How to reduce coefficients to their lowest possible integers using Matlab - Balancing Chemical Equations

Question

I am attempting to develop a Matlab program to balance chemical equations. I am able to balance them via solving a system of linear equations. Currently my output is a column vector with the coefficients.

My problem is that I need to return the smallest integer values of these coefficients. For example, if [10, 20, 30] was returned. I want [1, 2, 3] to be returned.

What is the best way to accomplish this?

I want this program to be fully autonomous once it is fed a matrix with the linear system. Thus I can not play around with the values, I need to automate this from the code. Thanks!

% Chemical Equation in Matrix Form
Chem = [1 0 0 -1 0 0 0; 1 0 1 0 0 -3 0; 0 2 0 0 -1 0 0; 0 10 0 0 0 -1 0; 0 35 4 -4 0 12 1; 0 0 2 -1 -3 0 2]

%set x4 = 1 then Chem(:, 4) = b and 
b = Chem(:, 4);     % Arbitrarily set x4 = 1 and set its column equal to b
Chem(:,4) = []      % Delete the x4 column from Chem and shift over
g = 1;              % Initialize variable for LCM 
x = Chem\b          % This is equivalent to the reduced row echelon form of 
                    % Chem | b

% Below is my sad attempt at factoring the values, I divide by the smallest decimal to raise all the values to numbers greater than or equal to 1
for n = 1:numel(x)
   g = x(n)*g
    M = -min(abs(x))
    y = x./M
end


I want code that will take some vector with coefficients, and return an equivalent coefficient vector with the lowest possible integer coefficients. Thanks!

What is the meaning of "return the smallest integer values of these coefficients"? The smallest value of [10, 20, 30] should be 10. — Chan, Oct 17 '19 at 01:32
When balancing chemical equations, the goal is to find the right "combination" of molecules on each side of the reaction that yields the right proportions. Say you are given CO2 + H2O → C6H12O6 + O2. One needs to find the lowest integer coefficients that make this equation true. In this case, the answer would be: 6 CO2 + 6 H2O → C6H12O6 + 6 O2. Referring to my example of [10, 20, 30], we could think of it as 10x + 20y + 30z = 0, where I want the smallest integer coefficients possible that make the equation true. I am free to divide them, so in this case my answer would be x + 2y + 3z =0 — Joshua, Oct 17 '19 at 02:55
It seems to me that you are doing somethings like integer programming. You may take a look on https://www.mathworks.com/discovery/integer-programming.html. — Chan, Oct 17 '19 at 03:05
Thanks for the resource, it seems this may require the Optimization Toolbox, which I do not have access to... — Joshua, Oct 17 '19 at 03:53
After you get an integer vector as result, you need to compute the GCD of the entries and divide all entries by it. Either matlab has such a function, or you have to implement a vector Euclidean algorithm reducing the vector elements by the smallest non-zero component (of course not reducing this component) until only one non-zero component is left over. — Lutz Lehmann, Oct 17 '19 at 09:26
@RodrigodeAzevedo You are correct, sorry I am new to stack overflow and wasn't sure where to post the question, since I thought there could be a built-in function on matlab for this. — Joshua, Oct 18 '19 at 20:37
@LutzL Thank you! I was able to do this with the built-in lcm (least common multiple) function in Matlab. — Joshua, Oct 18 '19 at 20:39

score 0 · Accepted Answer · answered Oct 18 '19 at 20:34

I was able to find a solution without using integer programming. I converted the non-integer values to rational expressions, and used a built-in matlab function to extract the denominator of each of these expressions. I then used a built in matlab function to find the least common multiples of these values. Finally, I multiplied the least common multiple by the matrix to find my answer coefficients.

    % Chemical Equation in Matrix Form
clear, clc
% Enter chemical equation as a linear system in matrix form as Chem
Chem = [1 0 0 -1 0 0 0; 1 0 1 0 0 -3 0; 0 2 0 0 -1 0 0; 0 10 0 0 0 -1 0; 0 35 4 -4 0 -12 -1; 0 0 2 -1 -3 0 -2];
% row reduce the system
C = rref(Chem);
% parametrize the system by setting the last variable xend (e.g. x7) = 1
x = [C(:,end);1];
% extract numerator and denominator from the rational expressions of these
% values
[N,D] = rat(x);

% take the least common multiple of the first pair, set this to the
% variable least
least = lcm(D(1),D(2));

% loop through taking the lcm of the previous values with the next value
% through x
for n = 3:numel(x)
  least = lcm(least,D(n));
end

% give answer as column vector with the coefficients (now factored to their
% lowest possible integers
coeff = abs(least.*x)

How to reduce coefficients to their lowest possible integers using Matlab - Balancing Chemical Equations

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