Accurate computations with Wronskian matrices

In this paper we provide algorithms for computing the bidiagonal decomposition of the Wronskian matrices of the monomial basis of polynomials and of the basis of exponential polynomials. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these Wronskian matrices, such as the calculation of their inverses, their eigenvalues or their singular values and the solutions of some linear systems. Numerical experiments illustrate the results

Polynomial Total Positivity and High Relative Accuracy Through Schur Polynomials

In this paper, Schur polynomials are used to provide a bidiagonal decomposition of polynomial collocation matrices. The symmetry of Schur polynomials is exploited to analyze the total positivity on some unbounded intervals of a relevant class of polynomial bases. The proposed factorization is used to achieve relative errors of the order of the unit round-off when solving algebraic problems involving the collocation matrix of relevant polynomial bases, such as the Hermite basis. The numerical experimentation illustrates the accurate results obtained when using the findings of the paper

On the stability of the representation of finite rank operators

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered

Accurate Computations with Collocation and Wronskian Matrices of Jacobi Polynomials

In this paper an accurate method to construct the bidiagonal factorization of collocation and Wronskian matrices of Jacobi polynomials is obtained and used to compute with high relative accuracy their eigenvalues, singular values and inverses. The particular cases of collocation and Wronskian matrices of Legendre polynomials, Gegenbauer polynomials, Chebyshev polynomials of the first and second kind and rational Jacobi polynomials are considered. Numerical examples are included. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

Accurate computations with Wronskian matrices of Bessel and Laguerre polynomials

This paper provides an accurate method to obtain the bidiagonal factorization of Wronskian matrices of Bessel polynomials and of Laguerre polynomials. This method can be used to compute with high relative accuracy their singular values, the inverse of these matrices, as well as the solution of some related systems of linear equations. Numerical examples illustrating the theoretical results are included. © 2022 The Author

High relative accuracy through Newton bases

Bidiagonal factorizations for the change of basis matrices between monomial and Newton polynomial bases are obtained. The total positivity of these matrices is characterized in terms of the sign of the nodes of the Newton bases. It is shown that computations to high relative accuracy for algebraic problems related to these matrices can be achieved whenever the nodes have the same sign. Stirling matrices can be considered particular cases of these matrices, and then computations to high relative accuracy for collocation and Wronskian matrices of Touchard polynomial bases can be obtained. The performed numerical experimentation confirms the accurate solutions obtained when solving algebraic problems using the proposed factorizations, for instance, for the calculation of their eigenvalues, singular values, and inverses, as well as the solution of some linear systems of equations associated with these matrices

Total positivity and accurate computations with Gram matrices of Said-Ball bases

In this article, it is proved that Gram matrices of totally positive bases of the space of polynomials of a given degree on a compact interval are totally positive. Conditions to guarantee computations to high relative accuracy with those matrices are also obtained. Furthermore, a fast and accurate algorithm to compute the bidiagonal factorization of Gram matrices of the Said-Ball bases is obtained and used to compute to high relative accuracy their singular values and inverses, as well as the solution of some linear systems associated with these matrices. Numerical examples are included

Total positivity and accurate computations with Gram matrices of Bernstein bases

In this paper, an accurate method to construct the bidiagonal factorization of Gram (mass) matrices of Bernstein bases of positive and negative degree is obtained and used to compute with high relative accuracy their eigenvalues, singular values and inverses. Numerical examples are included

Accurate computations with Gram and Wronskian matrices of geometric and Poisson bases

In this paper we deduce a bidiagonal decomposition of Gram and Wronskian matrices of geometric and Poisson bases. It is also proved that the Gram matrices of both bases are strictly totally positive, that is, all their minors are positive. The mentioned bidiagonal decompositions are used to achieve algebraic computations with high relative accuracy for Gram and Wronskian matrices of these bases. The provided numerical experiments illustrate the accuracy when computing the inverse matrix, the eigenvalues or singular values or the solutions of some linear systems, using the theoretical results

Accurate computations with matrices related to bases {tie¿t}

The total positivity of collocation, Wronskian and Gram matrices corresponding to bases of the form (eλt,teλt,…,tneλt) is analyzed. A bidiagonal decomposition providing the accurate numerical resolution of algebraic linear problems with these matrices is derived. The numerical experimentation confirms the accuracy of the proposed methods