Computing hermite forms of polynomial matrices

@inproceedings{Gupta2011ComputingHF,
  title={Computing hermite forms of polynomial matrices},
  author={Somi Gupta and Arne Storjohann},
  booktitle={ISSAC '11},
  year={2011}
}
This paper presents a new algorithm for computing the Hermite form of a polynomial matrix. Given a nonsingular <i>n</i> x <i>n</i> matrix <i>A</i> filled with degree <i>d</i> polynomials with coefficients from a field, the algorithm computes the Hermite form of <i>A</i> using an expected number of (<i>n</i>3<i>d</i>)<sup>1+o(1)</sup> field operations. This is the first algorithm that is both softly linear in the degree <i>d</i> and softly cubic in the dimension <i>n</i>. The algorithm is… 

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References

SHOWING 1-10 OF 34 REFERENCES
A linear space algorithm for computing the hermite normal form
TLDR
The presented algorithm has the same time complexity of the asymptotically fastest (but space inefficient) algorithms and a heuristic algorithm for HNF that achieves a substantial speedup when run on randomly generated input matrices.
On the complexity of polynomial matrix computations
TLDR
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 Popov and Hermite forms of polynomial matrices
TLDR
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.
Normal forms for general polynomial matrices
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
TLDR
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].
On computing the determinant and Smith form of an integer matrix
TLDR
A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix by computing the Smith form of the integer matrix an extremely useful canonical form in itself.
A uniform approach for the fast computation of Matrix-type Padé approximants
TLDR
A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants, which generalizes previous work by Van Barel and Bultheel and, in a more general form, by Beckermann.
A course in computational algebraic number theory
  • H. Cohen
  • Computer Science, Mathematics
    Graduate texts in mathematics
  • 1993
TLDR
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Shifted normal forms of polynomial matrices
TLDR
lIotl gives a fractiorl-frw algorithm for computing niatris riormal forms and is able to c11lbct1 tlic probleni of conqmtiug il normal forni into 0Iic of deterniiJIing a sliift.
...
...