7,187 research outputs found
Tensor decomposition and homotopy continuation
A computationally challenging classical elimination theory problem is to
compute polynomials which vanish on the set of tensors of a given rank. By
moving away from computing polynomials via elimination theory to computing
pseudowitness sets via numerical elimination theory, we develop computational
methods for computing ranks and border ranks of tensors along with
decompositions. More generally, we present our approach using joins of any
collection of irreducible and nondegenerate projective varieties
defined over . After computing
ranks over , we also explore computing real ranks. Various examples
are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix
multiplication with zeros. (26 pages, 1 figure
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
From graphs to tensegrity structures: Geometric and symbolic approaches
A form-finding problem for tensegrity structures is studied; given an
abstract graph, we show an algorithm to provide a necessary condition for it to
be the underlying graph of a tensegrity in (typically )
with vertices in general position. Furthermore, for a certain class of graphs
our algorithm allows to obtain necessary and sufficient conditions on the
relative position of the vertices in order to underlie a tensegrity, for what
we propose both a geometric and a symbolic approach.Comment: 17 pages, 8 figures; final versio
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Fast Computation of Smith Forms of Sparse Matrices Over Local Rings
We present algorithms to compute the Smith Normal Form of matrices over two
families of local rings.
The algorithms use the \emph{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 \nxn matrix over the ring \Fzfe, where is a power of an
irreducible polynomial f \in \Fz of degree , our algorithm requires
\bigO(\eta de^2n) operations in \F, where our black-box is assumed to
require \bigO(\eta) operations in \F to compute a matrix-vector product by
a vector over \Fzfe (and is assumed greater than \Pden). The
algorithm only requires additional storage for \bigO(\Pden) elements of \F.
In particular, if \eta=\softO(\Pden), then our algorithm requires only
\softO(n^2d^2e^3) operations in \F, which is an improvement on known dense
methods for small and .
For the ring \ZZ/p^e\ZZ, where 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 where is the number of operations in \ZZ/p\ZZ to evaluate the
black-box (assumed greater than ) and 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.Comment: Preliminary version to appear at ISSAC 201
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