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Section: New Results

Algebraic computing and high-performance kernels

Linear differential equations as a data-structure

A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be computed. A variety of algorithms has thus been designed in recent years that do not aim at “solving”, but at computing with this representation. Many of these results are surveyed in [11].

Absolute root separation

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic. [5]

Improving the complexity of block low-rank factorizations with fast matrix arithmetic

We consider in [9] the LU factorization of an $n\times n$ matrix represented as a block low-rank (BLR) matrix: most of its off-diagonal blocks are approximated by matrices of small rank $r$, which reduces the asymptotic complexity of computing the LU factorization down to $𝒪\left(n^{2}r\right)$. Even though lower complexities can be achieved with hierarchical matrices, the BLR format allows for a very simple and efficient implementation. In this article, our aim is to further reduce the BLR complexity without losing its nonhierarchical nature by exploiting fast matrix arithmetic, that is, the ability to multiply two $n\times n$ full-rank matrices together for $𝒪\left(n^{\omega }\right)$ flops, where $\omega <3$. We devise a new BLR factorization algorithm whose cost is $𝒪\left(n^{(\omega +1)/2}{r}^{(\omega -1)/2}\right)$, which represents an asymptotic improvement compared with the standard BLR factorization as soon as $\omega <3$. In particular, for Strassen's algorithm, $\omega \approx 2.81$ yields the cost $𝒪\left(n^{1.904}{r}^{0.904}\right)$. Our numerical experiments are in good agreement with this analysis.

Fast computation of approximant bases in canonical form

In [10] we design fast algorithms for the computation of approximant bases in shifted Popov normal form. For $𝖪$ a commutative field, let $F$ be a matrix in $𝖪{\left[x\right]}^{m\times n}$ (truncated power series) and $\overrightarrow{d}$ be a degree vector, the problem is to compute a basis $P\in 𝖪{\left[x\right]}^{m\times m}$ of the $𝖪\left[x\right]$-module of the relations $p\in 𝖪{\left[x\right]}^{1\times m}$ such that $p\left(x\right)·F\left(x\right)\equiv 0mod{x}^{\overrightarrow{d}}$. We obtain improved complexity bounds for handling arbitrary (possibly highly unbalanced) vectors $\overrightarrow{d}$. We also improve upon previously known algorithms for computing $P$ in normalized shifted form for an arbitrary shift. Our approach combines a recent divide and conquer strategy which reduces the general case to the case where information on the output degree is available, and partial linearizations of the involved matrices.