For questions about defining recursive functions, recursive algorithms and solving recursive equations.

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4
votes
1answer
243 views

Programming a recursive FFT with Mathematica

I'm trying to program a FFT using a recursive function in Mathematica, though my program is not getting me anywhere at the moment. Could you say what's wrong with it and what I can do to make it work? ...
0
votes
2answers
131 views

Solving an Equation (Recursively)

I have the following Values: $a[0]=1;a[1]=0;a[2]=0;a[3]=0;a[4]=\frac{g_2}{20};a[5]=0$. I need to compute the value of $b[n]$ in the following equation $$a[n]=\sum _{m=0}^{n} (m-2) \;\;a[m] ...
6
votes
2answers
395 views
4
votes
3answers
507 views

How to solve an integral equation by iteration method? [duplicate]

How I can obtain $n^{th}$ approximation of the following equation $f(t)=t+\int_0^tds f(s)$ by iteration method?
5
votes
1answer
256 views

Setting $RecursionLimit crashes Mathematica 10

I have encountered what I shall presume is a bug: setting $RecursionLimit repeatably crashes the Mathematica 10 kernel: ...
3
votes
1answer
162 views

Find exact limit of recurrence relation $x_{n+1}=\cos(x_n)+1, x_0=1$

I did this: b[x_] := Cos[b[x - 1]] + 1; b[0] = 1; num = SequenceLimit[N[Table[b[n], {n, 1, 100}], 50]] I also did the listplot which shows a convergent value. ...
13
votes
3answers
418 views

ToExpression fails with long nested string expressions

Consider the following conversion from String to Mathematica Expression: ...
1
vote
2answers
91 views

Print argument of function that exceeds $RecursionLimit

Is there a way to print the argument of a function which results in a $RecursionLimit::reclim error? As an example, cosider the code ...
1
vote
0answers
79 views

Calculating the recursion at the Fourier domain

I would like to calculate the following recursion with mathematica $$f_n(x)=\int_{B}^{A}f_{n-1}(x-\omega)f(\omega)\mbox{d}\omega\quad\quad f_1(w):=f(w)$$ This is simply the convolution of $f$ with ...
29
votes
3answers
2k views

Smooth Peter de Jong attractor

Today I was playing with Peter de Jong attractor. At the bottom of the page I've linked there are beautiful examples like: My attempts are not so great: It is around 10^5 points. For more than ...
1
vote
1answer
47 views

Replacing variables in an RSolve expression

I have a recursive expression, which is something like: $$ a(n,x)=f(x,y,m)\,a(n-1,x)+g(x,y)\,a(n-1,y) $$ where $f(x,y,m)$ and $g(x,y)$ are known. So the second $a$ depends on $y$ rather than $x$. I ...
4
votes
1answer
360 views

How do I built a zoomable Koch curve?

I'm new to Mathematica and my goal is to write a simple program in order to demonstrate self-similarity of the Koch curve by zooming in. Here is a good example of what I mean (it's a Java applet). I ...
0
votes
1answer
682 views

Can't plot recursive function

I am new to Mathematica and now I could use some help. Trying to plot any of these functions generates a $RecursionLimit exception and I am wondering why. ...
5
votes
1answer
410 views

Variable number of nested variable-range sums

I would like to express the following nested sum in Mathematica: $$ S(m,j,N) = \sum_{k_1=m+j-1}^{N-1} f(N,k_1) \sum_{k_2=m+j-2}^{k_1-1} f(k_1,k_2) \cdots \sum_{k_m=j}^{k_{m-1}-1} f(k_{m-1},k_m) $$ ...
0
votes
4answers
160 views

RSolve fails on a two-equation non-linear system

I have a system of just two recurrence equations, I need to solve them and so I use RSolve: ...
1
vote
1answer
107 views

Limit as n -> ∞ for RSolve with Multivariate Recurrence Relation

I'm trying to solve the following when n -> ∞: ...
1
vote
0answers
23 views

Solving a pair of recurrence relations [duplicate]

