i[n]\\ ⋮\end{array}\right]=\left[\begin{array}{c}\sum _{i=1}^n{c}_i{\lambda }_i^a{\lambda }_i^{n-1}\\ \sum _{i=1}^n{c}_i{\lambda }_i^b{\lambda }_i^{n-1}\\ ⋮\end{array}\right]=$

$=\left[\begin{array}{c}\sum c_i{\lambda }_i^{a+n-1}\\ \sum c_i{\lambda }_i^{b+n-1}\\ ⋮\end{array}\right]=\left[\begin{array}{cccc}{\lambda }_1^{a+n-1}& {\lambda }_2^{a+n-1}& \cdots & {\lambda }_n^{a+n-1}\\ {\lambda }_1^{b+n-1}& {\lambda }_2^{b+n-1}& \cdots & {\lambda }_n^{b+n-1}\\ ⋮& ⋮& \ddots & ⋮\end{array}\right]\left[\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_n\end{array}\right].$

This description is really no different from general method above, however it is more succinct. It also works nicely for situations like

$\{\begin{array}{l}{a}_n=a_{n-1}-b_{n-1}\\ b_n=2a_{n-1}+b_{n-1}.\end{array}$

where there are several linked recurrences Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park..

Solving with z-transforms

Certain difference equations - in particular, linear constant coefficient difference equations - can be solved using z-transforms. 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.

Theorem

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p(t) be the characteristic polynomial (also "auxiliary polynomial")

$t^d-c_1{t}^{d-1}-c_2{t}^{d-2}-\cdots -c_d=0$

such that each c_i corresponds to each c_i in the original recurrence relation (see the general form above). Suppose λ is a root of p(t) having multiplicity r. This is to say that (t−λ)^r divides p(t). The following two properties hold:

1. Each of the r sequences ${\lambda }^n,n{\lambda }^n,n^2{\lambda }^n,\dots ,n^{r-1}{\lambda }^n$ satisfies the recurrence relation.
2. Any sequence satisfying the recurrence relation can be written uniquely as a linear combination of solutions constructed in part 1 as λ varies over all distinct roots of p(t).

As a result of this theorem a linear homogeneous recurrence relation with constant coefficients can be solved in the following manner:

1. Find the characteristic polynomial p(t).
2. Find the roots of p(t) counting multiplicity.
3. Write a_n as a linear combination of all the roots (counting multiplicity as shown in the theorem above) with unknown coefficients b_i.

$a_n=(b_1{\lambda }_1^n+b_{2}n{\lambda }_1^n+b_3{n}^2{\lambda }_1^n+\cdots +b_r{n}^{r-1}{\lambda }_1^n)+\cdots +(b_{d-q+1}{\lambda }_{*}^n+\cdots +b_d{n}^{q-1}{\lambda }_{*}^n)$

This is the general solution to the original recurrence relation. (q is the multiplicity of λ_*)

4. Equate each $a_0,a_1,\dots ,a_d$ from part 3 (plugging in n = 0, ..., d into the general solution of the recurrence relation) with the known values $a_0,a_1,\dots ,a_d$ from the original recurrence relation. However, the values a_n from the original recurrence relation used do not usually have to be contiguous: excluding exceptional cases, just d of them are needed (i.e., for an original linear homogeneous recurrence relation of order 3 one could use the values a₀, a₁, a₄). This process will produce a linear system of d equations with d unknowns. Solving these equations for the unknown coefficients $b_1,b_2,\dots ,b_d$ of the general solution and plugging these values back into the general solution will produce the particular solution to the original recurrence relation that fits the original recurrence relation's initial conditions (as well as all subsequent values $a_0,a_1,a_2,\dots$ of the original recurrence relation).

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 e^λx where λ is a complex number that is determined by substituting the guess into the differential equation.

This is not a coincidence. Considering the Taylor series of the solution to a linear differential equation:

$\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

it can be seen that the coefficients of the series are given by the n^th derivative of f(x) evaluated at the point a. The differential equation provides a linear difference equation relating these coefficients.

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:

$y^{[k]}\to f[n+k]$

and more generally

$x^m*y^{[k]}\to n(n-1)(n-m+1)f[n+k-m]$

Example: The recurrence relationship for the Taylor series coefficients of the equation:

$(x^2+3x-4)y^{[3]}-(3x+1)y^{[2]}+2y=0$

is given by

$n(n-1)f[n+1]+3nf[n+2]-4f[n+3]-3nf[n+1]-f[n+2]+2f[n]=0$

or

$-4f[n+3]+2nf[n+2]+n(n-4)f[n+1]+2f[n]=0.$

This example shows how problems generally solved using the power series solution method taught in normal differential equation classes can be solved in a much easier way.

