• Corpus ID: 6688527

# A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix

@article{Labahn2016AFD,
title={A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix},
author={George Labahn and Wei Zhou},
journal={ArXiv},
year={2016},
volume={abs/1602.02049}
}
• Published 5 February 2016
• Mathematics
• ArXiv
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity $O^{\sim}\left(n^{\omega}d\right)$ where $d$ is the degree of $\mathbf{F}$. Here soft-$O$ notation is Big-$O$ with log factors removed and $\omega$ is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal…
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 m x m nonsingular polynomial matrix of degree d in expected ~O(mω d) field operations, where ω is the exponent of matrix
Bases of relations in one or several variables: fast algorithms and applications. (Bases de relations en une ou plusieurs variables : algorithmes rapides et applications)
In this thesis, we study algorithms for a problem of finding relations in one or several variables. It generalizes that of computing a solution to a system of linear modular equations over a

## References

SHOWING 1-10 OF 24 REFERENCES
Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix
The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in [SIAM J. Comput., 8 (1979), pp. 499–507].
Efficient algorithms for order basis computation
• Mathematics, Computer Science
J. Symb. Comput.
• 2012
Computing Popov and Hermite forms of polynomial matrices
These results are obtamed by applying in the matrix case, the techniques used in the scalar case of the gcd of polynomials to the Hermite normal form.
A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants
• Computer Science
• 1994
A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants, and these methods result in fast (and superfast) reliable algorithms for the inversion of stripedHankel, layered Hankel, and (rectangular) block-Hankel matrices.
Computing hermite forms of polynomial matrices
• Computer Science
ISSAC '11
• 2011
This paper presents a new algorithm for computing the Hermite form of a polynomial matrix that is both softly linear in the degree d and softly cubic in the dimension n, and is randomized of the Las Vegas type.
On the complexity of polynomial matrix computations
• Computer Science
ISSAC '03
• 2003
Under the straight-line program model, it is shown that multiplication is reducible to the problem of computing the coefficient of degree <i>d</i> of the determinant and algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication are proposed.
Computing minimal nullspace bases
• Computer Science
ISSAC
• 2012
A deterministic algorithm for the computation of a minimal nullspace basis of an input matrix of univariate polynomials over a field K with <i>m</i> ≤ <i*n</i>.
Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs
• Mathematics, Computer Science
SIAM J. Matrix Anal. Appl.
• 2000
A new set of algorithms for computation of matrix rational interpolants and one-sided matrix greatest common divisors, suitable for computation in exact arithmetic domains where growth of coefficients in intermediate computations is a central concern.