# 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} }

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…

## 2 Citations

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

- Mathematics, Computer ScienceISSAC
- 2016

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)

- Computer Science, Mathematics
- 2016

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

- Computer ScienceSIAM J. Comput.
- 1989

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 ScienceJ. Symb. Comput.
- 2012

Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x]

- Computer Science, MathematicsJ. Symb. Comput.
- 2012

Computing Popov and Hermite forms of polynomial matrices

- MathematicsISSAC '96
- 1996

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 ScienceISSAC '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 ScienceISSAC '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 ScienceISSAC
- 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 ScienceSIAM 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.

Essentially optimal computation of the inverse of generic polynomial matrices

- MathematicsJ. Complex.
- 2005