Example: The differential equation

$a{y}^{″}+b{y}^{\prime }+cy=0$

has solution

$y=e^{ax}.$

The conversion of the differential equation to a difference equation of the Taylor coefficients is

$af[n+2]+bf[n+1]+cf[n]=0.$

It is easy to see that the nth derivative of e^ax evaluated at 0 is aⁿ

Solving non-homogeneous recurrence relations

If the recurrence is inhomogeneous, a particular solution can be found by the method of undetermined coefficients and the solution is the sum of the solution of the homogeneous and the particular solutions. Another method to solve an inhomogeneous recurrence is the method of symbolic differentiation. For example, consider the following recurrence:

$a_{n+1}=a_n+1$

This is an inhomogeneous recurrence. If we substitute n ↦ n+1, we obtain the recurrence

$a_{n+2}=a_{n+1}+1$

Subtracting the original recurrence from this equation yields

$a_{n+2}-a_{n+1}=a_{n+1}-a_n$

or equivalently

$a_{n+2}=2a_{n+1}-a_n$

This is a homogeneous recurrence which can be solved by the methods explained above. In general, if a linear recurrence has the form

$a_{n+k}={\lambda }_{k-1}{a}_{n+k-1}+{\lambda }_{k-2}{a}_{n+k-2}+\cdots +{\lambda }_1{a}_{n+1}+{\lambda }_0{a}_n+p(n)$

where ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_{k-1}$ are constant coefficients and p(n) is the inhomogeneity, then if p(n) is a polynomial with degree r, then this inhomogeneous recurrence can be reduced to a homogeneous recurrence by applying the method of symbolic differencing r times.

If

$P(x)=\sum _{n=0}^{\mathrm{\infty }}{p}_n{x}^n$

is the generating function of the inhomogeneity, the generating function

$A(x)=\sum _{n=0}^{\mathrm{\infty }}a(n)x^n$

of the inhomogeneous recurrence

$a_n=\sum _{i=1}^s{c}_i{a}_{n-i}+p_n,n\ge n_r,$

with constant coefficients Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is derived from

$(1-\sum _{i=1}^s{c}_i{x}^i)A(x)=P(x)+\sum _{n=0}^{n_r-1}[a_n-p_n]x^n-\sum _{i=1}^s{c}_i{x}^i\sum _{n=0}^{n_r-i-1}{a}_n{x}^n.$

If P(x) is a rational generating function, A(x) is also one. The case discussed above, where p_n = K is a constant, emerges as one example of this formula, with P(x) = K/(1−x). The solution of homogeneous recurrences is incorporated as p = P = 0.

Moreover, for the general first-order linear inhomogeneous recurrence relation with variable coefficient(s)

$a_{n+1}=f_n{a}_n+g_n,f_n\ne 0,$

there is also a nice method to solve it:

$a_{n+1}-f_n{a}_n=g_n$

$\frac{a_{n+1}}{\prod _{k=0}^n{f}_k}-\frac{f_n{a}_n}{\prod _{k=0}^n{f}_k}=\frac{g_n}{\prod _{k=0}^n{f}_k}$

$\frac{a_{n+1}}{\prod _{k=0}^n{f}_k}-\frac{a_n}{\prod _{k=0}^{n-1}{f}_k}=\frac{g_n}{\prod _{k=0}^n{f}_k}$

Let

$A_n=\frac{a_n}{\prod _{k=0}^{n-1}{f}_k},$

Then

$A_{n+1}-A_n=\frac{g_n}{\prod _{k=0}^n{f}_k}$

$\sum _{m=0}^{n-1}(A_{m+1}-A_m)=A_n-A_0=\sum _{m=0}^{n-1}\frac{g_m}{\prod _{k=0}^m{f}_k}$

$\frac{a_n}{\prod _{k=0}^{n-1}{f}_k}=A_0+\sum _{m=0}^{n-1}\frac{g_m}{\prod _{k=0}^m{f}_k}$

$a_n=\left(\prod _{k=0}^{n-1}{f}_k\right)(A_0+\sum _{m=0}^{n-1}\frac{g_m}{\prod _{k=0}^m{f}_k})$

General linear homogeneous recurrence relations

Many linear homogeneous recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. For example, the solution to

$J_{n+1}=\frac{2n}{z}{J}_n-J_{n-1}$

is given by

$J_n=J_n(z),$

the Bessel function, while

$(b-n)M_{n-1}+(2n-b-z)M_n-n{M}_{n+1}=0$

is solved by

$M_n=M(n,b;z)$

the confluent hypergeometric series.

Solving a first order rational difference equation

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

A first order rational difference equation has the form $w_{t+1}=\frac{a{w}_t+b}{c{w}_t+d}$. Such an equation can be solved by writing $w_t$ as a nonlinear transformation of another variable $x_t$ which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in $x_t$.

Stability

Stability of linear higher-order recurrences

The linear recurrence of order d,

$a_n=c_1{a}_{n-1}+c_2{a}_{n-2}+\dots +c_d{a}_{n-d},$

has the characteristic equation

${\lambda }^d-c_1{\lambda }^{d-1}-c_2{\lambda }^{d-2}-\dots -c_d{\lambda }^0=0.$

The recurrence is stable, meaning that the iterates converge asymptotically to a fixed value, if and only if the eigenvalues (i.e., the roots of the characteristic equation), whether real or complex, are all less than unity in absolute value.

