Fast computation of approximant bases in canonical form

Computing Canonical Bases of Modules of Univariate Relations

We study the computation of canonical bases of sets of univariate relations (p1, . . . ,pm ) ∈ K[x]m such that p1 f1 + · · · + pm fm = 0; here, the input elements f1, . . . , fm are from a quotient K[x]n/M, whereM is a K[x]-module of rank n given by a basis M ∈ K[x]n×n in Hermite form. We exploit the triangular shape of M to generalize a divide-and-conquer approach which originates from fast mi...

Determining the order of minimal realization of descriptor systems without use of the Weierstrass canonical form

A common method to determine the order of minimal realization of a continuous linear time invariant descriptor system is to decompose it into slow and fast subsystems using the Weierstrass canonical form. The Weierstrass decomposition should be avoided because it is generally an ill-conditioned problem that requires many complex calculations especially for high-dimensional systems. The present ...

Certification of minimal approximant bases

Considering a given computational problem, a certificate is a piece of additional data that one attaches to the output in order to help verifying that this output is correct. Certificates are often used to make the verification phase significantly more efficient than the whole (re-)computation of the output. Here, we consider the minimal approximant basis problem, for which the fastest known al...

Fast Computation of Orthonormal Basis for Rbf Spaces through Krylov Space Methods

In the last years, in the setting of Radial Basis Function (RBF), the study of approximation algorithms has particularly focused on the construction of (stable) bases for the associated Hilbert spaces. One of the way of describing such spaces and their properties is the study of a particular integral operator and its spectrum. We proposed in a recent work the so-called WSVD basis, which is stri...

A uniform approach for the fast computation of Matrix-type Padé approximants