Why does this RSolve[{y[t] == a*y[t - 1]}, {y[t]}, t] work, while this ...
1
vote
1answer
234 views

How to plot particular piecewise defined functions in mathematica

How to plot such functions in Mathematica? Let $a_0, p,g,c$ be any positive integers, defining: $$a_{n+1} = \begin{cases}\frac{a_n}{p} &, a_n \text{ divisible by p}\\ ga_n +c &, a_n \text{ ...
4
votes
1answer
388 views

Creating a dynamic plot directly from the recurrence relation

I have the following recurrence relation that has no general solution: $$x(t+1) = \frac{x^2 + x(1-x)(1-sh)}{x^2 + 2x(1-x)(1-sh) + (1-x)^2(1-s)}$$ In Mathematica language it gives: ...
2
votes
0answers
268 views

Faster Ways to compute recursive summation

It takes a long time to compute the summation below, and I'd like to know if there are some better ways to compute things faster. I have used $3$ ways to calculate, but they are very unsatisfactory. I ...
7
votes
3answers
486 views

Nested list to graph

The following nested list can be regarded as a representation of a (tree) graph: ...
0
votes
1answer
132 views

How to define a function that operate with a base case recursively? [duplicate]

Let's start from the simplest case: say I have a function $f(x)=x$, and let's say I want to define a new function (of $x$) to be its $n$-th power $(f(x))^n$. How should I write the code? I tried: ...
1
vote
2answers
324 views

Summing Over a Variable Number of Indices

I am trying to figure out the best way to include a variable number of indexes of summation in a program. For example: Suppose I want to define a function $g[n,s]$ that returns the sum ...
2
votes
2answers
206 views

Solving a dynamic program via recursion

I'm trying to use recursion to solve a joint inventory/ dynamic pricing problem as in monahan, petruzzi and zhao 2004. I tried to solve for y[t] and ...
3
votes
2answers
159 views

Unset while running recursion

I am trying to run a recursive definition while at the same time clearing previously found values, so my memory is not completely consumed. I found this: How to clear parts of a memoized function? ...
6
votes
1answer
310 views

What is the most elegant way to specifiy a start case for recursion?

When writing recursive code that computes the least value at which a condition is true and various other scenarios it is often necessary to specify a base case. What I have done to supply this is ...
1
vote
1answer
104 views

Defining a formal Primitive Recursion

I'm searching for a way to define the function this wikipedia article call Primitive Recursion. It should have two functions as argument and it should return a function. I need this as helping in ...
1
vote
1answer
204 views

Defining a recursive integral sequence

I have a recursive integral sequence as follows: $$y_0(x)=1+r,$$ $$y_1(x)=-\frac{1}{\Gamma(a)}\int_0^x (x-t)^{a-1} y_0(t) dt,$$ $$\vdots$$ $$y_n(x)=-\frac{1}{\Gamma(a)}\int_0^x (x-t)^{a-1} y_{n-1}(t) ...
0
votes
1answer
113 views

How to solve the recursion equation that include the uncertain value i in it?

I write the following code of a recursion equation, but it can not work correctly. ...
2
votes
1answer
342 views

calculating a sequence of functions using iteration

I am trying to compute a sequence of functions using iteration and keep running into problems trying to use built in looping commands because of the recursive nature of the definition. The code below ...
13
votes
7answers
427 views

Concatenation of lists in a Fibonacci-like pattern

I am trying to create a List of elements that follow the general pattern: $$X_{n+1} = X_n X_{n-1}$$ where the operation on the right hand side is concatenation, i.e., joining. I want to achieve the ...
0
votes
1answer
160 views

Explicit, closed formula for recursive integral as a function of the recursive parameter

This is a follow-up of this question/answer. I'm working on a recursive integral more complicated than the one on the linked question, but this one can be used as an example. I would like to obtain ...
4
votes
1answer
268 views

Computing a series in terms of exponential function

Is there any way to compute the following series in terms of exponential function ? $$\sum_{k=0}^\infty Y_1(k)\;x^k$$ where $$Y_1(k) = \frac{(k - 1)!}{k!}Y_3(k - 1)$$ $$Y_2(k) = \frac{(k - ...
1
vote
1answer
2k views

Why do I get an error message when for this type of recursion?

