Numerical treatment of a generalized Vandermonde system of equations

Vel, H. Van de

doi:10.1016/0024-3795(77)90035-0

Cited by 16 publications

(11 citation statements)

References 5 publications

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“…We demonstrate that Björck-Pereyra-type methods are in fact applicable to G. To derive those, one must use the bidiagonal decomposition of G −1 and not Newton's method. Since the Björck-Pereyra-type methods yield more accurate solutions than LU-based algorithms (see section 7), this paper represents a substantial advance over the results in [29].…”

Section: Related Work Björck and Pereyra Proposed An O(nmentioning

confidence: 98%

“…Explicit formulas for the entries of the LU decomposition of an ordinary Vandermonde matrix were obtained in [28] and for the entries of the LU decomposition and the inverse of a generalized Vandermonde matrix in [29].…”

Section: Related Work Björck and Pereyra Proposed An O(nmentioning

confidence: 99%

“…The numerical solution of a generalized Vandermonde linear system was discussed in [29]. In that paper Van de Vel shows how to compute an LU decomposition of a TP generalized Vandermonde matrix G without suffering any subtractive cancellation, thus computing each entry of L and U to high relative accuracy.…”

Section: Related Work Björck and Pereyra Proposed An O(nmentioning

confidence: 99%

“…The "hats" indicate the omission of the symbols they cover. The formula (5.1) was also obtained in [29], where the Schur function is referred to as a "D-determinant." Using (5.1) we can compute G −1 to high relative accuracy componentwise by employing the techniques from [7] for computing the Schur function.…”

Section: The Inverse Of Gmentioning

confidence: 99%

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The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System

Demmel

Koev²

2005

SIAM J. Matrix Anal. & Appl.

129

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Abstract. Vandermonde, Cauchy, and Cauchy-Vandermonde totally positive linear systems can be solved extremely accurately in O(n 2 ) time using Björck-Pereyra-type methods. We prove that Björck-Pereyra-type methods exist not only for the above linear systems but also for any totally positive linear system as long as the initial minors (i.e., contiguous minors that include the first row or column) can be computed accurately. Using this result we design a new O(n 2 ) Björck-Pereyra-type method for solving generalized Vandermonde systems of equations by using a new algorithm for computing the Schur function. We present explicit formulas for the entries of the bidiagonal decomposition, the LDU decomposition, and the inverse of a totally positive generalized Vandermonde matrix, as well as algorithms for computing these entries to high relative accuracy.

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Section: Related Work Björck and Pereyra Proposed An O(nmentioning

confidence: 98%

Section: Related Work Björck and Pereyra Proposed An O(nmentioning

confidence: 99%

Section: Related Work Björck and Pereyra Proposed An O(nmentioning

confidence: 99%

Section: The Inverse Of Gmentioning

confidence: 99%

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The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System

Demmel

Koev²

2005

SIAM J. Matrix Anal. & Appl.

129

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“…In 1977 Van de Vel [34] proposed a subtraction-free algorithm for the LDU decomposition of G. While accuracy was clearly guaranteed, efficiency was not. Recently, motivated by this result and some theoretical arguments [6, section 9.1(2)], Demmel and the current author presented an accurate algorithm for computing BD(G) [8,9].…”

Section: The Schur Complementmentioning

confidence: 99%

Accurate Computations with Totally Nonnegative Matrices

Koev¹

2007

SIAM J. Matrix Anal. & Appl.

125

165

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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.

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Mutually disjoint Steiner systems from BCH codes

Yan,

Zhou

2023

Des. Codes Cryptogr.

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Numerical treatment of a generalized Vandermonde system of equations

Cited by 16 publications

References 5 publications

The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System

The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System

Accurate Computations with Totally Nonnegative Matrices

Mutually disjoint Steiner systems from BCH codes

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