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Of course, Maple already has this command built in, but we,re going to use it as an example of defining our own procedures. The procedure should be entered in its entirety at one command prompt. The name of the procedure is fact. It receives a value of n and if that value is 0 , it returns the value of 1.

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Otherwise, it returns n times the value of the procedure evaluated at n - 1. Lets see what happens when we use this procedure. As you can see, this procedure apparently computes the factorial of what ever number we give it. Here is another version of the procedure.

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This version makes the same computation, but also prints out the intermediate values and the current value of n. This provides a window to what is going on inside the procedure. Here is an example of a procedure to create the Fibonacci sequence which shows its inner working. Related to this concept, there are also expressions which are recursively defined such as This can be thought of as a recursively defined sequence such as , Unfortunately, the rsolve command can not resolve this kind of recurrence. However, we can deal with this in another way. We can simply define a function based on this recurrence, then iterate the function - that is repeatedly compose the function with itself and evaluate at 1.

We can iterate the function 10 times by repeatedly composing it with itself in this way. In fact, we can see what is happening by doing this in a loop. It appears that the values are converging to a number around 1. But how can we find the exact value o the expression? Suppose for a moment that the values of f n where identical. If we simply solve that equation for x, we will have our answer. An example of a recursive formula is Newton's Method for finding roots of a function. This method requires that you be familiar with a little bit of Calculus.

Newton's Method is to start with a guess for what a root may be. The initial guess is x0 , then repeatedly apply this formula. In many cases, the formula will converge to the value of a root fairly quickly. Lets start with a function f x , and then define the function N x to give the next guess according to Newton's method. The values seem to converge quite quickly. The actual solution is this. The subject of fixed points of recursions is an interesting one for many reasons and one we can investigate algebraically, numerically, and geometrically.

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We will need the following special plot function. It is now immediate to write 2 in a slightly different way:. This is nothing else but a Euclidean division of power series with remainder: The formal power series K x is divided by the polynomial Q x yielding the quotient F s x and the remainder U x.

Since we are dealing with multivariate power series we have to fix a monomial order. We interpret the division by a power series as a perturbation of the division by its initial monomial. Let us have a short look on a special case:. In our setting where the division 3 arises from a recurrence, we are in fact not interested in performing the division explicitly, because we can obtain its result i.

Instead we are interested in deducing properties of the generating function. The generating function F s x is rational if and only if the initial condition function K x is a rational function. We only formulate the theorem in the case of the division by one series:.

In order to deal with this case one can employ the Lafon—Hironaka division theorem:. In particular, the theorem implies that in the division 3 the quotient F s x and the remainder U x are algebraic, provided that K x is algebraic and the initial monomial of Q x involves only one variable. Hence, if the apex p of 1 has exactly one nonzero component and if the initial conditions constitute an algebraic power series K x , then the generating function F s x is algebraic.

The step set immediately gives rise to the recurrence. For these reasons the recurrence is rewritten as follows:.

An easy calculation yields. We are now turning to P -finite recurrences, i. If it happens that c 0 n does have an integer root there, the whole recursion would break down. This situation can often be avoided by an adequate choice of the starting point s. In the following we will always assume that the recurrence fulfills the conditions of the corollary.

It is a well-known fact that a P -finite recurrence translates to a differential equation for the generating function. We want to recall briefly how this can be performed. We first rewrite the polynomial coefficients c t n using shifted falling factorials:.

Then the recurrence 4 rewrites as follows:. The power series. Analogously to Eq. A closer inspection of the differential operator D will reveal an important property: perfectness. Before doing so, we want to review briefly the theory of perfect differential operators and their division cf. A differential operator is called a monomial operator if all its summands have the same shift r ; this is equivalent to saying that the operator maps monomials to monomials. This operator is not perfect since, e. Note that the concept of perfect operators is more subtle than these two examples suggest.

Multivariate linear recurrences and power series division

Then we have the direct sum decompositions. We conclude that the division 5 has always a unique solution. On the use of matrix functions for fractional partial differential equations. Mathematics and Computers in Simulation , 81 5 —, Noye, M. New explicit finite difference schemes for two-dimensional diffusion subject to specification of mass.

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Recurrence Relations & Generating Functions

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Efficient recurrence relations for univariate and multivariate Taylor series coefficients

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