Grobner Basis Computation of Drazin Inverses with Multivariate Rational Function Entries

In this paper we show how to apply Grobner bases to compute the Drazin inverse of a matrix with multivariate rational functions as entries. When the coeficients of the rational functions depend on parameters, we give suficient conditions for the Drazin inverse to specialize properly. In addition, we extend the method to weighted Drazin inverses. We present an empirical analysis that shows a good timing performance of the method

Computation of Moore-Penrose generalized inverses of matrices with meromorphic function entries

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)In this paper, given a field with an involutory automorphism, we introduce the notion of Moore-Penrose field by requiring that all matrices over the field have Moore-Penrose inverse. We prove that only characteristic zero fields can be Moore-Penrose, and that the field of rational functions over a Moore-Penrose field is also Moore-Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K, we find sufficient conditions for the elements in K to ensure that the specialization of the Moore-Penrose inverse is the Moore-Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromeorphic functions being invariant by the involutory automorphism, computes its Moore-Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Cissoid constructions of augmented rational ruled surfaces

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)Given two real affine rational surfaces we derive a criterion for deciding the rationality of their cissoid. Furthermore, when one of the surfaces is augmented ruled and the other is either an augmented ruled or an augmented Steiner surface, we prove that the cissoid is rational. Furthermore, given rational parametrizations of the surfaces, we provide a rational parametrization of the cissoid.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Distance bounds of ϵ-points on hypersurfaces

ϵ-points were introduced by the authors (see [S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2–3) (2004) 627–650 (Special issue); S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147–181; S. Pérez-Díaz, J.R. Sendra, J. Sendra, Distance properties of ϵ-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45–61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of ϵ-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an ϵ-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the ϵ-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an ϵ-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds.Ministerio de Educación y CienciaComunidad de MadridUniversidad de Alcal

Parametrization of aproximate algebraic surfaces by lines

In this paper we present an algorithm for parametrizing approximate algebraic surfaces by lines. The algorithm is applicable to ²-irreducible algebraic surfaces of degree d having an ²–singularity of multiplicity d−1, and therefore it generalizes the existing approximate parametrization algorithms. In particular, given a tolerance ² > 0 and an ²-irreducible algebraic surface V of degree d, the algorithm computes a new algebraic surface V , that is rational, as well as a rational parametrization of V . In addition, in the error analysis we show that the output surface V and the input surface V are close. More precisely, we prove that V lies in the offset region of V at distance, at most, O(² 1 2d )

Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis

An algorithm to parametrize approximately space curves

This is the author’s\ud version of a work that was accepted for publication in\ud Journal of Symbolic Computation. Changes resulting from the publishing\ud process, such as peer review, editing, corrections,\ud structural formatting, and other quality control mechanisms may not be\ud reflected in this document.\ud Changes may have been made to this work since it was submitted for\ud publication.\ud A definitive version was subsequently published in Journal of Symbolic\ud Computation vol. 56 pp. 80-106 (2013).\ud DOI: 10.1016/j.jsc.2013.04.002We present an algorithm that, given a non-rational irreducible\ud real space curve, satisfying certain conditions, computes a rational\ud parametrization of a space curve near the input one. For a given\ud tolerance \epsilon > 0, the algorithm checks whether a planar projection\ud of the given space curve is \epsilon -rational and, in the affirmative\ud case, generates a planar parametrization that is lifted to a space\ud parametrization. This output rational space curve is of the same\ud degree as the input curve, both have the same structure at infinity,\ud and the Hausdorff distance between their real parts is finite.\ud Moreover, in the examples we check that the distance is small.This work has been developed, and partially supported, by the Spanish “Ministerio de Ciencia e\ud Innovación” under the Project MTM2008-04699-C03-01, and by the “Ministerio de Economía y Competitividad”\ud under the project MTM2011-25816-C02-01. All authors belong to the Research Group\ud ASYNACS (Ref. CCEE2011/R34)

Bounding and Estimating the Hausdorff distance between real space algebraic curves

This is the author’s version of a work that was accepted for publication in Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Rueda S.L., Sendra J., Sendra J.R., (2014). "Bounding and Estimating the Hausdorff distance\ud between real space algebraic curves ". Computer Aided Geometric Design. vol 31 (2014)\ud 182-198; DOI 10.1016/j.cagd.2014.02.005In this paper, given two real space algebraic curves, not necessarily bounded,\ud whose Hausdor distance is nite, we provide bounds of their distance. These\ud bounds are related to the distance between the projections of the space curves onto\ud a plane (say, z = 0), and the distance between the z-coordinates of points in the\ud original curves. Using these bounds we provide an estimation method for a bound\ud of the Hausdor distance between two such curves and we check in applications that\ud the method is accurate and fas

Moore-Penrose approach in the Hough transform framework

Maria-Laura Torrente is a member of GNAMPA - Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni of INDAM. J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683).Let F(x, a) be a real polynomial in two sets of variables, x and a, that is linear with respect to one of the variable sets, say a. In this paper, we deal with two of the main steps of the Hough transform framework for the pattern recognition technique to detect loci in images. More precisely, we present an algorithmic process, based on the Moore–Penrose pseudo-inverse, to provide a region of analysis in the parameter space. In addition, we state an upper bound for the sampling distance of the discretization of the parameter space region.Agencia Estatal de Investigació