a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

analytic subvarieties with many rational points we give a generalization of the classical bombieri schneider lang criterion in transcendence theory we give a local notion of lg germ which is similar to the notion of e function and gevrey condition and which generalize and replace the condition on derivatives in the theorem quoted above let k subset mathbb c be a number field and x a quasi projective variety defined over k let m x be an holomorphic map of finite order from a parabolic riemann surface to x such that the zariski closure of the image of it is strictly bigger then one suppose that for every p in x k cap gamma m the formal germ of m near p is an lg germ then we prove that x k cap gamma m is a finite set then we define the notion of conformally parabolic kahler varieties this generalize the notion of parabolic riemann surface we show that on these varieties we can define a value distribution theory the complementary of a divisor on a compact kahler manifold is conformally parabolic in particular every quasi projective variety is suppose that a is conformally parabolic variety of dimension m over mathbb c with kahler form and a x is an holomorphic map of finite order such that the zariski closure of the image is strictly bigger then m suppose that for every p in x k cap gamma a the image of a is an lg germ then we prove that there exists a current t on a of bidegree 1 1 such that int at wedge omega m 1 explicitly bounded and with lelong number bigger or equal then one on each point in 1 x k in particular if a is affine 1 x k is not zariski dense

preconditioned progressive iterative approximation for triangular bezier patches and its application abstract in this paper we propose a preconditioned progressive iterative approximation for triangular bezier patches firstly the diagonally compensated reduction is used in designing the preconditioner some properties of the preconditioner are shown the validity of the presented iterative method by using lu factorization of the preconditioner is discussed then the preconditioned progressive iterative approximation is used to approximate rational triangular bezier patches the given numerical examples show that the proposed method achieves a great improvement on the convergence rate compared to the existing progressive iterative approximation

error analysis of the bjo rck pereyra algorithms for solving vandermonde systems a forward error analysis is presented for the bjorck pereyra algorithms used for solving vandermonde systems of equations this analysis applies to the case where the points defining the vandermonde matrix are nonnegative and are arranged in increasing order it is shown that for a particular class of vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only being independent of the condition number of the coefficient matrix by comparing appropriate condition numbers for the vandermonde problem with the forward error bounds it is shown that the bjorck pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right hand side vector on the computer a technique for computing running a posteriori error bounds is derived several numerical experiments are presented and it is observed that the ordering of the points can greatly affect the solution accuracy

fast and accurate algorithms for jacobi stirling matrices abstract jacobi stirling numbers of first and second kind and their corresponding associated matrices play an important role in combinatorics this paper presents an efficient method that guarantees that the computation of the singular values and the inverses of these matrices can be performed with high relative accuracy numerical examples are included

limit theorems for self similar tilings we study deviation of ergodic averages for dynamical systems given by self similar tilings on the plane and in higher dimensions the main object of our paper is a special family of finitely additive measures for our systems an asymptotic formula is given for ergodic integrals in terms of these finitely additive measures and as a corollary limit theorems are obtained for dynamical systems given by self similar tilings

a fast parallel bjorck pereyra type algorithm for solving cauchy linear equations abstract we propose a new fast o n 2 parallel algorithm for solving cauchy systems of linear equations we perform an a priori rounding error analysis and obtain componentwise bounds for the forward backward and residual errors these bounds indicate that for the class of totally positive cauchy matrices the new algorithm is forward and backward stable producing a remarkably high relative accuracy in particular hilbert linear systems often considered to be too ill conditioned to be attacked can be rapidly solved with high precision the results indicate a close resemblance between the numerical properties of cauchy matrices and the much studied vandermonde matrices in fact our proposed cauchy solver is an analog of the well known bjorck pereyra algorithm for vandermonde systems as a by product we obtain favorably backward error bounds for the original bjorck pereyra algorithm several computed examples illustrate a connection of high relative accuracy to the concepts of effective well conditioning and total positivity

accurate computations with collocation matrices of rational bases the rational bernstein basis plays a crucial role in computer aided geometric design the collocation matrices of this basis are called rational bernstein vandermonde matrices in this paper we provide algorithms for computing the bidiagonal decomposition of rational bernstein vandermonde matrices and their inverses with high relative accuracy similar results are obtained for the collocation matrices of another important rational basis the rational said ball basis it is also shown that these algorithms can be used to perform accurately some computations with these matrices such us the calculation of their inverses their eigenvalues or their singular values numerical experiments illustrate the results

