Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

Pachon, Ricardo; Gonnet, Pedro; Deun, Joris Van

doi:10.1137/100797291

Cited by 34 publications

(32 citation statements)

References 39 publications

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“…Rational interpolation has a propensity, well-known to practitioners, for rounding errors to cause spurious pole-zero pairs (sometimes called Froissart doublets) to appear when the interpolant is computed in the obvious way [42]. To combat this, we use the algorithm presented in [13], which combines the earlier algorithm of [29] with a regularization technique based on the SVD to detect and remove these spurious pairs. This algorithm is implemented in the MATLAB code ratinterp within the freely available Chebfun software package [6].…”

Section: Rational Interpolation On a Real Intervalmentioning

confidence: 99%

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Austin¹,

Trefethen²

2015

SIAM J. Sci. Comput.

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Powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals; best known are the works of Sakurai and coauthors and Polizzi and coauthors. Even if the matrices are real symmetric, most such methods rely on complex arithmetic, leading to expensive linear systems to solve. An appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems, for which there is no need to leave the real axis. Investigation shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective. This article presents a technique of this kind that is related to previous work published in Japanese by Murakami. Introduction. LetA ∈ C N ×N be a large square matrix, and consider the problem of computing the eigenvalues of A that lie within a given region of the complex plane. Some of the most successful techniques for attacking this problem are based on projecting the matrix A onto an approximately invariant subspace associated with the eigenvalues of interest and computing the eigenvalues of the projection. Of these, perhaps the best known are the Krylov subspace techniques such as the implicitly restarted Arnoldi method [39], which is implemented in the widely used software package ARPACK [20].Recently, a new class of algorithms has been proposed which derive their projections from complex contour integrals. Though early traces of these ideas can be found in the works of Goedecker on linear-scaling methods for electronic structure calculation [11,12], the best-known algorithms of this type are the Sakurai-Sugiura (SS) method [37] and the FEAST algorithm, due to Polizzi [32]. A major computational advantage offered by these algorithms is that they are very easily parallelizable.Some of the most important eigenvalue problems involve matrices A that are real and symmetric. For large such problems, it is natural to want to take advantage of the parallelism offered by contour integral methods; however, by their very nature, these methods require complex arithmetic even though the sought-after eigenvalues are real. Aside from the fact that the need to use complex arithmetic to solve a real problem is conceptually jarring, this means that methods based on contour integrals suffer roughly a factor of two penalty in both time and storage costs, compared with approaches that rely only on real arithmetic.

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Section: Rational Interpolation On a Real Intervalmentioning

confidence: 99%

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Austin¹,

Trefethen²

2015

SIAM J. Sci. Comput.

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“…To do this we use the Chebfun command ratinterp, which is a robust implementation of linearised rational interpolation. Here we will explain mathematically how ratinterp works, based on the algorithms described in [9] and [16].…”

Section: The Proposed Methodmentioning

confidence: 99%

“…The method explained above generalises to arbitrary points in the complex plane, and the user can specify whatever nodes they want ratinterp to use. The general approach for the linearised interpolation problem is discussed in [16]. The case where the nodes are the N th roots of unity is particularly elegant because the polynomials satisfying a discrete orthogonality relation are simply the monomials (powers of roots of unity make up the Fourier basis).…”

Section: The Proposed Methodmentioning

confidence: 99%

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Computing Complex Singularities of Differential Equations with Chebfun

2013

SIURO

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Abstract. Given a solution to an ordinary differential equation (ODE) on a time interval, the solution for complex-valued time may be of interest, in particular whether the solution is singular at some complex time value. How can the solution be approximated in the complex plane using only the data on the interval? A polynomial approximation of the solution always fails to capture singularities; to extrapolate solutions with singularities, approximation with rational functions is more appropriate. In this paper, a robust form of rational interpolation and least-squares approximation, due to Pachón, Gonnet et al., is discussed and tested. It is found that the method avoids the issue of spurious poles found by many standard rational approximations, but that it is not suitable when a high degree of accuracy is required.

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“…The second complication with rational methods is the appearance of spurious pole-zero pairs, known as Froissart doublets, that prevent pointwise convergence. 3 Numerous algorithms have been proposed to compute numerator and denominator coefficients for univariate rationals (see Pachón, 4 Hesthaven 5 and van Deun 6 ), however, very few have been extended to two dimensions. Two algorithms that do offer such an extension are by Chantrasmi et al 1 and Gonnet et al 7 Both approaches effectively attempt to regularize the ill-posed problem of obtaining numerator and denominator coefficients, and do so by resorting to a singular value decomposition.…”

Section: Introductionmentioning

confidence: 99%

Sparse Robust Rational Interpolation for Parameter-dependent Aerospace Models

Seshadri

Constantine

Gonnet

et al. 2013

54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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A multivariate, robust, rational interpolation method for propagating uncertainties in several dimensions is presented. The algorithm for selecting numerator and denominator polynomial orders is based on recent work that uses a singular value decomposition approach. In this paper we extend this algorithm to higher dimensions and demonstrate its efficacy in terms of convergence and accuracy, both as a method for response suface generation and interpolation. To obtain stable approximants for continuous functions, we use an L2 error norm indicator to rank optimal numerator and denominator solutions. For discontinous functions, a second criterion setting an upper limit on the approximant value is employed. Analytical examples demonstrate that, for the same stencil, rational methods can yield more rapid convergence compared to pseudospectral or collocation approaches for certain problems.

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Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

Cited by 34 publications

References 39 publications

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic

Computing Complex Singularities of Differential Equations with Chebfun

Sparse Robust Rational Interpolation for Parameter-dependent Aerospace Models

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