updating gaussian elimination to be more general

[leios]

Hey guys, following the discussion in #193 and #201 , we looked at making the Gaussian Elimination code more general. This change does not change the computational complexity of the gaussian_elimination function, but it does for the back-substitution.

In the previous julia version, we were literally plugging the values in for the upper diagonal matrix. In this version, we are actually doing the full Gauss-Jordan elimination and creating a reduced row eschelon matrix. The definition of "back-substitution" can mean either case.

As we mentioned before, the previous code we had submitted was not "wrong"; however, this is technically more general. I am curious to see what you guys think about the back-substitution step. In the worst-case, we can add a condition to switch between the methods if the matrix is not singular (which means that it admits a regular set of solutions). As a note: the solution vector of this method is the last column of the output matrix.

Please let me know what you think about this implementation. We can always change it.

[jiegillet]

I want to apologize for not fully grasping your original implementation. I had not realized that the complexity was so different.
I support the idea of keeping both functions. Checking if a matrix is singular after it goes through gaussian_elimination is as simple as checking A[rows, cols-1] == 0 (or by stopping the computation as you were doing before). After that, one could decide to use your old back_substitution or, say, to_reduced_echelon.

[leios]

I'm alright with having multiple implementations that switch off depending on the matrix. It should be easy enough to implement and we can highlight the difference in the text.

[jiegillet]

I don't even think you need to implement it, you just need to make both functions available and let people decide which method they want. Someone who only wants to solve a system wouldn't be interested in looking at the reduced echelon form anyway.

[leios]

Yeah, I just meant that we re-write the code like you suggested and mention the differences in the text.

[jiegillet]

God I really wish I had not put text and code in the same PR. Do you want me to do it or you want to PR my branch?

[leios]

I will PR your branch (tomorrow, though)

[leios]

Alright, sorry this took so long. I was busy this weekend with other commitments. Here is an attempt to solve the issues and PR's for Gaussian Elimination.

I split the chapter to have both Gauss-Jordan elimination and back-substitution, which seemed to be the fairest thing to do. This allows the programmer to choose which method best suits their use-case. I am sure there will be discussion, so this is a "rough" edit.

[jiegillet]

This is a good change. I'll get rid of the chapter edits in my PR.
A small nitpick is the inconsistent capitalization of Echelon Form/echelon form.

[jiegillet]

Sorry, closed it by mistake

[leios]

No prob bob. I'll review for typos and update this PR soon.

[jiegillet]

Those were the only nitpicks I found

[jiegillet]

This is not needed anymore.

[jiegillet]

The capitalization of [R|r]ow [E|e]echelon is all over the place. After some thoughts, I suggest dropping the capital letter everywhere (since row echelon is not a name, it's like saying diagonal), except maybe when it's first introduced. Same deal for RRE.

[leios]

Cool, I'll make the changes as soon as I get home tonight!

EDIT: Made them now.