Algorithms in Matlab or Octave

The functions for Matlab or Octave can be used for free via the web or by installing Octave.

There are two versions of the functions:

Version 1, in the compressed file matlaboctave.zip (updated on May 17, 2014); detailed instructions (in PDF or without a reader) are available.
Version 2, improved but in testing, is in the compressed file matlaboctave2.zip (updated on December 5, 2024); a reference (in PDF or without a reader) can be consulted that complements the detailed instructions of the previous version.

The functions allow you to solve the Von Neumann model (VN) and other problems with several different algorithms, with processes defined in multiple time steps or in continuous time, analyze the distance to maximum conditions, obtain perturbation series, etc. With the improved version, you can also solve VN with variable or exact precision and with internal variables. For more advanced use, you can consult the online help for the functions in the first and improved versions. New versions of the algorithms will be added soon, as well as functions for other problems, for analyzing solutions, etc.

Brief Instructions for Solve the Von Neumann Model

The Von Neumann Model (VN)

For an introduction to the model for non-mathematicians, see Chapter 1 of Computing Von Neumann.

Suppose we have three processes: the first process uses 280 quarters of wheat and 12 tons of iron as inputs to produce 575 quarters of wheat; the second process consumes 120 quarters of wheat and 8 tons of iron to produce 20 tons of iron; the third process, like the first, uses 280 quarters of wheat and 12 tons of iron but produces 400 quarters of wheat. The outputs are obtained at a time after the inputs are consumed. We will write these processes in tables or matrices of inputs A and outputs B, where each process occupies a row and each material a column:

Imagen con las matrices de insumos y productos

Solving VN consists of calculating how much each process must operate so that the system grows proportionally as much as possible. In proportional growth, the processes exhibit the same rate of expansion, and for each material, the total consumption at a time is equal to the total production from the previous time.

Solving VN in Matlab or Octave

To install the functions on a computer, but not on the web, unzip matlaboctave.zip (or matlaboctave2.zip, if you're using the improved version for testing) into a directory like c:\matlaboctave and type the command in Octave or Matlab:

cd c:\matlaboctave

On both a computer and the web, in Matlab or Octave we'll first define the processes, specifying the flows at the time they occur and setting the consumption with a negative sign and the production with a positive sign. Then, we'll type the command to solve VN. For the processes above, we have (press the "Enter" key after typing each command):

F(:,:,1)=[-280,-12;-120,-8;-280,-12];
F(:,:,2)=[575,0;0,20;400,0];
[k,X,Y]=vn(F)

The Matlab or Octave output is:

Algoritmo 1: vnautovalor
k =
1.2500
X =
0.4000 0.6000 0
Y =
0.0038
0.0572

The output informs us of the algorithm used; the expansion factor k is 1.25, the system grows by 25% at each time step; the intensities X at which the processes operate are [0.4, 0.6, 0], the first process operates 0.4 times, the second 0.6 times, and the third does not operate; the prices Y are [0.0038, 0.0572], wheat is worth 0.0038, and iron is worth 0.0572.

Here we have set up VN assuming that free elimination is not available; if it were available, which is how John von Neumann defined the model in the original article, we would write the command

[k,X,Y]=vn(F,1,1)

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