The Fast Generalized Parker–Traub Algorithm for Inversion of Vandermonde and Related Matrices

Gohberg, Israel; Olshevsky, Vadim

doi:10.1006/jcom.1997.0442

Cited by 42 publications

(16 citation statements)

References 24 publications

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“…The data in Table 1 and in numerous other examples suggest that for the nodes of both signs, x k ∈ (−1, 1), the use of Leja ordering is a stabilizing technique for the both GE and BP algorithms, i.e., it provides a smaller size for the residual errors. This confirms our experience and the computational experience of many others that Leja ordering enhances the accuracy of various algorithms for Vandermonde matrices [Hig90], [R90b], [RO91], [GK93], [GO94a], [GO97].…”

Section: Condition Number Monotonic Orderingsupporting

confidence: 73%

Pivoting for structured matrices and rational tangential interpolation

Olshevsky

2003

Fast Algorithms for Structured Matrices: Theory and Applications

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Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point operation. This problem motivated a number of interesting and important studies in modern numerical linear algebra; for our purposes in this paper we only mention that starting with the breakthrough work of Wilkinson, several pivoting techniques have been proposed to stabilize the numerical behavior of Gaussian elimination. Interestingly, matrix interpretations of many known and new algorithms for various applied problems can be seen as a way of computing triangular fac-torizations for the associated structured matrices, where different patterns of structure arise in the context of different physical problems. The special structure of such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] often allows one to speed-up the computation of its triangular factorization, i.e., to efficiently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under different names, e.g., fast Cholesky, fast Gaussian elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they are efficient fast implementations of a numerically inaccurate [for indefinite matrices] method. In this paper we survey recent results on the fast implementation of various pivoting techniques which allowed us to improve numerical accuracy for a variety of fast algorithms. This approach led us to formulate new more accurate numerical methods for factorization of general and of J-unitary rational matrix functions, for solving various tangential interpolation problems, new Toeplitz-like and Toeplitz-plus-Hankel-like solvers, and new divided differences schemes. We beleive that similar methods can be used to design accurate fast algorithm for the other applied problems and a recent work of our colleagues supports this anticipation.

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Section: Condition Number Monotonic Orderingsupporting

confidence: 73%

Pivoting for structured matrices and rational tangential interpolation

Olshevsky

2003

Fast Algorithms for Structured Matrices: Theory and Applications

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“…In its form, determinant (7) resembles the well-known Van der Monde determinant. This determinant was the subject of a very intensive analysis [15][16][17][18]. However, all of these papers were proposing only the procedures that relied on program procedures (mostly iterative ones) in determining the value of the original and inverted Van der Monde matrix with an improved efficiency.…”

Section: Problem Formulationmentioning

confidence: 99%

New method and circuit for processing of band-limited periodic signals

Petrovič

2010

SIViP

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The paper presents a reconstruction of analogue multi-harmonic signals, from a number of integrated values of input signals. Based on the value of the integral of the original input signal, with a known frequency spectrum but unknown amplitudes and phases, a reconstruction of its basic parameters is done by the means of derived analytical and summarized expressions. It is applied to signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. The proposed method of processing can be used for precise RMS measurements (or power and energy) of periodic signal based on the presented signal reconstruction. Subsequent calculation of all relevant indicators related to the monitoring and processing of ac voltage and current signals is provided in this manner. The paper investigates the errors related to the signal parameter estimation, and there is a computer simulation that demonstrates the accuracy of these algorithms.

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“…In [9] fast triangularization and fast Gaussian elimination algorithm with partial pivoting were given for the displacement equation of the form (5.1) (for details, see [2,9,14]), therefore, the algorithm in [9] is applied to the confluent polynomial Vandermonde-like matrix.…”

Section: Block Gaussian Eliminationmentioning

confidence: 99%

Confluent polynomial Vandermonde-like matrices: displacement structures, inversion formulas and fast algorithm

Yang

2004

Linear Algebra and its Applications

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In the present paper we use the displacement structure approach to introduce a new class of what are called confluent polynomial Vandermonde-like matrices, which generalize the ordinary simple nodes polynomial Vandermonde matrices studied earlier by different authors. The displacement structure approach leads to the explicit inversion formulas for confluent polynomial Vandermonde-like matrices and fast algorithms for inversion and for solving the associated linear systems.

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The Fast Generalized Parker–Traub Algorithm for Inversion of Vandermonde and Related Matrices

Cited by 42 publications

References 24 publications

Pivoting for structured matrices and rational tangential interpolation

Pivoting for structured matrices and rational tangential interpolation

New method and circuit for processing of band-limited periodic signals

Confluent polynomial Vandermonde-like matrices: displacement structures, inversion formulas and fast algorithm

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