$$ \newcommand{\tp}{\thinspace .} $$

Sequences and difference equations

Hans Petter Langtangen [1, 2]

[1] Center for Biomedical Computing, Simula Research Laboratory [2] Department of Informatics, University of Oslo

Jun 14, 2016

From mathematics you probably know the concept of a sequence, which is nothing but a collection of numbers with a specific order. A general sequence is written as $$ \begin{equation*} x_0,\ x_1,\ x_2,\ \ldots,\ x_n,\ldots, \end{equation*} $$ One example is the sequence of all odd numbers: $$ \begin{equation*} 1, 3, 5, 7, \ldots, 2n+1,\ldots \end{equation*} $$ For this sequence we have an explicit formula for the $ n $-th term: $ 2n+1 $, and $ n $ takes on the values 0, 1, 2, $ \ldots $. We can write this sequence more compactly as $ (x_n)_{n=0}^\infty $ with $ x_n=2n+1 $. Other examples of infinite sequences from mathematics are $$ \begin{equation} 1,\ 4,\ 9,\ 16,\ 25,\ \ldots\quad (x_n)_{n=0}^\infty,\ x_n=(n+1)^2, \tag{1} \end{equation} $$ $$ \begin{equation} 1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\ \ldots\quad (x_n)_{n=0}^\infty,\ x_n={1\over n+1}\tp \tag{2} \end{equation} $$

The former sequences are infinite, because they are generated from all integers $ \geq 0 $ and there are infinitely many such integers. Nevertheless, most sequences from real life applications are finite. If you put an amount $ x_0 $ of money in a bank, you will get an interest rate and therefore have an amount $ x_1 $ after one year, $ x_2 $ after two years, and $ x_N $ after $ N $ years. This process results in a finite sequence of amounts $$ \begin{equation*} x_0, x_1, x_2,\ldots, x_N,\quad (x_n)_{n=0}^N\tp\end{equation*} $$ Usually we are interested in quite small $ N $ values (typically $ N\leq 20-30 $). Anyway, the life of the bank is finite, so the sequence definitely has an end.

For some sequences it is not so easy to set up a general formula for the $ n $-th term. Instead, it is easier to express a relation between two or more consecutive elements. One example where we can do both things is the sequence of odd numbers. This sequence can alternatively be generated by the formula $$ \begin{equation} x_{n+1} = x_{n} + 2\tp \tag{3} \end{equation} $$ To start the sequence, we need an initial condition where the value of the first element is specified: $$ \begin{equation*} x_0 = 1\tp\end{equation*} $$ Relations like (3) between consecutive elements in a sequence is called recurrence relations or difference equations. Solving a difference equation can be quite challenging in mathematics, but it is almost trivial to solve it on a computer. That is why difference equations are so well suited for computer programming, and the present document is devoted to this topic. Necessary background knowledge is programming with loops, arrays, and command-line arguments and visualization of a function of one variable.

The program examples regarding difference equations are found in the folder src/diffeq.

This chapter is taken from the book A Primer on Scientific Programming with Python by H. P. Langtangen, 5th edition, Springer, 2016.

Sequences and difference equations

Jun 14, 2016

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