# On Efficient Sparse Integer Matrix Smith Normal Form Computations

@article{Dumas2001OnES, title={On Efficient Sparse Integer Matrix Smith Normal Form Computations}, author={Jean-Guillaume Dumas and B. David Saunders and Gilles Villard}, journal={J. Symb. Comput.}, year={2001}, volume={32}, pages={71-99} }

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (elimination and/or black box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in…

## 92 Citations

### Efficient computation of the characteristic polynomial

- Computer Science, MathematicsISSAC
- 2005

A probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices.

### Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms

- MathematicsAlgebra, Geometry, and Software Systems
- 2003

Alternative approaches to the calculation of simplicial homology are described and motivating examples and actual experiments with the GAP package that was implemented by the authors are described.

### Computing the Rank of Large Sparse Matrices over Finite Fields

- Computer Science
- 2002

It is proved here that the probabilistic, blackbox, Wiedemann algorithm is the fastest iterative variant of the Krylov methods to compute the minimal polynomial or the rank of a sparse matrix.

### Computing the Rank of Large Sparse Matrices over Finite Fields

- Computer Science
- 2022

It is proved here that the probabilistic, blackbox, Wiedemann algorithm is the fastest iterative variant of the Krylov methods to compute the minimal polynomial or the rank of a sparse matrix.

### On finding multiplicities of characteristic polynomial factors of black-box matrices

- Computer Science, MathematicsISSAC '09
- 2009

Altered in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.

### Finite field linear algebra subroutines

- Mathematics, Computer ScienceISSAC '02
- 2002

Very efficient implementations of finite field dot products, matrix-vector products and matrix-matrix products (namely the symbolic equivalent of level 1, 2 and 3 BLAS) are presented.

### Smith Normal Form over Local Rings and Related Problems

- Computer Science, Mathematics
- 2017

The ultimate goal is to extend the applications of numerical algorithms for computing eigenvalues to computing the invariant factors of symbolic matrices as well as to design an algorithm for computing uniform samples from the nullspace.

### Sheafhom: Software for Sparse Integer Matrices

- Computer Science, Mathematics
- 2007

The quotient of the free abelian group on 26 letters by sets of words in the dictionary is found, in the spirit of [8].

### Bounds on the coefficients of the characteristic and minimal polynomials

- Computer Science, MathematicsArXiv
- 2006

Algorithms are presented to compute more precise input-dependant bounds on these co- efficients to perform deterministic Chinese remaindering of the characteristic or minimal polynomial of an integer matrix.

### Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages

- Computer Science, MathematicsTOMS
- 2008

This article studies high performance implementations of basic linear algebra routines over word size prime fields: especially matrix multiplication, with the goal being to provide an exact alternate to the numerical BLAS library.

## References

SHOWING 1-10 OF 54 REFERENCES

### Integer Smith form via the valence: experience with large sparse matrices from homology

- Computer ScienceISSAC
- 2000

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo…

### Probabilistic Computation of the Smith Normal Form of a Sparse Integer Matrix

- Computer ScienceANTS
- 1996

A new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ ℤm×n, which suffers from no “fill-in” or intermediate value explosion, and uses very little additional space.

### Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix

- MathematicsSIAM J. Comput.
- 1979

Recently, Frumkin [9] pointed out that none of the well-known algorithms that transform an integer matrix into Smith [16] or Hermite [12] normal form is known to be polynomially bounded in its runn...

### 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].

### Near optimal algorithms for computing Smith normal forms of integer matrices

- Computer Science, MathematicsISSAC '96
- 1996

We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d For the case of matrices over Z d we present an algorithm that computes the…

### Efficient parallel solution of sparse systems of linear diophantine equations

- Computer Science, MathematicsPASCO '97
- 1997

We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This…

### Analysis of Coppersmith's Block Wiedemann Algorithm for the Parallel Solution of Sparse Linear Systems

- Computer ScienceAAECC
- 1993

It is proved that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors.

### Effective polynomial computation

- Mathematics, Computer ScienceThe Kluwer international series in engineering and computer science
- 1993

1. Euclid's Algorithm. 2. Continued Fractions. 3. Diophantine Equations. 4. Lattice Techniques. 5. Arithmetic Functions. 6. Residue Rings. 7. Polynomial Arithmetic. 8. Polynomial GCD's: Classical…

### Solving sparse linear equations over finite fields

- Computer Science, MathematicsIEEE Trans. Inf. Theory
- 1986

A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described and a probabilistic algorithm is shown to exist for finding the determinant of a square matrix.

### Finite Field Arithmetic Using the Connection Machine

- Computer ScienceCAP
- 1990

A Connection Machine (model CM-2) with 32K processors has been used to carry out calculations in finite fields with as many as 221 elements and of various characteristics; a typical calculation is to…