# More Improvements to the Simplification Method

I was out of town for the beginning of this week, so I don’t have as much to report, nevertheless in the last few days, I have made some good improvements to the cg_simp method, allowing it to simplify cases other than just the simple sums, tho it still only handles the same case as before involving the sum of . While it is entirely possible I’m doing something stupid in performing all the checks, as far as I can tell, it works for the given case. Because the code itself is not yet very clear and it is not always straightforward what is happening, I’ll explain what I have implemented.

First note that for the simplifications that will be made, it is required to have a sum of terms. The first for loop in the method constructs two lists cg_part and other_part, the former consisting of all terms of the sum containing a CG coefficient and the latter the other terms.

Next, we iterate over the list cg_part. The number of CG coefficients is computed, which determines which simplifications can be made (currently, the only implemented simplification uses terms with 1 CG coefficient in each sum term). Those terms with the proper number of CG coefficients are then considered for the simplification. The way this will work is: based on the properties of the CG coefficient in the term, it will search the rest of the list for other terms that can be used in the simplification, and if all the terms exist, will simplify the terms.

Turning to the implementation, when iterating through the list, the first thing to do is determine the properties of the CG coefficient terms, that is to extract from the term in the sum the CG coefficient itself and the numerical leading terms. Here it is also noted the sign of these numerical terms.

Next, the rest of the list is checked to see if a simplification can be made using the determined term. To keep track of this, a list, cg_index, is initialized with False as the element of the list. In checking the later terms, we preform a similar decomposition as with the first term, that is splitting up the CG coefficient from the other components of the term, determining the CG coefficient, the leading terms and the sign of the terms. If the properties of these are correct, then the corresponding element of cg_index is updated with a tuple (term, cg, coeff) where term is the term in cg_part (so this element can be removed later), the CG coefficient and the leading numerical coefficient of the coefficient.

Now, if all the elements of cg_index are changed, the simplification is preformed. When this happens, first we find the minimum coefficient of the chosen CG coefficients, which determines the number of times we can apply the simplification. Then the replacing happens; for each element in cg_index (which is a tuple) the first element of the tuple is popped off cg_part, then, if the term is not eliminated by the simplification, a new term is created and added to cg_part, and finally a constant is added to other_part, completing the simplification.

Looking at the code, this method is very straightforward, but should be robust and scalable for treating cases of sums with numerical leading coefficients, and now that the i’s have been dotted and the t’s have been crossed on testing this method, implementing new cases should come rapidly in the next couple days. However, one place where this will still need some work is in implementing symbolic simplification, both in dealing with symbolic leading terms on the CG coefficients and symbolic CG coefficients themselves. This will take a bit of thought and likely a bit of help to complete, but this is one thing I hope to work on in the next week. In addition, as the simplification comes into place, I’ll work on polishing out the last of the details to get the classes for the Wigner3j/CG coefficients working properly.