accurate bidiagonal decomposition of totally positive h bernstein vandermonde matrices and applications abstract an algorithm for the computation of the bidiagonal decomposition of strictly totally positive h bernstein vandermonde matrices is presented these matrices are collocation matrices of h bernstein bases a generalization of the bernstein basis of the space of polynomials of degree less than or equal to n the algorithm is fast its computational complexity is o n 2 and has high relative accuracy the computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices linear system solving eigenvalue computation and inverse computation numerical experiments are also included

factorizations of cauchy vandermonde matrices abstract this paper analyzes the factorization of the inverse of a cauchy vandermonde matrix as a product of bidiagonal matrices factorizations of cauchy vandermonde matrices are also considered the results are applied to derive a fast algorithm for solving linear systems whose coefficient matrices are cauchy vandermonde matrices

extremal and optimal properties of b bases collocation matrices totally positive matrices are related with the shape preserving representations of a space of functions the normalized b basis of the space has optimal shape preserving properties bernstein polynomials b splines and rational bernstein bases are examples of normalized b bases it is proven that the minimal eigenvalue and singular value of a collocation matrix of a normalized b basis is bounded below by the minimal eigenvalue and singular value of the corresponding collocation matrix of any normalized totally positive basis of the same space the optimal conditioning for the infty norm of a collocation matrix of a normalized b basis among all the normalized totally positive bases of a space of functions is also shown numerical examples confirm the theoretical results and answer related questions

accurate computations with said ball vandermonde matrices abstract a generalization of the vandermonde matrices which arise when the power basis is replaced by the said ball basis is considered when the nodes are inside the interval 0 1 then those matrices are strictly totally positive an algorithm for computing the bidiagonal decomposition of those said ball vandermonde matrices is presented which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is shown to be fast and to guarantee high relative accuracy some numerical experiments which illustrate the good behaviour of the algorithm are included

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

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

bernstein bezoutian matrices and curve implicitization a new application of bernstein bezoutian matrices a type of resultant matrices constructed when the polynomials are given in the bernstein basis is presented in particular the approach to curve implicitization through sylvester and bezout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational parametric equations of the curve are expressed in the bernstein basis avoiding the basis conversion from the bernstein to the power basis the coefficients of the implicit equation are computed in the bivariate tensor product bernstein basis and their computation involves the bidiagonal factorization of the inverses of certain totally positive matrices

polynomial least squares fitting in the bernstein basis the problem of polynomial least squares fitting in which the usual monomial basis is replaced by the bernstein basis is considered the coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular bernstein vandermonde matrix in order to use the method based on the qr decomposition of a the first stage consists of computing the bidiagonal decomposition of the coefficient matrix a starting from that bidiagonal decomposition an algorithm for obtaining the qr decomposition of a is the applied finally a triangular system is solved by using the bidiagonal decomposition of the r factor of a some numerical experiments showing the behavior of this approach are included

accurate computation of the moore penrose inverse of strictly totally positive matrices abstract the computation of the moore penrose inverse of structured strictly totally positive matrices is addressed since these matrices are usually very ill conditioned standard algorithms fail to provide accurate results an algorithm based on the q r factorization and which takes advantage of the special structure and the totally positive character of these matrices is presented the first stage of the algorithm consists of the accurate computation of the bidiagonal decomposition of the matrix numerical experiments illustrating the good behavior of our approach are included

bernstein bezoutian matrices several computational and structural properties of bezoutian matrices expressed with respect to the bernstein polynomial basis are shown the exploitation of such properties allows the design of fast algorithms for the solution of bernstein bezoutian linear systems without never making use of potentially ill conditioned reductions to the monomial basis in particular we devise an algorithm for the computation of the greatest common divisor gcd of two polynomials in bernstein form a series of numerical tests are reported and discussed which indicate that bernstein bezoutian matrices are much less sensitive to perturbations of the coefficients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the bernstein and the power basis

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

linearizations of matrix polynomials in bernstein bases we discuss matrix polynomials expressed in a bernstein basis and the associated polynomial eigenvalue problems using mobius transformations of matrix polynomials large new families of strong linearizations are generated matrix polynomials that are structured with respect to a bernstein basis together with their associated spectral symmetries are also investigated the results in this paper apply equally well to scalar polynomials and include the development of new companion pencils for polynomials expressed in a bernstein basis

richardson method and totally nonnegative linear systems we show that modified richardson method converges for any nonsingular totally nonnegative stochastic matrix for any choice of the parameter between 0 and 2 we present a variant of the modified richardson method that is convergent for any nonsingular totally nonnegative matrix we obtain the optimal parameter value for this method and give a procedure for estimating it numerical experiments are presented