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Exploiting parity in converting to and from Bernstein polynomials and orthogonal polynomials. (English) Zbl 1138.65024

Summary: The coefficients of a polynomial in the Bernstein basis can be converted to the coefficients of a Legendre or Chebyshev series by a simple matrix-vector multiply at a cost of O(\(2[N + 1]^{2}\)) operations where \(N\) is the degree of the polynomial. In this note, we show that by exploiting parity with respect to the center of the interval \(x \in\) [0, 1], is possible to halve the cost. In \(d\) dimensions with a tensor product basis, the savings are a factor of two independent of \(d\).

MSC:

65D20 Computation of special functions and constants, construction of tables
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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[1] Barrio, R.; Pena, J. M., Basic conversions among univariate polynomial representations, C.R. Acad. Sci., Paris, Ser. I, 339, 293-298 (2004) · Zbl 1055.65035
[2] Boyd, J. P., Chebyshev and Fourier Spectral Methods (2001), Dover: Dover Mineola, NY, 665 p · Zbl 0987.65122
[3] Boyd, J. P., A test, based on conversion to the Bernstein polynomial basis, for an interval to be free of zeros applicable to polynomials in Chebyshev form and to transcendental functions approximated by Chebyshev series, Appl. Math. Comput., 188, 1780-1789 (2007) · Zbl 1121.65049
[4] Farouki, R. T., Legendre-Bernstein basis transformations, J. Comput. Appl. Math., 119, 145-160 (2000) · Zbl 0962.65042
[5] Li, Y.; Zhang, X., Basis conversion among Bézier, Tchebyshev and Legendre, Comput. Aided Geometric Des., 15, 637-642 (1998) · Zbl 0905.68145
[6] Rababah, A., Jacobi-Bernstein basis transformation, Comput. Meth. Appl. Math., 4, 206-214 (2004) · Zbl 1071.41006
[7] Rababah, A., Transformation of Chebyshev-Bernstein polynomial basis, Comput. Meth. Appl. Math., 3, 608-622 (2004) · Zbl 1046.41005
[8] Watkins, M. A.; Worsey, A. J., Degree reduction of Bézier curves, Comput. Aided Des., 20, 398-405 (1988) · Zbl 0658.65014
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