Stability of linear first-order matrix recurrences

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

In the first-order matrix difference equation

$[x_t-x^{*}]=A[x_{t-1}-x^{*}]$

with state vector x and transition matrix A, x converges asymptotically to the steady state vector x* if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value which is less than 1.

Stability of nonlinear first-order recurrences

Consider the nonlinear first-order recurrence

$x_n=f(x_{n-1}).$

This recurrence is locally stable, meaning that it converges to a fixed point x* from points sufficiently close to x*, if the slope of f in the neighborhood of x* is smaller than unity in absolute value: that is,

$|f^{\prime }(x^{*})|<1.$

A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable; for continuous f two adjacent fixed points cannot both be locally stable.

A nonlinear recurrence relation could also have a cycle of period k for k > 1. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function

$g(x):=f\circ f\circ \cdot \cdot \cdot \circ f(x)$

with f appearing k times is locally stable according to the same criterion:

$|g^{\prime }(x^{*})|<1,$

where x* is any point on the cycle.

In a chaotic recurrence relation, the variable x stays in a bounded region but never converges to a fixed point or an attracting cycle; any fixed points or cycles of the equation are unstable. See also logistic map, dyadic transformation, and tent map.

Relationship to differential equations

When solving an ordinary differential equation numerically, one typically encounters a recurrence relation. For example, when solving the initial value problem

$y^{\prime }(t)=f(t,y(t)),y(t_0)=y_0,$

with Euler's method and a step size h, one calculates the values

$y_0=y(t_0),y_1=y(t_0+h),y_2=y(t_0+2h),\dots$

by the recurrence

$y_{n+1}=y_n+hf(t_n,y_n).$

Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the discretization article.

Applications

Biology

Some of the best-known difference equations have their origins in the attempt to model population dynamics. For example, the Fibonacci numbers were once used as a model for the growth of a rabbit population.

The logistic map is used either directly to model population growth, or as a starting point for more detailed models. In this context, coupled difference equations are often used to model the interaction of two or more populations. For example, the Nicholson-Bailey model for a host-parasite interaction is given by

$N_{t+1}=\lambda N_t{e}^{-a{P}_t}$

$P_{t+1}=N_t(1-e^{-a{P}_t}),$

with N_t representing the hosts, and P_t the parasites, at time t.

Integrodifference equations are a form of recurrence relation important to spatial ecology. These and other difference equations are particularly suited to modeling univoltine populations.

Digital signal processing

In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filters.

For example, the equation for a "feedforward" IIR comb filter of delay T is:

$y_t=(1-\alpha )x_t+\alpha y_{t-T}$

Where $x_t$ is the input at time t, $y_t$ is the output at time t, and α controls how much of the delayed signal is fed back into the output. From this we can see that

$y_t=(1-\alpha )x_t+\alpha ((1-\alpha )x_{t-T}+\alpha y_{t-2T})$

$y_t=(1-\alpha )x_t+(\alpha -{\alpha }^2)x_{t-T}+{\alpha }^2{y}_{t-2T})$

etc.

Economics

Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics. In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which some agents' actions depend on lagged variables. The model would then be solved for current values of key variables (interest rate, real GDP, etc.) in terms of exogenous variables and lagged endogenous variables. See also time series analysis.

Computer Science

Recurrence relations are also of fundamental importance in Analysis of Algorithms. If an algorithm is designed so that it will break a problem into smaller subproblems, its running time is described by a recurrence relation.

A simple example is the time an algorithm takes to search an element in an ordered vector with $n$ elements, in the worst case.

A naive algorithm will search from left to right, one element at a time. The worst possible scenario is when the required element is the last, so the number of comparisons is $n$.

A better algorithm is called binary search. It will first check if the element is at the middle of the vector. If not, then it will check if the middle element is greater or lesser than the sought element. At this point, half of the vector can be discarded, and the algorithm can be run again on the other half. The number of comparisons will be given by

$c_1=1$

$c_n=1+c_{n/2}$

which will be close to ${\log }_2(n)$.

See also

• Iterated function
• Matrix difference equation
• Orthogonal polynomials
• Recursion
• Recursion (computer science)
• Lagged Fibonacci generator
• Master theorem
• Circle points segments proof
• Continued fraction
• Time scale calculus
• Integrodifference equation
• Combinatorial principles
• Infinite impulse response

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 1990. ISBN 0-262-03293-7. Chapter 4: Recurrences, pp. 62–90.
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 chapter 7.
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 9.1: Difference Equations.
• Template:Cite web at EqWorld - The World of Mathematical Equations.
• Template:Cite web at EqWorld - The World of Mathematical Equations.

External links

• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/
• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
• Template:Cite web
• Introductory Discrete Mathematics
• Template:Cite web OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)