Fast computation of Smith forms of sparse matrices over local rings

@inproceedings{Elsheikh2012FastCO,
  title={Fast computation of Smith forms of sparse matrices over local rings},
  author={Mustafa Elsheikh and Mark Giesbrecht and Andrew Novocin and B. David Saunders},
  booktitle={ISSAC},
  year={2012}
}
We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the <i>black-box</i> model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. For an <i>n</i> x <i>n</i> matrix <i>A</i> over the ring F[<i>z</i>]/(<i>f</i><sup><i>e</i></sup>), where <i>f</i><sup… 
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References

SHOWING 1-10 OF 38 REFERENCES
Parallel algorithms for matrix normal forms
Fast computation of the Smith form of a sparse integer matrix
TLDR
A new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix and an asymptotically fast algorithm for dense matrices that return the correct answer with a controllable, exponentially small probability of error.
Algorithms for matrix canonical forms
Computing canonical forms of matrices over rings is a classical math¬ ematical problem with many applications to computational linear alge¬ bra. These forms include the Frobenius form over a field,
Faster inversion and other black box matrix computations using efficient block projections
TLDR
The correctness of the algorithm to find rational solutions for sparse systems of linear equations is established by proving the existence of efficient block projections for arbitrary non-singular matrices over sufficiently large fields by incorporating them into existing black-box matrix algorithms to derive improved bounds for the cost of several matrix problems.
Large matrix, small rank
TLDR
It is conjecture that the sequence of ranks of the D(3,k) satisfies a linear recurrence of degree three, and a software infrastructure is built to improve efficiency of matrix operations over a small finite field.
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.
Solving sparse linear equations over finite fields
TLDR
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.
On Efficient Sparse Integer Matrix Smith Normal Form Computations
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
On finding multiplicities of characteristic polynomial factors of black-box matrices
TLDR
Altered in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.
Reliable Krylov-based algorithms for matrix null space and rank
TLDR
This report presents a block Lanczos algorithm that is somewhat simpler than block algorithms that are presently in use and provably reliable for computations over large fields and a randomized (Monte Carlo) black box algorithm for matrix rank that is asymptotically faster, in the small field case, than any other that is presently known.
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