Monday, 11 August 2014

Week of 8/4 Update

This week I've been working on on fleshing out the algorithm I described last week for computing the last invariant factor of a polynomial matrix. I've written a high-level implementation in Sage to demonstrate proof of concept. There are a few subtleties I've uncovered:
• The degree of $\bar{x}$, $N$, needs to be bigger than I originally thought. A safe bound is $N\geq{}2n*deg(A)+deg(B)+1$ although I'm not yet sure whether this bound is sharp.
• The degree of $\vec{b}$, $d_{b}$, can be fairly small, about $2\log_{q}n$. This ensures that the components are probably distinct.
• When computing $\frac{n_{i}}{d}$ such that $\frac{n_{i}}{d}\equiv{}\bar{x_{i}}(mod\;x^{N})$ there's one such for any given choice of $k=deg(d)$. However only one such choice of $k$ will actually solve the system. To figure out which choice is correct I run the extended euclidean algorithm for several rows of $\bar{x}$ simultaneously and mark values of $d$ that occur in more than one row as candidates.
• Given a candidate $d$ the other $n_{i}$'s can be determined quickly by dividing $\frac{\bar{x}_{i}}{d}$
• Any particular $\bar{x}_{i}$ has a chance of producing the wrong $d$ due to cancellation in the Cramer's rule formula $\bar{x}_{i}=\frac{|A_{i}|}{|A|}$. This occurs with probability $\frac{1}{q}$. Checking more $\bar{x}_{i}$'s for candidate $d$'s fixes this. To ensure that we get the correct $d$ as a candidate with probability $p$ we need to check $qbinom(p,n,1-\frac{1}{q})$ where qbinom is the binomial quantile function.
• Sometimes there will be no $\alpha$ such that $|A(-\alpha)|\neq{}0$, even if $A$ is non-singular (over the rational functions). In this case we can still find an answer by finding a random solution to the Block-Toeplitz system: $\begin{pmatrix}A_{d_{b}}&A_{d_{b-1}}&...&A_{0}&0&...\\A_{d_{b+1}}&A_{d_{b}}&...&A_{0}&0&...\\&&\ddots{}&&\\0&...&A_{d_{A}}&A_{d_{A}-1}&...&A_{0}\end{pmatrix}\begin{pmatrix}\bar{x}^{(0)}\\\bar{x}^{(1)}\\\vdots{}\\\bar{x}^{N}\end{pmatrix}=\vec{0}$
Next week I'll start wrapping things up as the Google Summer of Code draws to a close.