# 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} }

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 ScienceISSAC
- 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.

### Computation of the nearest non-prime polynomial matrix: Structured low-rank approximation approach

- Mathematics, Computer Science
- 2021

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

- Computer Science, MathematicsISSAC
- 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.

### Computing Lower Rank Approximations of Matrix Polynomials

- Computer Science, MathematicsJ. Symb. Comput.
- 2020

### 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

- 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, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

- Computer Science, MathematicsJ. Complex.
- 2017

### Computing column bases of polynomial matrices

- Computer ScienceISSAC '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 ScienceJ. Symb. Comput.
- 2012

### 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>.

## References

SHOWING 1-10 OF 42 REFERENCES

### 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.

### Essentially optimal computation of the inverse of generic polynomial matrices

- MathematicsJ. Complex.
- 2005

### On fast multiplication of polynomials over arbitrary algebras

- Computer Science, MathematicsActa Informatica
- 2005

This paper generalizes the well-known Sch6nhage-Strassen algorithm for multiplying large integers to an algorithm for dividing polynomials with coefficients from an arbitrary, not necessarily commutative, not always associative, algebra d, and obtains a method not requiring division that is valid for any algebra.

### A pencil approach for embedding a polynomial matrix into a unimodular matrix

- Mathematics, Computer Science
- 1988

A new method for constructing the unimodular embedding of a polynomial matrix $P( \lambda )$ is derived, which uses a fast variant of the staircase algorithm and only requires $O( p^3 )$ operations in contrast to the methods proposed up to now.

### On computing determinants of matrices without divisions

- Mathematics, Computer ScienceISSAC '92
- 1992

An algorithm ~ given that computes the determinant of an n x n matrix with entries fr~rn an arbitrary commutative ring in 0(n3~ ring additions, subtractions, and multiplications; the ‘soft-O” O…

### Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems

- Mathematics, Computer Science
- 1975

It is shown how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization.

### On Wiedemann's Method of Solving Sparse Linear Systems

- Computer ScienceAAECC
- 1991

Douglas Wiedemann’s (1986) landmark approach to solving sparse linear systems over finite fields provides the symbolic counterpart to non-combinatorial numerical methods for solving sparse linear…

### Computation of structural invariants of generalized state-space systems

- Mathematics, Computer ScienceAutom.
- 1994