# Computing the rank and a small nullspace basis of a polynomial matrix

@article{Storjohann2005ComputingTR,
title={Computing the rank and a small nullspace basis of a polynomial matrix},
author={Arne Storjohann and Gilles Villard},
journal={Proceedings of the 2005 international symposium on Symbolic and algebraic computation},
year={2005}
}
• Published 11 May 2005
• Mathematics, Computer Science
• Proceedings of the 2005 international symposium on Symbolic and algebraic computation
We reduce the problem of computing the rank and a null-space basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree, d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d. If the latter multiplication is done in MM(n,d)= O~(nωd operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O~MM(n,d…
44 Citations

### Computing the Nearest Rank-Deficient Matrix Polynomial

• Mathematics, Computer Science
ISSAC
• 2017
An iterative algorithm which, on given input sufficiently close to a rank-deficient matrix, produces that matrix and is proven to converge quadratically given a sufficiently good starting point.

### Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix

• Computer Science, Mathematics
ISSAC
• 2022
This work gives an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann and provides a second algorithm which computes the column rank profile of F with a rank-sensitive complexity of O~ (rw-2n(m+d) operations in K.

### Ecient Algorithms for Order Bases Computation

• Computer Science, Mathematics
• 2011
This paper presents two algorithms for the computation of a shifted order basis of an m n matrix of power series over a field K with m n and presents a second algorithm which balances the high degree rows and computes an order basis in the case that the shift is unbalanced but satisfies the condition P n i=1(max(~) ~si) m .

### Hermite form computation of matrices of differential polynomials

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.

### Computing column bases of polynomial matrices

• Computer Science
ISSAC '13
• 2013
This paper presents a deterministic algorithm for the computation of a column basis of an input matrix with m x n input matrix, and shows that the average column degree is bounded by the commonly used matrix degree that is also the maximum column degree of the input matrix.

### Efficient algorithms for order basis computation

• Mathematics, Computer Science
J. Symb. Comput.
• 2012

### 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>.