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Inversion formulas and fast algorithms for Löwner-Vandermonde matrices

Karla Rost, Zdenĕk Vavr̆ín
1998 Linear Algebra and its Applications  
Kewords: Fast algorithm; LGwner-Vandermonde matrices '  ...  rational interpolation problems.  ...  Special cases Let us shortly discuss formulas for the inverse of Vandermonde, Liiwner, Cauchy, and Cauchy-Vandermonde matrices.  ... 
doi:10.1016/s0024-3795(97)10011-8 fatcat:wxpwqteyefg3plxugz4hamerfq

Factorizations of Cauchy-Vandermonde matrices

J.J. Martínez, J.M. Peña
1998 Linear Algebra and its Applications  
This Paper analyzes the factorization of the inverse of a Cauchy-Vandermonde matrix as a product of bidiagonal matrices. Factorizations of Cauchy-Vandermonde matrices are also considered.  ...  The results are applied to derive a fast algorithm for solving linear Systems whose coefficient matrices are Cauchy-Vandermonde matrices. 0 1998 Elsevier Science Inc. All rights reserved.  ...  In contrast, in [13] the main tool is the construction of the triangular matrices L and U by using the connection between a Cauchy-Vandermonde linear System and the rational interpolation Problem associated  ... 
doi:10.1016/s0024-3795(98)10073-3 fatcat:2v4xz4uoxvfpdl5lnu55ewoykm

Quasiexponential polynomial bases in analytic spaces

S. S. Linchuk, N. I. Nagnibida
1975 Siberian mathematical journal  
Special attention is given to duality. Numerical results show that the matrices of values of the new bases have smaller condition numbers than the common monomial bases.  ...  It can also be pointed out that the new introduced bases are good candidates for interpolation.  ...  Polynomial interpolation theory has a number of important uses.  ... 
doi:10.1007/bf00969808 fatcat:vvr6rcxka5e5fmkwbgelj5pwlm

Bjorck-Pereyra-type methods and total positivity [article]

Jose-Javier Martinez
2018 arXiv   pre-print
Vandermonde 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.  ...  interpolation could be interpreted in connection with a factorization of the inverse of the Vandermonde matrix as a product of bidiagonal matrices.  ... 
arXiv:1811.08406v1 fatcat:xvld2sfrrrdpxpvrzdkfw7b3tq

Structured inversion of the Bernstein-Vandermonde Matrix [article]

Larry Allen, Robert C. Kirby
2020 arXiv   pre-print
In this paper, we use properties of the Bernstein-Bezout matrix to factor the inverse of the Bernstein-Vandermonde matrix into a difference of products of Hankel, Toeplitz, and diagonal matrices.  ...  Previously, we used the relationship between monomial Bezout matrices and the inverse of Hankel matrices to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz  ...  Properties of Bernstein polynomials lead to special recursive blockwise-structure for finite element matrices [13, 14] .  ... 
arXiv:2005.03251v1 fatcat:zkcvjuaorndf7bga4ttrjvafza

Generalized confluent Cauchy–Vandermonde matrices: displacement structures, inversion formulas and tangential interpolations

Zheng-Hong Yang
2003 Journal of Computational and Applied Mathematics  
By using displacement structure theory fast inversion formulas for these matrices are derived. The tangential interpolation interpretations for associated linear systems with such matrices are given.  ...  Generalized con uent Cauchy and Cauchy-Vandermonde matrices are introduced.  ...  Cauchy and Cauchy-Vandermonde matrices are connected with rational interpolation problem with prescribed poles.  ... 
doi:10.1016/s0377-0427(02)00858-0 fatcat:k5xyb4aexrdztdyu4k3baqifda

Numerical differentiation and the solution of multidimensional Vandermonde systems

G. Galimberti, V. Pereyra
1970 Mathematics of Computation  
We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices.  ...  These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit algorithm is produced to solve systems of linear equations with MV matrices of coefficients.  ...  They are obtained as some special submatrices of Kronecker products of standard Vandermonde matrices.  ... 
doi:10.1090/s0025-5718-1970-0275668-2 fatcat:2yxal5xrgjhbxiotco3flldrau

Numerical Differentiation and the Solution of Multidimensional Vandermonde Systems

G. Galimberti, V. Pereyra
1970 Mathematics of Computation  
We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices.  ...  These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit algorithm is produced to solve systems of linear equations with MV matrices of coefficients.  ...  They are obtained as some special submatrices of Kronecker products of standard Vandermonde matrices.  ... 
doi:10.2307/2004482 fatcat:c7d3zgh6bzcdll2lyeefpawpny

