598 research outputs found
Error analysis, perturbation theory and applications of the bidiagonal decomposition of rectangular totally-positive h-Bernstein-Vandermonde matrices
A fast and accurate algorithm to compute the bidiagonal decomposition of rectangular totally positive h-Bernstein-Vandermonde matrices is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices are addressed. The computation of this bidiagonal decomposition is used as the first step for the accurate and efficient computation of the singular values of rectangular totally positive h-Bernstein-Vandermonde matrices and for solving least squares problems whose coefficient matrices are such matrices.Agencia Estatal de InvestigaciĂł
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 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
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
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
Quaternion Singular Value Decomposition based on Bidiagonalization to a Real Matrix using Quaternion Householder Transformations
We present a practical and efficient means to compute the singular value
decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to
a real bidiagonal matrix B using quaternionic Householder transformations.
Computation of the svd of B using an existing subroutine library such as lapack
provides the singular values of A. The singular vectors of A are obtained
trivially from the product of the Householder transformations and the real
singular vectors of B. We show in the paper that left and right quaternionic
Householder transformations are different because of the noncommutative
multiplication of quaternions and we present formulae for computing the
Householder vector and matrix in each case
Computing the complete CS decomposition
An algorithm is developed to compute the complete CS decomposition (CSD) of a
partitioned unitary matrix. Although the existence of the CSD has been
recognized since 1977, prior algorithms compute only a reduced version (the
2-by-1 CSD) that is equivalent to two simultaneous singular value
decompositions. The algorithm presented here computes the complete 2-by-2 CSD,
which requires the simultaneous diagonalization of all four blocks of a unitary
matrix partitioned into a 2-by-2 block structure. The algorithm appears to be
the only fully specified algorithm available. The computation occurs in two
phases. In the first phase, the unitary matrix is reduced to bidiagonal block
form, as described by Sutton and Edelman. In the second phase, the blocks are
simultaneously diagonalized using techniques from bidiagonal SVD algorithms of
Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical
features.Comment: New in v3: additional discussion on efficiency, Wilkinson shifts,
connection with tridiagonal QR iteration. New in v2: additional figures and a
reorganization of the tex
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
- …