How can I create a matrix that represents all possible "recipes" of 3 chemical components? (i.e., the 3 chemical components must be positive and sum to 100 wt%)

Hello! I am trying to create a matrix that represents all possible recipie combinations of 3 chemical components. The 3 chemical components must sum to 100 wt%, and individually have an abundace between 0 and 100 wt% . Eventually I want to create a giant matrix of all possible combinations of 10 chemical components, but I'll start off easy with 3 chemical ingredidents.

I want to create the matrix in small increments of around 0.1 wt%.

The end result I am going for would have 3 columns (one for each ingreident), and each row would contain a recipe (e.g., 10.3 wt% ingredient 1, 70.9 wt% ingredient 2, and 18.8 wt% of inregident 3).

Thanks!

Answer 1

[Davide Masiello]

I am going to assume that the percentage of each compound can be rounded to the first decimal digit.

Then I would recommend using the following function:

https://uk.mathworks.com/matlabcentral/fileexchange/17818-all-permutations-of-integers-with-sum-criteria

Given a number n of compunds, the matrix A of all possible recipes is found by

A = allVL1(n,1000)/10;

For n = 3, the first ten rows of the outputs are

A =

         0         0  100.0000
         0    0.1000   99.9000
         0    0.2000   99.8000
         0    0.3000   99.7000
         0    0.4000   99.6000
         0    0.5000   99.5000
         0    0.6000   99.4000
         0    0.7000   99.3000
         0    0.8000   99.2000
         0    0.9000   99.1000
         ...
         ...    

In this case, the size of A is 501501x3, so be careful not to run out of memory when you scale it up to 10 compounds!

[Stephen23]

Indeed, 10 compounds is going to be rather challenging to store, unless you happen to have access to a computer with ... umm... around two hundred sextillion bytes of memory. Just to hold it, not even to do any calculations on it.

R = allVL1(3, 1000, '==', NaN)

R = 501501

R = allVL1(4, 1000, '==', NaN)

R = 167668501

R = allVL1(5, 1000, '==', NaN)

R = 4.2085e+10

R = allVL1(10, 1000, '==', NaN)

Warning: Result may not be exact. Coefficient is greater than 9.007199e+15 and is only accurate to 15 digits

R = 2.8822e+21

Shameless plug of NUM2WORDS:

num2words(R*10*64/8)

ans = 'two hundred and thirty sextillion, five hundred and seventy-three quintillion, eighty-four quadrillion, nine hundred and ninety-six trillion, two hundred and sixty-three billion'

[Davide Masiello]

I thought so! XD

Thanks for putting the numbers down.

@Sterling Vanderzee you could think about the practicality of having all the possible "recipes". Perhaps your problem could be reduced to a simpler one.

[Sterling Vanderzee]

Thank you @Davide Masiello and @Stephen!

Wow I was expecting a giagantic "recipie book" for all possible combinations of 10 compounds, but "two hundred sextillion bytes of memory" is beyond anything I imagined lol.

Do you know of a more efficient way of creating a histogram of all possible solutions to underdetermined systems of linear equations?

i.e., If I know the bulk chemical composition of a mixture of known chemical compunds (let's call them A,B,C and D) is there a computationally efficient way to create a histogram of all possible concentrations of chemical compund A?

I have used MATLAB's fmincon to solve for the maximium and minimum possible concentrations of chemcial comound A, however I am more interested in the most probable concentrations of compound A, rather than the extremes.

The system of equations I'm working with is underdetermined, therefore I cannot solve for a unique solution. And I also am considering the range of solutions that are acceptable within measurement error of the bulk chemical composition of the mixture of compounds.

Thanks!

[Walter Roberson]

is there a computationally efficient way to create a histogram of all possible concentrations of chemical compund A?

Generate random combinations and histogram the results.

Are we to understand there are constraints, and you want to have a look at ranges of values that can satisfy the constraints? Randomly generating values and asking them to satisfy constraints can be pretty slow if the constraints are difficult to satisfy.

[Sterling Vanderzee]

Hi Walter! Thank you for your insight.

Yes I am trying to generate a histogram of possible solutions that satisfy linear constraints.

I am not clear on how to make fmincon output random solutions to the problem. I currently using an objective function (f = compound A) to calculate the minimum possible abundnace of chemical compound A, where the solutions must satisfy linear constraints. I currently have a working fmincon code to do this, but I am more interested in the most probable solution(s), rather than the minumum or maximum.

I am given the bulk chemical compositon of mixure of chemical compounds, I know what the possible chemical compounds are within the mixture, and I want to calculate the most probable concentration of a specific chemical compund of interest (or at least see a histogram of random possible solutions).

Here's a simplified example of the contraints, in reality it's an underdetermined system of linear equations because there are other chemical compounds I need to include:

The constraints are that each chemical compound [Mg(OH)2, Mg2SiO4, Mg3Si2O5(OH)4 and CaMgSi2O6] must have a concentraion ranging from 0 to 100 wt%, and the sum of the chemical compounds must equal 100%. There are also constraints (defined by two inequalities for each chemical element) that the bulk chemical compsition of the modelled mixture must agree with the measured value to within 2% relative error. i.e., the wt% Mg content of the modelled mixture must be within 2% relative error of the measured Mg content of the mixure, and so on for the other chemical elements.

Thanks!

[Sterling Vanderzee]

@Walter Roberson based on your suggestion of generating random combinations, perhaps you mean I could create a series of vectors that represent random guesses for the abundnace of each chemical compound, and then within fmincon I would create an objective function that minimizes the difference between the randomly guessed abundnaces, and the modelled abundance?

Is there a code to randomly generate vectors that are subject to linear constraints? i.e. the randomly generated values in the vector must sum to 100, and perhaps also specifiy that certain rows of the vector must be between 0 and 12? I'd want a lot of random vectors, otherwise I'd do it manually.

[Sterling Vanderzee]

@Davide Masiello and @Stephen, do you think it would be possible to create a reasonalby efficient code that modies the AllVL1 code to only generate chemical compound "recipies" that satisfy a linear constraint on the bulk chemical composition of the mixture (i.e., in cases where I know the bulk chemical composition and want to know the possible recipies)? Maybe that would drastically reduce the amount of memory needed by only focusing on a small subset of the billions of possible recipes?

[Davide Masiello]

@Sterling Vanderzee I am slightly confused by your terminology.

For instance, when you say the most probable concentration of a specific chemical compund of interest, how is that probability determined?

Are we to understand that there are equations modelling the probability that a certain compound has a certain concentration in your mixture?

If the only constraint is that the sum of all concentrations equals 100, then you are back to the combinatoric problem of all the possible recipes, which is unfeasible for the reasons mentioned above.