RecurrenceTable - Introduction

RecurrenceTable defines a sequence by setting f[n+1] equal to some function of f[n], and an initial condition for f[1]. Recurrence relations are also called difference equations and the method of finite differences uses recurrence relations. These are discrete versions of differential calculus. See the excellent articles in Wolfram Mathworld.

In our computer simulations of neural circuits, while the equations governing the neurons' behavior are phrased as differential equations, in reality we compute them as difference equations. For one thing we add noise to them, and for another, in practice they contain singularities. The "forward Euler method" et cetera won't work. I suspect that many prima facie differential equations are, in practice, unsolvable, and mask an underlying reality actually described by difference equations.

Here is a comparison between Table and RecurrenceTable that simply shows how to use the syntax of RecurrenceTable.

RecurrenceTable

In[21]:= powersOfOneHalf = Table[2^-i, {i, 1, 25}] // N

Out[21]= {0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, \
0.00195313, 0.000976563, 0.000488281, 0.000244141, 0.00012207, 0.0000610352, \
0.0000305176, 0.0000152588, 7.62939*10^-6, 3.8147*10^-6, 1.90735*10^-6,
 9.53674*10^-7, 4.76837*10^-7, 2.38419*10^-7, 1.19209*10^-7, 5.96046*10^-8,
 2.98023*10^-8}

In[23]:= RecurrenceTable[{a[n + 1] == .5 a@n, a@1 == 0.5}, a, {n, 25}]

Out[23]= {0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, \
0.00195313, 0.000976563, 0.000488281, 0.000244141, 0.00012207, 0.0000610352, \
0.0000305176, 0.0000152588, 7.62939*10^-6, 3.8147*10^-6, 1.90735*10^-6,
 9.53674*10^-7, 4.76837*10^-7, 2.38419*10^-7, 1.19209*10^-7, 5.96046*10^-8,
 2.98023*10^-8}

We can define f[n] in terms of more than one preceding term. If so, we need to supply the initial conditions for as many terms as specify the recurrence relations. Here is a Fibonacci-type series, a different flavor defined using three preceding terms rather than two. The initial conditions can also be thought of as boundary conditions.

In[4]:= RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + a[n - 3], a[1] == 1, a[2] == 1,
   a@3 == 1}, a,
   {n, 10}]

Out[4]= {1, 1, 1, 3, 5, 9, 17, 31, 57, 105}