Interpolation function of a general triangular mid-edge finite element

Claude Lacombe, Claude Bédard
1986 Computers and Mathematics with Applications  
An interpolation function for triangular mid-edge finite elements is developed using an algebraic interpolation approach.  ...  A convenient method for deriving shape functions of serendipity type directly and explicitly arises also from the interpolation function proposed above. t Research Student ttAssistant Professor  ...  Interpolation function of a general triangular mid-edge finite element  ... 
doi:10.1016/0898-1221(86)90195-1 fatcat:mhpkykedtrbuhlibkf5uz4ns74

Some Results on Vandermonde Matrices with an Application to Time Series Analysis

André Klein, Peter Spreij
2003 SIAM Journal on Matrix Analysis and Applications  
In this paper we study Stein equations where the coefficient matrices are in companion form.  ...  Solutions to such equations are relatively easy to compute as soon as one knows how to invert a Vandermonde matrix (in the generic case where all eigenvalues have multiplicity one) or a confluent Vandermonde  ...  We thank one of the referees for many (detailed) suggestions that helped us in writing the revision.  ... 
doi:10.1137/s0895479802402892 fatcat:exnjhvosj5gqjbsq4xo3ukgqm4

Interpolation theory and λ-matrices

K Hariche, E.D Denman
1989 Journal of Mathematical Analysis and Applications  
This paper shows how interpolation theory can be used in the structural analysis of I-matrices, which often occur in the description of the dynamics of multivariable systems.  ...  Using scalar interpolatory polynomials for the linear I-matrix case and interpolatory I-matrices for the general case, a set of spectral projectors is derived.  ...  Using f(A)= Mi,o(;l) Mi,,(I) in (1.7) leads to property (iii). As a special case of (iii) for I = 0, the idempotency property is obtained.  ... 
doi:10.1016/0022-247x(89)90058-9 fatcat:ofusnajq35crbeuii3b2xqfhdy

Displacement structure approach to Cauchy and Cauchy–Vandermonde matrices: inversion formulas and fast algorithms

Zheng-Hong Yang, Yong-Jian Hu
2002 Journal of Computational and Applied Mathematics  
We show that the Cauchy and Cauchy-Vandermonde matrices satisfy some special type of matrix equations.  ...  In this paper, we use another way called displacement structure approach to deal with matrices of this kind.  ...  With c and d we associate two classes of matrices. Deÿnition 1.  ... 
doi:10.1016/s0377-0427(01)00378-8 fatcat:pdlkni2645b2ll2g2ast457fim

Recursive solution of Löwner-Vandermonde systems of equations. I

Karla Rost, Zdeněk Vavřín
1996 Linear Algebra and its Applications  
In the special case wi = 0 for all i < 0 LK(W, 2) = != (I ,=I and A is a Cauchy-Vandermonde matrix. C:auch~-Vandermonde matrices were considered for the first time in [4] . Linear (see [6] ).  ...  INTRODUCTION Throughout this paper we associate two given complex vectors [ (1.4) (In case L = 0, A is a Liiwner matrix; m case K = 0, A is a Vandermonde matrix.)  ... 
doi:10.1016/0024-3795(94)00040-9 fatcat:lyjcs66f7bg4bmef4ypdwsecua

Non-Unit Bidiagonal Matrices for Factorization of Vandermonde Matrices

R. Purushothaman Nair
2015 Bulletin of Mathematical Sciences and Applications  
Also it is demonstrated that factors of Vandermonde and inverse Vandermonde matrices can be conveniently presented using the matrices introduced here.  ...  Since a Vandermonde matrix can also be constructed using given n quantities, it is established in this paper that Vandermonde matrices can be factorized in a simple way by applying these bidiagonal matrices  ...  This is a pointer to the association of the eigen vectors of matrices (2.9) to such special matrices which are basically generated out of given n distinct quantities.  ... 
doi:10.18052/www.scipress.com/bmsa.12.1 fatcat:6epx2moa5fagzjfzghczmeylzm

Parallel computation of determinants of matrices with polynomial entries

Ana Marco, José-Javier Martı́nez
2004 Journal of symbolic computation  
It is based on classical multivariate Lagrange polynomial interpolation, and it exploits the Kronecker product structure of the coefficient matrix of the linear system associated with the interpolation  ...  Since these matrices are Vandermonde matrices we use the Björck-Pereyra algorithm to solve those linear systems.  ...  Vandermonde matrices).  ... 
doi:10.1016/j.jsc.2003.11.002 fatcat:aswpcu2u5nda7cte23ebvyiwwi
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