26,559 research outputs found
Fast Order Basis and Kernel Basis Computation and Related Problems
In this thesis, we present efficient deterministic algorithms
for polynomial matrix computation problems, including the computation
of order basis, minimal kernel basis, matrix inverse, column basis,
unimodular completion, determinant, Hermite normal form, rank and
rank profile for matrices of univariate polynomials over a field.
The algorithm for kernel basis computation also immediately provides
an efficient deterministic algorithm for solving linear systems. The
algorithm for column basis also gives efficient deterministic algorithms
for computing matrix GCDs, column reduced forms, and Popov normal
forms for matrices of any dimension and any rank.
We reduce all these problems to polynomial matrix multiplications.
The computational costs of our algorithms are then similar to the
costs of multiplying matrices, whose dimensions match the input matrix
dimensions in the original problems, and whose degrees equal the average
column degrees of the original input matrices in most cases. The use
of the average column degrees instead of the commonly used matrix
degrees, or equivalently the maximum column degrees, makes our computational
costs more precise and tighter. In addition, the shifted minimal bases
computed by our algorithms are more general than the standard minimal
bases
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices
It is well known that if a matrix solves the
matrix equation , where is a linear bivariate polynomial,
then is normal; and can be simultaneously reduced in a finite
number of operations to tridiagonal form by a unitary congruence and, moreover,
the spectrum of is located on a straight line in the complex plane. In this
paper we present some generalizations of these properties for almost normal
matrices which satisfy certain quadratic matrix equations arising in the study
of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure
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
Vertex operators arising from Jacobi-Trudi identities
We give an interpretation of the boson-fermion correspondence as a direct
consequence of Jacobi-Trudi identity. This viewpoint enables us to construct
from a generalized version of the Jacobi-Trudi identity the action of Clifford
algebra on polynomial algebras that arrives as analogues of the algebra of
symmetric functions. A generalized Giambelli identity is also proved to follow
from that identity. As applications, we obtain explicit formulas for vertex
operators corresponding to characters of the classical Lie algebras, shifted
Schur functions, and generalized Schur symmetric functions associated to linear
recurrence relations.Comment: 23 page
On integer programing with bounded determinants
Let be an integral matrix, and let
be an -dimensional polytope. The width of is defined as . Let and denote the greatest and the
smallest absolute values of a determinant among all
sub-matrices of , where is the rank of a matrix . We prove that if
every sub-matrix of has a determinant equal to or and , then contains
affine independent integer points. Also we have similar results for the case of
\emph{-modular} matrices. The matrix is called \emph{totally
-modular} if every square sub-matrix of has a determinant in the set
. When is a simplex and , we describe a polynomial time algorithm for finding an integer
point in . Finally we show that if is \emph{almost unimodular}, then
integer program can be
solved in polynomial time. The matrix is called \emph{almost unimodular} if
and any sub-matrix has a
determinant from the set .Comment: The proof of Lemma 4 has been fixed. Some minor corrections has been
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