13 research outputs found
A (hopefully) friendly introduction to the complexity of polynomial matrix computations
This paper aims at a friendly introduction to the field of fast algorithms for polynomial matrices, and surveys the results of the ISSAC 2003 paper 'On the Complexity of Polynomial Matrix Computations' by Pascal Giorgi, Claude-Pierre Jeannerod, and Gilles Villard
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of
higher order q-recurrence equations with rational coefficients. We extend a
method for finding a bound on the maximal power of t in the denominator of
arbitrary rational solutions y(t) as well as a method for bounding the degree
of polynomial solutions from the scalar case to the systems case. The approach
is direct and does not rely on uncoupling or reduction to a first order system.
Unlike in the scalar case this usually requires an initial transformation of
the system.Comment: 8 page
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
Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations
We give a Las Vegas algorithm which computes the shifted Popov form of an nonsingular polynomial matrix of degree in expected
field operations, where is the
exponent of matrix multiplication and
indicates that logarithmic factors are omitted. This is the first algorithm in
for shifted row reduction with arbitrary
shifts.
Using partial linearization, we reduce the problem to the case where is the generic determinant bound, with bounded from above by both the average row degree and the average column
degree of the matrix. The cost above becomes , improving upon the cost of the fastest previously
known algorithm for row reduction, which is deterministic.
Our algorithm first builds a system of modular equations whose solution set
is the row space of the input matrix, and then finds the basis in shifted Popov
form of this set. We give a deterministic algorithm for this second step
supporting arbitrary moduli in
field operations, where is the number of unknowns and is the sum
of the degrees of the moduli. This extends previous results with the same cost
bound in the specific cases of order basis computation and M-Pad\'e
approximation, in which the moduli are products of known linear factors.Comment: 8 pages, sig-alternate class, 5 figures (problems and algorithms
Computing Popov Forms of Polynomial Matrices
This thesis gives a deterministic algorithm to transform a row reduced matrix to canon-
ical Popov form. Given as input a row reduced matrix R over K[x], K a field, our algorithm
computes the Popov form in about the same time as required to multiply together over
K[x] two matrices of the same dimension and degree as R. Randomization can be used to
extend the algorithm for rectangular input matrices of full row rank. Thus we give a Las
Vegas algorithm that computes the Popov decomposition of matrices of full row rank. We also show that the problem of transforming a row reduced matrix to Popov form is at least
as hard as polynomial matrix multiplication
Implicitizing rational curves by the method of moving quadrics
International audienceA new technique for finding implicit matrix-based representations of rational curves in arbitrary dimension is introduced. It relies on the use of moving quadrics following curve parameterizations, providing a high-order extension of the implicit matrix representations built from their linear counterparts, the moving planes. The matrices we obtain offer new, more compact, implicit representations of rational curves. Their entries are filled by linear and quadratic forms in the space variables and their ranks drop exactly on the curve. Typically, for a general rational curve of degree d we obtain a matrix whose size is half of the size of the corresponding matrix obtained with the moving planes method. We illustrate the advantages of these new matrices with some examples, including the computation of the singularities of a rational curve
Hermite form computation of matrices of differential polynomials
Given a matrix A in F(t)[D;\delta]^{n\times n} over the ring of differential polynomials, we first prove the existence of the Hermite form H of A over this ring. Then we determine degree bounds on U and H such that UA=H. Finally, based on the degree bounds on U and H, we compute the Hermite form H of A by reducing the problem to solving a linear system of equations over F(t). The algorithm requires a polynomial number of operations in F in terms of the input sizes: n, deg_{D} A, and deg_{t} A. When F=Q it requires time polynomial in the bit-length of the rational coefficients as well