Explicit factorization of the Vandermonde matrix

Oruç, Halil; Phillips, G. M.

doi:10.1016/s0024-3795(00)00124-5

Cited by 58 publications

(46 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is well known that Vandermonde and confluent Vandermonde matrices V can be factorized into a lower triangular matrix L and an upper triangular matrix U where V = LU , see for instance [42,43]. In what follows, we show that the same applies for the functional confluent Vandermonde matrix (7)-(9) by establishing explicit formulas for L and U where Υ = LU .…”

Section: On Functional Confluent Vandermonde Matricesmentioning

confidence: 56%

Characterizing the Codimension of Zero Singularities for Time-Delay Systems

Boussaada

Niculescu

2016

Acta Appl Math

View full text Add to dashboard Cite

The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author's knowledge, the bounds of such a multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters' algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.

show abstract

Section: On Functional Confluent Vandermonde Matricesmentioning

confidence: 56%

Characterizing the Codimension of Zero Singularities for Time-Delay Systems

Boussaada

Niculescu

2016

Acta Appl Math

View full text Add to dashboard Cite

show abstract

“…In this section we report some results concerning the factorization of a Vandermonde matrix (actually, its transpose), to be used later: though most of them are known (see, for example, [1,9,14,15]), nevertheless, they are here cast in the most general and appropriate form for subsequent reference.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Economical error estimates for block implicit methods for ODEs via deferred correction

Brugnano¹,

Magherini²

2006

Applied Numerical Mathematics

View full text Add to dashboard Cite

Deferred correction is a widely used tool for improving the numerical approximation to the solution of ODE problems [10,11,12,13,16,17,18,19,20,21,23]. Indeed, it allows to estimate the error due to the use of discrete methods. Such an estimate may be a global one, in the case of continuous BVPs, or a local one, when IVPs are to be approximated [2,7]. Recently, it has been implemented in the computational code BiM [5] for the numerical solution of stiff ODE-IVPs. In this paper we analyze deferred correction in connection with the methods used in that code, resulting in an overall simplification of the procedure, due to the properties of the underlying methods. The analysis is then extended to more general methods.

show abstract

“…One can use the formulas (2.4) to compute explicit formulas for the bidiagonal decompositions of Vandermonde [10,23,33] Neville elimination is just one of eight analogous, but slightly different methods to eliminate a TN matrix A using only adjacent rows and columns [19, section 4]. Each method yields a decomposition of A as a product of nonnegative bidiagonal matrices with analogous but different nonzero patterns.…”

Section: Bidiagonal Decompositions Of Tn Matricesmentioning

confidence: 99%

Accurate Computations with Totally Nonnegative Matrices

Koev¹

2007

SIAM J. Matrix Anal. & Appl.

125

165

View full text Add to dashboard Cite

Abstract. We consider the problem of performing accurate computations with rectangular (m × n) totally nonnegative matrices. The matrices under consideration have the property of having a unique representation as products of nonnegative bidiagonal matrices. Given that representation, one can compute the inverse, LDU decomposition, eigenvalues, and SVD of a totally nonnegative matrix to high relative accuracy in O(max(m 3 , n 3 )) time-much more accurately than conventional algorithms that ignore that structure. The contribution of this paper is to show that the high relative accuracy is preserved by operations that preserve the total nonnegativity-taking a product, re-signed inverse (when m = n), converse, Schur complement, or submatrix of a totally nonnegative matrix, any of which costs at most O (max(m 3 , n 3 )). In other words, the class of totally nonnegative matrices for which we can do numerical linear algebra very accurately in O(max(m 3 , n 3 )) time (namely, those for which we have a product representation via nonnegative bidiagonals) is closed under the operations listed above.

show abstract

Explicit factorization of the Vandermonde matrix

Cited by 58 publications

References 11 publications

Characterizing the Codimension of Zero Singularities for Time-Delay Systems

Characterizing the Codimension of Zero Singularities for Time-Delay Systems

Economical error estimates for block implicit methods for ODEs via deferred correction

Accurate Computations with Totally Nonnegative Matrices

Contact Info

Product

Resources

About