5,049 research outputs found
Fast systematic encoding of multiplicity codes
We present quasi-linear time systematic encoding algorithms for multiplicity
codes. The algorithms have their origins in the fast multivariate interpolation
and evaluation algorithms of van der Hoeven and Schost (2013), which we
generalise to address certain Hermite-type interpolation and evaluation
problems. By providing fast encoding algorithms for multiplicity codes, we
remove an obstruction on the road to the practical application of the private
information retrieval protocol of Augot, Levy-dit-Vehel and Shikfa (2014)
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
Model Reduction of Descriptor Systems by Interpolatory Projection Methods
In this paper, we investigate interpolatory projection framework for model
reduction of descriptor systems. With a simple numerical example, we first
illustrate that employing subspace conditions from the standard state space
settings to descriptor systems generically leads to unbounded H2 or H-infinity
errors due to the mismatch of the polynomial parts of the full and
reduced-order transfer functions. We then develop modified interpolatory
subspace conditions based on the deflating subspaces that guarantee a bounded
error. For the special cases of index-1 and index-2 descriptor systems, we also
show how to avoid computing these deflating subspaces explicitly while still
enforcing interpolation. The question of how to choose interpolation points
optimally naturally arises as in the standard state space setting. We answer
this question in the framework of the H2-norm by extending the Iterative
Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical
examples are used to illustrate the theoretical discussion.Comment: 22 page
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