Juan Manuel Peña

Total positivity and Neville elimination

Abstract Neville elimination is described in terms of Schur complements of matrices and used to improve some well-known characterizations of totally positive and strictly totally positive matrices.

Shape preserving representations and optimality of the Bernstein basis

This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bézier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ∞) is also totally positive.

Shape preserving alternatives to the rational Bézier model

Abstract We discus several alternatives to the rational Bezier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding B-bases in general spaces. We also revisit the C-Bezier curves presented by Zhang (1996), which coincide with the helix spline segments developed by Pottmann and Wagner (1994), and are nothing else than curves expressed in the normalized B-basis of the space P 1 = span {1,t, cos t, sin t} . Such curves provide a valuable alternative to the rational Bezier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2 = span {1,t, cos t, sin t, cos 2t, sin 2t} , Q= span {1,t,t 2 , cos t, sin t} and I = span {1,t, cos t, sin t,t cos t,t sin t} are also suitable for shape preserving design, and we find their normalized B-basis.

Totally positive bases for shape preserving curve design and optimality of B-splines

Abstract Normalized totally positive (NTP) bases present good shape preserving properties when they are used in Computer Aided Geometric Design. Here we characterize all the NTP bases of a space and obtain a test to know if they exist. Furthermore, we construct the NTP basis with optimal shape preserving properties in the sense of (Goodman and Said, 1991), that is, the shape of the control polygon of a curve with respect to the optimal basis resembles with the highest fidelity the shape of the curve among all the control polygons of the same curve corresponding to NTP bases. In particular, this is the case of the B-spline basis in the space of polynomial splines. Further examples are given.

Shape preserving representations for trigonometric polynomial curves

Abstract This paper has two main goals. Firstly, we show that the space of trigonometric polynomials Tm = span{1, cost, sint, ..., cosmt, sinmt} is not suitable for those methods of CAGD which use control polygons. It is well-known that the bases with good shape preserving properties are the normalized totally positive bases and we prove here that Tm does not possess normalized totally positive bases. Secondly, we show that the space Cm = span{1, cost, ..., cosmt} is suitable for design purposes using control polygons. In fact, we construct a basis Cm of Cm with optimal shape preserving properties and analyze some aspects for the computation of the corresponding curves.

On Factorizations of Totally Positive Matrices

Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given.

A matricial description of Neville elimination with applications to total positivity

Abstract The Neville elimination process, used by the authors in some previous papers in connection with totally positive matrices, is studied in detail in the case of nonsingular matrices. A wide class of matrices is found where Neville elimination has a lower computational cost than Gauss elimination. Finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices, in terms of their Neville elimination and that of their inverses.

A Class of P -Matrices with Applications to the Localization of the Eigenvalues of a Real Matrix

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row means is called a B-matrix. It is proved that the class of B-matrices is a subset of the class of P-matrices. Properties of B-matrices are used to localize the real eigenvalues of a real matrix and the real parts of all eigenvalues of a real matrix.

Total positivity, QR factorization, and Neville elimination

A well-known characterization of nonsingular totally positive matrices is improved: Only the sign of minors with consecutive initial rows or consecutive initial columns has to be checked. On the other hand, a new characterization of such matrices by their

A shape preserving representation with an evaluation algorithm of linear complexity

QR