Abstract
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.
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References
Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)
Carnicer, J.M., Peña, J.M.: Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1, 173–196 (1993)
Delgado, J., Peña, J.M.: Accurate computations with collocation matrices of rational bases. Appl. Math. Comput. 219, 4354–4364 (2013)
Delgado, J., Peña, J.M.: Accurate computations with collocation matrices of q-Bernstein polynomials. SIAM J. Matrix Anal. Appl. 36, 880–893 (2015)
Demmel, J., Koev, P.: The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl. 27, 42–52 (2005)
Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, Princeton (2011). Princeton Series in Applied Mathematics
Gasca, M., Peña, J.M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25–44 (1992)
Gasca, M., Peña, J.M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–53 (1994)
Gasca, M., Peña, J.M.: On factorizations of totally positive matrices. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and Its Applications, pp. 109–130. Kluver Academic Publishers, Dordrecht (1996)
Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)
Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)
Marco, A., Martínez, J.J.: A fast and accurate algorithm for solving Bernstein–Vandermonde linear systems. Linear Algebra Appl. 422, 616–628 (2007)
Marco, A., Martínez, J.J.: Accurate computations with Said–Ball–Vandermonde matrices. Linear Algebra Appl. 432, 2894–2908 (2010)
Marco, A., Martinez, J.J.: Accurate computation of the Moore–Penrose inverse of strictly totally positive matrices. J. Comput. Appl. Math. 350, 299–308 (2019)
Pinkus, A.: Totally Positive Matrices, Cambridge Tracts in Mathematics, 181. Cambridge University Press, Cambridge (2010)
Acknowledgements
This work was partially supported through the Spanish research grant PGC2018-096321-B-I00 (MCIU/AEI), by Gobierno de Aragón (E41\(\_\)17R ) and by Feder 2014–2020 “Construyendo Europa desde Aragón”.
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Mainar, E., Peña, J.M. & Rubio, B. Accurate computations with Wronskian matrices. Calcolo 58, 1 (2021). https://doi.org/10.1007/s10092-020-00392-4
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DOI: https://doi.org/10.1007/s10092-020-00392-4