Factorizations of Cauchy-Vandermonde matrices

Martínez, José‐Javier; Peña, J. M.

doi:10.1016/s0024-3795(98)10073-3

Cited by 36 publications

(37 citation statements)

References 13 publications

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“…In section 7 we prove that BD(V ) and BD(C), and in turn the eigenvalues and the singular values of V and C, will be computed accurately. Similar accurate formulas for BD(A) may be obtained when A is a TP generalized Vandermonde [13] or a Cauchy-Vandermonde [33] matrix.…”

Section: Therefore Being Able To Compute All N 2 Initial Minors Of Amentioning

confidence: 65%

Accurate Eigenvalues and SVDs of Totally Nonnegative Matrices

Koev¹

2005

SIAM J. Matrix Anal. & Appl.

134

205

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Abstract. We consider the class of totally nonnegative (TN) matrices-matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices.We present new O(n 3 ) algorithms that, given the bidiagonal factors of a nonsingular TN matrix, compute its eigenvalues and SVD to high relative accuracy in floating point arithmetic, independent of the conventional condition number. All eigenvalues are guaranteed to be computed to high relative accuracy despite arbitrary nonnormality in the TN matrix.We prove that the entries of the bidiagonal factors of a TN matrix determine its eigenvalues and SVD to high relative accuracy.We establish necessary and sufficient conditions for computing the entries of the bidiagonal factors of a TN matrix to high relative accuracy, given the matrix entries.In particular, our algorithms compute all eigenvalues and the SVD of TN Cauchy, Vandermonde, Cauchy-Vandermonde, and generalized Vandermonde matrices to high relative accuracy.Key words. eigenvalue, singular value, high relative accuracy, totally positive matrix, totally nonnegative matrix, oscillatory matrix, sign regular matrix, bidiagonal matrix AMS subject classifications. 65F15, 15A18 DOI. 10.1137/S08954798034382251. Introduction. The matrices with all minors nonnegative are called totally nonnegative (TN). They appear in a wide area of problems and applications [6,16,19,21,22,31] and can be notoriously ill-conditioned-the Hilbert matrix and the Vandermonde matrix with increasing nonnegative nodes are two examples of many.When traditional algorithms are used to compute the eigenvalues or the singular values of an ill-conditioned TN matrix, only the largest eigenvalues and the largest singular values are computed with guaranteed relative accuracy. The tiny eigenvalues and singular values may be computed with no relative accuracy at all, even though they may be the only quantities of practical interest [8]. Their accurate computation using traditional algorithms is then only possible through an increase in the working precision, which may lead to a drastic increase in the computational time.As our first major contribution we present new O(n 3 ) algorithms that compute all eigenvalues and singular values of a nonsingular TN matrix to high relative accuracy. In particular, all computed eigenvalues and singular values, including the tiniest ones, must have correct sign and leading digits: In contrast, traditional SVD and symmetric eigenvalue algorithms guarantee only small absolute errors in the computed singular values and eigenvalues of TN matricesPreviously proposed nonsymmetric eigenvalue algorithms guarantee only the Perron root of a TN matrix to be computed to high relative accuracy [15]. The accuracy in the smaller eigenvalues is also affected by the angle between left and right eigenvectors.Relative accuracy is lost in the traditional eigenvalue and SVD algorithms because of certain round...

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Section: Therefore Being Able To Compute All N 2 Initial Minors Of Amentioning

confidence: 65%

Accurate Eigenvalues and SVDs of Totally Nonnegative Matrices

Koev¹

2005

SIAM J. Matrix Anal. & Appl.

134

205

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“…Linear system solving. The fast and accurate solution of structured linear systems is a problem that has been studied in the field of numerical linear algebra for different types of structured matrices (see, for example, [5,6,14,33]). Now we Let Ax = b be a linear system whose coefficient matrix A is a square BernsteinVandermonde matrix of order n + 1 generated by the nodes {x i } 1≤i≤n+1 , where 0 < x 1 < · · · < x n+1 < 1.…”

Section: ) Bidiagonal Matrices Of the Formmentioning

confidence: 99%

Accurate computations with totally positive Bernstein-Vandermonde matrices

Marco¹,

Martínez²

2013

ELA

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Abstract. The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein-Vandermonde matrix is considered. Bernstein-Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to n the Bernstein basis instead of the monomial basis.The approach in this paper is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix or of its inverse. The explicit expressions obtained for the determinants involved in the process make the algorithm both fast and accurate. The error analysis of this algorithm for computing this bidiagonal factorization and the perturbation theory for the bidiagonal factorization of totally positive Bernstein-Vandermonde matrices are also carried out.Several applications of the computation with this type of matrices are also pointed out.

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“…The following theorem, which is analogous to Theorem 2.1 in [11] for the case of Cauchy-Vandermonde matrices, gives us the bidiagonal factorization of the inverse of a Bernstein-Vandermonde matrix: Theorem 3.1 Let A = (a i,j ) 1≤i,j≤n+1 be a Bernstein-Vandermonde matrix whose nodes satisfy 0 < x 1 < x 2 < . .…”

Section: Bernstein-vandermonde Matricesmentioning

confidence: 99%

“…Factorizations in terms of bidiagonal matrices are very useful when working with Vandermonde [2], Cauchy [3], Cauchy-Vandermonde [11] and generalized Vandermonde matrices [5].…”

Section: Introductionmentioning

confidence: 99%

Bidiagonal decomposition of totally positive Bernstein‐Vandermonde matrices

Martínez

Marco

2007

Proc Appl Math and Mech

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The class of Bernstein-Vandermonde matrices (a generalization of Vandermonde matrices arising when the monomial basis is replaced by the Bernstein basis) is considered. A convenient ordering of their rows makes these matrices strictly totally positive. By using results related to total positivity and Neville elimination, an algorithm for computing the bidiagonal decomposition of a Bernstein-Vandermonde matrix is constructed. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate. One of the applications of our algorithm is the design of fast and accurate algorithms for solving Lagrange interpolation problems when using the Bernstein basis, an approach useful for the field of Computer Aided Geometric Design since it avoids the stability problems involved with basis transformations between the Bernstein and the monomial bases. A different application consists of the use of the bidiagonal decomposition as an intermediate step of the computation of the eigenvalues and the singular value decomposition of a totally positive Bernstein-Vandermonde matrix.

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Factorizations of Cauchy-Vandermonde matrices

Cited by 36 publications

References 13 publications

Accurate Eigenvalues and SVDs of Totally Nonnegative Matrices

Accurate Eigenvalues and SVDs of Totally Nonnegative Matrices

Accurate computations with totally positive Bernstein-Vandermonde matrices

Bidiagonal decomposition of totally positive Bernstein‐Vandermonde matrices

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