B. Saunders - Academia.edu (original) (raw)
Papers by B. Saunders
Lecture Notes in Computer Science, 1985
ABSTRACT An implementation of packages is described. The implementation is based on viewing packa... more ABSTRACT An implementation of packages is described. The implementation is based on viewing packages as substitutions. Using a few operations on packages it is possible to build any package, from a small collection of basic packages. Generic packages can be built, that can be used as building blocks for new packages. The packages defined by the model lend themselves well to algebraic manipulation.
Proceedings of the 2009 international symposium on Symbolic and algebraic computation, 2009
We present algorithms and heuristics to compute the characteristic polynomial of a matrix given i... more We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.
Lecture Notes in Computer Science, 2010
To maximize efficiency in time and space, allocations and deallocations, in the exact linear alge... more To maximize efficiency in time and space, allocations and deallocations, in the exact linear algebra library LinBox, must always occur in the founding scope. This provides a simple lightweight allocation model. We present this model and its usage for the rebinding of matrices between different coefficient domains. We also present automatic tools to speed-up the compilation of template libraries and a software abstraction layer for the introduction of transparent parallelism at the algorithmic level.
Proceedings of the 2000 international symposium on Symbolic and algebraic computation, 2000
We present algorithms to compute the Smith Normal Form of matrices over two families of local rin... more We present algorithms to compute the Smith Normal Form of matrices
over two families of local rings. The algorithms use the black-box
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 n × n matrix A over the ring F[z]/(f
e
), where f
e
is a power
of an irreducible polynomial f ∈ F[z] of degree d, our algorithm requires
O(ηde2n) operations in F, where our black-box is assumed to
require O(η) operations in F to compute a matrix-vector product by
a vector over F[z]/(f
e
) (and η is assumed greater than nde). The algorithm
only requires additional storage for O(nde) elements of F. In
particular, if η = O˜(nde), then our algorithm requires only O˜(n
2d
2
e
3
)
operations in F, which is an improvement on known dense methods
for small d and e.
For the ring Z/peZ, where p is a prime, we give an algorithm which
is time- and memory-efficient when the number of nontrivial invariant
factors is small. We describe a method for dimension reduction while
preserving the invariant factors. The time complexity is essentially
linear in µnre log p, where µ is the number of operations in Z/pZ to
evaluate the black-box (assumed greater than n) and r is the total
number of non-zero invariant factors. To avoid the practical cost of
conditioning, we give a Monte Carlo certificate, which at low cost,
provides either a high probability of success or a proof of failure. The
quest for a time- and memory-efficient solution without restrictions
on the number of nontrivial invariant factors remains open. We offer
a conjecture which may contribute toward that end.
Lecture Notes in Computer Science, 1985
ABSTRACT An implementation of packages is described. The implementation is based on viewing packa... more ABSTRACT An implementation of packages is described. The implementation is based on viewing packages as substitutions. Using a few operations on packages it is possible to build any package, from a small collection of basic packages. Generic packages can be built, that can be used as building blocks for new packages. The packages defined by the model lend themselves well to algebraic manipulation.
Proceedings of the 2009 international symposium on Symbolic and algebraic computation, 2009
We present algorithms and heuristics to compute the characteristic polynomial of a matrix given i... more We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.
Lecture Notes in Computer Science, 2010
To maximize efficiency in time and space, allocations and deallocations, in the exact linear alge... more To maximize efficiency in time and space, allocations and deallocations, in the exact linear algebra library LinBox, must always occur in the founding scope. This provides a simple lightweight allocation model. We present this model and its usage for the rebinding of matrices between different coefficient domains. We also present automatic tools to speed-up the compilation of template libraries and a software abstraction layer for the introduction of transparent parallelism at the algorithmic level.
Proceedings of the 2000 international symposium on Symbolic and algebraic computation, 2000
We present algorithms to compute the Smith Normal Form of matrices over two families of local rin... more We present algorithms to compute the Smith Normal Form of matrices
over two families of local rings. The algorithms use the black-box
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 n × n matrix A over the ring F[z]/(f
e
), where f
e
is a power
of an irreducible polynomial f ∈ F[z] of degree d, our algorithm requires
O(ηde2n) operations in F, where our black-box is assumed to
require O(η) operations in F to compute a matrix-vector product by
a vector over F[z]/(f
e
) (and η is assumed greater than nde). The algorithm
only requires additional storage for O(nde) elements of F. In
particular, if η = O˜(nde), then our algorithm requires only O˜(n
2d
2
e
3
)
operations in F, which is an improvement on known dense methods
for small d and e.
For the ring Z/peZ, where p is a prime, we give an algorithm which
is time- and memory-efficient when the number of nontrivial invariant
factors is small. We describe a method for dimension reduction while
preserving the invariant factors. The time complexity is essentially
linear in µnre log p, where µ is the number of operations in Z/pZ to
evaluate the black-box (assumed greater than n) and r is the total
number of non-zero invariant factors. To avoid the practical cost of
conditioning, we give a Monte Carlo certificate, which at low cost,
provides either a high probability of success or a proof of failure. The
quest for a time- and memory-efficient solution without restrictions
on the number of nontrivial invariant factors remains open. We offer
a conjecture which may contribute toward that end.