When I type the following code: mus[] = 0; mus[x__] := First[List[x]] + mus[Rest[List[x]]] What one would think would happen is that if, ...
1
vote
1answer
675 views

Solving a nonlinear system of recurrence equations

I am having two problems regarding Mathematica, and both of them are happening because it does not accept such inputs: ...
5
votes
1answer
283 views

How can I use a recursive function in a Module

I have a question about using a recursive function in a module. I made the following example to illustrate the question. The followwing code works: ...
5
votes
3answers
456 views

Solving recurrence sequence using an ODE

I'm trying to solve the recurrence a[n] == 0 for n < l a[l] == 1 a[n] == (l + 2(n - l) a[n - l] + (n - l)(n - 1) a[n - 1])/n(n - l + 1) for n >= l using ...
6
votes
1answer
278 views

How to deal with too much recursion

I have put together a very simple climate model based around four equations which define the state at time t based on time t-1, ...
3
votes
2answers
172 views

Rook walk and RecurrenceTable in Mathematica

This appears in a combinatorics book: $a(m,n)=2 a(m,n-1)+2 a(m-1,n)-3 a(m-1,n-1)$ It is a recurrence equation for the number of rook walks from $(0,0)$ to $(m,n)$. The initial conditions are: $ ...
2
votes
2answers
82 views

Optimize recurrence output

I have recurrence and I need to calculate first $N$ members of the sequence ...
10
votes
2answers
428 views

Recursive Integral for Volume of $n$-Ball

The volume of an $n$-ball (the $(n+1)$-dimensional analogue of a disk) of radius $r$ can be found by the following integral recurrence: $$V_0(r)=2r$$ ...
3
votes
4answers
865 views

Recursive function for calculating the arithmetic mean of a vector

I want to write recursive function for calculating the arithmetic mean of a vector. For $X(n)=\{1,2,3,\ldots, n\}$, the formula is ${\rm Mean}(X(n))=((n-1)\, {\rm Mean}((X(n-1))+X_n)/n$. My ...
4
votes
3answers
233 views

Long waiting time for computing a summation

It takes a long time to compute the summation below, and I'd like to know if there are alternative ways to compute things faster. When replacing $15$ by $\infty$, then I should get $3^{1/3}$. I need ...
2
votes
2answers
549 views

Recursive formula involving a piecewise function

I am trying to construct a recursion, g[t], where g[t] = a[t] if g[t-1] = 1 ...
12
votes
1answer
565 views

How can I define a Listable function only apply to a vector?

Consider the function f which should behave as f[{{1, 2}, {3, 4}, {5, 6, 7}}] ...
0
votes
1answer
156 views

fit recursive function to data

I have a function $g(x, n)$, which is a recursive expression: $x$ is a variable and $n$ the number of iterations. In my case $n=3$ for all $x$. An example is the following: ...
10
votes
0answers
574 views

Compile recursive function modifying global variables

How to compile recursive formula when it relies on more than a few global variables (global to the topmost compiled function)? It is unreasonable to pass on all such variables to each recursive ...
3
votes
1answer
392 views

Transform recursion for coefficients into differential equation for generating function

Assume, one is given a linear recursion with polynomial coefficients for a sequence $(a_i)_i$, such as a[i] == i a[i-1] I would like to convert this recursion ...
7
votes
2answers
742 views

How can I define a sequence of functions?

I need to define a function fun, and then re-define this function iteratively. The code is given at the end. First, a function ...
1
vote
1answer
308 views

How can I solve my recursion equation?

How can I solve the recursion equation given below? I doubt there have no solution for the recursion equation because this is circulating recucrsion equation. ...