Accurate and efficient computations with Wronskian matrices of Bernstein and related bases

Mainar, E.; Peña, J. M.; Rubio, B.

doi:10.1002/nla.2423

Cited by 5 publications

(4 citation statements)

References 27 publications

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“…On the other hand, although Wronskian and Gram matrices are very different, interestingly several applications of these matrices are presented together in [57]. The use of the bidiagonal decomposition to achieve accurate computations with different classes of totally positive Wronskian and Gram matrices has been presented, for instance, in [55,[58][59][60].…”

Section: Recent Extensionsmentioning

confidence: 99%

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

Martínez

2024

Axioms

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The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

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Section: Recent Extensionsmentioning

confidence: 99%

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

Martínez

2024

Axioms

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“…where p i,i are the diagonal pivots of the Neville elimination of A, m i,j are the multipliers of the Neville elimination of A and mi,j are the multipliers of the Neville elimination of A T . Looking at the code of Algorithm 1 (which computes the entries of M given in formulas ( 13), (14), and ( 15) in an adequate way), we can assert that it preserves HRA because it satisfies the NIC condition. Its computational cost is of O(n 2 ) arithmetic operations, and it must be observed that the Lagrange-Vandermonde matrix A is not constructed, what implies an additional saving in storage space.…”

Section: Algorithm For the Bidiagonal Decompositionmentioning

confidence: 99%

“…differing) signs, and otherwise only adds or subtracts input data. Some classes of totally positive matrices for which there are fast and accurate algorithms for computing their bidiagonal decompositions (A) are, for instance, Cauchy-Vandermonde, 8 generalized Vandermonde, 9 Bernstein-Vandermonde, 10 Lupaş, 11 generalized Pascal matrices, 12 collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis, 13 Wronskian matrices, 14 or Gram matrices of Bernstein bases. 15 The Lagrange basis of the space Π n (t) of the polynomials of degree less than or equal to n, widely used in polynomial interpolation, is…”

Section: Introductionmentioning

confidence: 99%

“…Some classes of totally positive matrices for which there are fast and accurate algorithms for computing their bidiagonal decompositions

ℬ 𝒟 (A)

are, for instance, Cauchy–Vandermonde, 8 generalized Vandermonde, 9 Bernstein–Vandermonde, 10 Lupaş, 11 generalized Pascal matrices, 12 collocation matrices of the Lupaş‐type (p,q)‐analogue of the Bernstein basis, 13 Wronskian matrices, 14 or Gram matrices of Bernstein bases 15 …”

Section: Introductionmentioning

confidence: 99%

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Accurate bidiagonal decomposition of Lagrange–Vandermonde matrices and applications

Marco,

Martínez,

Viaña

2023

Numerical Linear Algebra App

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Lagrange–Vandermonde matrices are the collocation matrices corresponding to Lagrange‐type bases, obtained by removing the denominators from each element of a Lagrange basis. It is proved that, provided the nodes required to create the Lagrange‐type basis and the corresponding collocation matrix are properly ordered, such matrices are strictly totally positive. A fast algorithm to compute the bidiagonal decomposition of these matrices to high relative accuracy is presented. As an application, the problems of eigenvalue computation, linear system solving and inverse computation are solved in an efficient and accurate way for this type of matrices. Moreover, the proposed algorithms allow to solve fastly and to high relative accuracy some of the cited problems when the involved matrices are collocation matrices corresponding to the standard Lagrange basis, although such collocation matrices are not totally positive. Numerical experiments illustrating the good performance of our approach are also included.

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Total Positivity and Accurate Computations Related to q-Abel Polynomials

Khiar,

Mainar,

Royo-Amondarain

et al. 2024

J Sci Comput

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The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

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Accurate and efficient computations with Wronskian matrices of Bernstein and related bases

Cited by 5 publications

References 27 publications

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

Accurate bidiagonal decomposition of Lagrange–Vandermonde matrices and applications

Total Positivity and Accurate Computations Related to q-Abel Polynomials

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