Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

Caliari, Marco; Marchi, Stefano De; Sommariva, Alvise; Vianello, Marco

doi:10.1007/s11075-010-9373-1

Cited by 26 publications

(30 citation statements)

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“…In order to simulate the Xu points as well as the Padua points, the numerical algorithms presented in [6,9] are used. The maximum interpolation errors are computed on a uniform grid of 100×100 points defined in a region = [0, 1]×[0, 1].…”

Section: A Simple and Efficient Scheme For The Computation Of The Intmentioning

confidence: 99%

“…Listing 1 Matlab/Octave code for a fast computation of the coefficients C n, p of the interpolating polynomial L n, p f from the data matrix G f 1 function C = LScfsfft (n ,p , G ) 2 % Input n , p : parameters of Lissajous curve 3 % G : (2 n +2 p +1) x (2 n +1) data matrix 4 % Output C : (2 n +2 p +1) x (2 n +1) coefficient matrix Remark 4 The matrix formulations in (19) and (20) are almost identical to the formulation of the interpolating scheme of the Padua points given in [6,8]. Also, the fast algorithm for the computation of the coefficients C n, p presented in Listing 1 is a modification of a respective algorithm developed for the Padua points in [6].…”

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confidence: 99%

“…Also, the fast algorithm for the computation of the coefficients C n, p presented in Listing 1 is a modification of a respective algorithm developed for the Padua points in [6]. These structural similarities are due to the particular representation (17) of the Lagrange polynomials.…”

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confidence: 99%

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Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

et al. 2015

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Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant. Mathematics Subject Classification

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Section: A Simple and Efficient Scheme For The Computation Of The Intmentioning

confidence: 99%

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confidence: 99%

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Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

et al. 2015

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“…As the points for the discrete inner product we use Padua points. The software for working with Padua points is described in [8].…”

Section: Numerical Experimentsmentioning

confidence: 99%

A method to compute recurrence relation coefficients for bivariate orthogonal polynomials by unitary matrix transformations

Barel

Chesnokov

2010

Numer Algor

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We present an algorithm computing recurrence relation coefficients for bivariate polynomials, orthonormal with respect to a discrete inner product. These polynomials make it possible to give the solution of a discrete least squares approximation problem. To compute these polynomials, we pose the inverse eigenvalue problem and solve it efficiently and in a stable way, using a sequence of Givens rotations. We also show how to generalize the algorithm for the case of polynomials in more variables. Several numerical experiments show the validity of the approach.

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“…[3,5,9,10]. They are the first known optimal nodal set for total-degree multivariate polynomial interpolation, with a Lebesgue constant increasing like (log n) 2 , n being the polynomial degree.…”

Section: Introductionmentioning

confidence: 99%

Trivariate polynomial approximation on Lissajous curves

2016

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We study Lissajous curves in the 3-cube, that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single 1-d Fast Chebyshev Transform (by the Chebfun package), and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Applications could arise in the framework of Lissajous sampling for MPI (Magnetic Particle Imaging).2010 AMS subject classification: 41A05, 41A10, 41A63, 65D05.

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Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

Abstract: We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions for algebraic cubature.

Cited by 26 publications

References 10 publications

Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

A method to compute recurrence relation coefficients for bivariate orthogonal polynomials by unitary matrix transformations

Trivariate polynomial approximation on Lissajous curves

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