Defactoring algorithms: Difference between revisions

To find this common factor, we need to linearly combine two of the first row {{map| 6 5 -4 }} and negative three of the 2nd row {{map| 4 -4 1 }} to produce {{map| 0 22 -11 }}. So we can see here that its common factor is 11. But there is no clear relationship between the numbers 2 and 3 and the number 11. And so we can begin to see that the problem of identifying enfactored mapping may not be very simple or straightforward.

In this section, we discuss three methods that defactors the mapping: Smith defactoring, developed by Gene Ward Smith<ref>but the name comes from a different Smith: [[Wikipedia: Henry John Stephen Smith|Henry John Stephen Smith]], for whom the [[Smith normal form]] is named, which this method uses</ref>; Pernet-Stein defactoring, described by Clément Pernet and William Stein; and column Hermite defactoring, developed by Dave and Douglas (the name comes, of course, from Hermite normal form, which it uses).  

Column Hermite defactoring could not have been developed, however, were it not for Gene's pioneering work with the Smith defactoring (what he calls the process of "saturating" a mapping). At first Dave and Douglas had no idea what the right reducing matrix of the Smith decomposition (the process which also provides the Smith normal form) had to do with common factors, only that it somehow magically worked. So they analyzed the Smith decomposition until they isolated its key actions which actually effect the defactoring, and then honed their method down to do only these necessary actions. Again, they wouldn't have known where to start were it not for Gene.

Dave and Douglas did much of their work in [https://www.wolfram.com/language/ Wolfram Language] (formerly Mathematica), a popular programming language used for math problems. In this section we'll give examples using it.

And that result matches what Gene finds in that xen wiki article. Defactoring and putting into normal form is equivalent to canonicalization.  

This algorithm was described in the 2009 paper "Fast computation of HNF of random integer matrices"<ref>https://www.wstein.org/papers/hnf/pernet-stein-fast_computation_of_hnf_of_random_integer_matrices.pdf</ref> by Clément Pernet and William Stein. At the time Dave and Douglas wrote the first draft of this article and developed column Hermite defactoring, they were unaware of this algorithm. After publicizing column Hermite defactoring, they were referred by [[Graham Breed]] to a similar method in [http://x31eq.com/temper/ Graham's popular online regular temperament tool], implemented as <code>saturate</code><ref>https://bitbucket.org/x31eq/regular/src/9bc9b5bd8c8e0ced6223b29c3ea487719d120c43/kernel.py#lines-178</ref> since 2011, and which includes in its commented documentation a link to the aforementioned paper. Unable to reverse-engineer Gene Ward Smith's saturation algorithm, Graham had gone back to the same source Gene had supposedly gotten his inspiration from — the Sage software developed by William Stein, co-author of this paper — and came across this paper. Graham's implementation turned out to be much more similar to the original description by Pernet and Stein than Gene's, differing only by an additional unnecessary use of the HNF at the beginning (while Gene's, by virtue of using the Smith Normal Form, could be said to essentially use some variable number of extraneous uses of HNF). It is not clear how Gene derived his saturation algorithm from Pernet and Stein's work, however, if the fact that Dave and Douglas derived something almost identical to Pernet and Stein's algorithm from Gene's, it suggests that it's not unreasonable for development to lead someone in the opposite direction along the same path. The very close relationship between column Hermite defactoring and Pernet-Stein defactoring will be discussed shortly.

pernetSteinDefactor[m_] := Inverse[getGreatestFactorA[m].m;</nowiki>

Here is the implementation for column Hermite defactoring:

The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref>In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of 33 = 3 × 11, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref>It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with -3 of the 2nd row, i.e. 2{{map|6 5 -4}} + -3{{map|4 -4 1}} = {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: -1{{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} = {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.

When in development on an ideal defactoring method — the effort which culminated in column Hermite defactoring — Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated).

This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found here: [[Temperament addition#3. Addabiliziation defactoring]].

To find the greatest factor of a mapping, put it into column HNF, and remove the extra columns of zeros. This gives the "greatest factor matrix". Its determinant is the greatest factor. A similar operation can be used for a comma basis.