Added python gaussian elimination #200
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| # Check to make sure matrix is good! | ||
| if (A[max_index][k] == 0): | ||
| print("matrix is singular! End!") |
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Singular matrices have nothing to do with Gaussian elimination. If your pivot is zero, you continue with the next column. It's still a valid matrix in the echelon form.
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https://stackoverflow.com/questions/23985004/gaussian-elimination-of-a-singular-matrix
In my understanding a singular matrix was indicative of an improper set of systems of equations and should be underdetermined. This algorithm shouldn't work for this case (I don't think, anyway). Most example code I can find stops once the matrix is considered singular; however, I think you are right. We can also take the general case. I am just not sure the best version of the code here.
| for i in range(k+1,rows): | ||
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| # Step 3: finding fraction | ||
| fraction = A[i][k] / A[k][k] |
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The pivot is not always in the diagonal, in the case that a previous pivot was zero. You have to consider this case.
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This is actually related to the above discussion, I think. The code I have in the algorithm archive is meant to solve a system of equations, so these cases will be caught by the "singular matrix" condition.
| soln = [0]*rows | ||
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| # initialize the final element | ||
| soln[-1] = A[-1][-1] / A[-1][-2] |
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Again here, this is not general, in case the Echelon form is not a pure upper triangular matrix.
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Again, this is caught by the "not singular" condition.
Currently looking at the code and seeing how we should update things.
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Oh boy. All the mistakes I found in your code are also in @leios's Julia code. This might take a while :) |
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Yup. This is a direct port of the Julia code. Might create an issue to fix that? BTW, the exact same errors appear in #195 |
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Yeah I'll open an issue. Let's keep this PR in the fridge for a bit if you don't mind. |
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I sniped you and just opened an Issue |
june128
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Implementation
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@jiegillet is right, the form we have in the AAA is not general; however, it is the most common implementation of the algorithm I could find. The current implementation solves a system of equations, which is what the chapter was set up to do. It can also find the determinant; however, it lacks the capability of finding the ranks and bases of a matrix, which means that it cannot be used for all cases. I'm currently looking at how to update the code from here. |
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@leios I was thinking about it yesterday, and yes, if the problem you are working on is solving a system of equations, then the algorithm is perfectly adequate. However, Echelon and Reduced Echelon Forms have more applications than that, for example finding the rank of a matrix. It seems a little bit wasteful to implement the general case 90% of the way and stop there. |
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@jiegillet Yeah, the issue #201 is probably the best place to discuss this. |
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Yeah, sorry, I just saw your comment there. I'm gathering my thoughts and I'll answer there. |
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Update: the Julia code and chapter were updated and merged. Have a look and change your code accordingly. |
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Still alive? I will close this PR if it's stale. |
With no use of numpys
arraydatastructure. This might be a contested decision.