# Computing Popov and Hermite forms of polynomial matrices

@inproceedings{Villard1996ComputingPA, title={Computing Popov and Hermite forms of polynomial matrices}, author={Gilles Villard}, booktitle={International Symposium on Symbolic and Algebraic Computation}, year={1996} }

For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(log~+’ (ml)) pamlel arithmetic steps and O(L(nd) ) processors if the same processor bound holds with time O (logX (rid)) for determining the lexicographically first maximal linearly independent subset…

## 44 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…

### Hermite form computation of matrices of differential polynomials

- Mathematics, Computer Science
- 2009

The Hermite form H of A is computed by reducing the problem to solving a linear system of equations over F(t), which requires a polynomial number of operations in F in terms of the input sizes: n, degDA, and degtA.

### Fast Parallel Algorithms for Matrix Reduction to Normal Forms

- Mathematics, Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 1997

Abstract. We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in ℳn,n(K), where K is an arbitrary commutative field, we…

### Computing Popov Forms of Polynomial Matrices

- Computer Science, Mathematics
- 2012

A Las Vegas algorithm that computes the Popov decomposition of matrices of full row rank is given and it is shown that the problem of transforming a row reduced matrix to Popov form is at least as hard as polynomial matrix multiplication.

### Triangular Factorization of Polynomial Matrices

- Computer Science
- 2000

A simple algorithm to recover H when working over the polynomial ring Fx], F a eld is described, which requires O(n 3 d 2) eld operations when A 2 Fx] nn is nonsingular with degrees of entries bounded by d, and the matrix U is recovered in the same time.

### Algorithms for normal forms for matrices of polynomials and ore polynomials

- Mathematics, Computer Science
- 2003

A fraction-free algorithm for row reduction for matrices of Ore polynomials is obtained by formulating row reduction as a linear algebra problem and this algorithm is used as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov forms, and the Popov form of a polynomial matrix.

### Converting between the Popov and the Hermite form of matrices of differential operators using an FGLM-like algorithm

- Mathematics
- 2010

We consider matrices over a ring K [∂; σ, θ ] of Ore polynomials over a skew field K . Since the Popov and Hermite normal forms are both Gröbner bases (for term over position and position over term…

### 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 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.

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