This paper deals with efficient quadrature formulas involving functions that are observed only at fixed sampling points. The approach that we develop is derived from efficient continuous quadrature formulas, such as Gauss-Legendre or Clenshaw-Curtis quadrature. We select nodes at sampling positions that are as close as possible to those of the associated classical quadrature and we update quadrature weights accordingly. We supply the theoretical quadrature error formula for this new approach. We show on examples the potential gain of this approach.
[1] M. B. Allen and E. L. Isaacson, Numerical Analysis for Applied Science,
Wiley-Interscience, 1998.
[2] B. Fornberg and P. G. Ciarlet and A. Iserles, A Practical Guide to
Pseudospectral Methods, Cambridge University Press, 1998.
[3] J. P. Berrut and L. N. Trefethen, "Barycentric Lagrange interpolation,
SIAM rev.", pp. 501-517, 2004.
[4] H. Jeffreys and B. S. Jeffreys, Weierstrass-s Theorem on Approximation
by Polynomials, P 14.08 in Methods of Mathematical Physics, 3rd ed.
England: Cambridge University Press, 1988.
[5] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, N.Y.:
Academic Press, 1975.
[6] C. W. Clenshaw A. R. Curtis, "A method for numerical integration on an
automatic computer", Numer. Math., vol. 2, pp. 197-205, 1960.
[7] L. N. Trefethen, "Is Gauss Quadrature Better Than Clenshaw-Curtis ?",
SIAM Rev., Society for Industrial and Applied Mathematics, vol. 50(1),
pp. 67-87, 2008.
[8] P. K. Kythe and M. R. Schäferkotter, Handbook of Computational
Methods for Integration, CRC Press, 2005.
[9] G. H. Golub and J. H. Welsch, "Calculation of gauss quadrature rules",
Math. Comp., vol. 23, pp. 221-230, 1969.
[10] W. M. Gentleman, "Implementing clenshaw-curtis quadrature, i- computing
the cosine transformation", Communications of the ACM, vol.
15(5), pp. 337-342, 1972.
[11] V. Levanon, Radar Principles, New York: John Wiley & Sons, 1988.
[12] J. Graves and P. M. Prenter, "Numerical iterative filters applied to first
kind Fredholm integral equation", Numer. Math., vol. 30, pp. 281-299,
1978.
[13] K. C. Aas and K. A. Duell and C. T. Mullis, "Synthesis of extremal
wavelet-generating filters using Gaussian quadrature", IEEE Transaction
on signal processing, vol. 43(5), pp. 1045-1057, 1995.
[14] B. R. Mahafza, Radar Systems Analysis ans Design Using Matlab, New
York: Chapman & Hall/CRC, 2000.
[15] P. M. Woodward, Probability and information Teory, with Application
to Radar, 2nd ed. Oxford: Pergamon Press, 1964.
[16] A. Papoulis, Signal Analysis, New York: McGraw-Hill, 1977.
[17] T. Shan and R. Tao, and R. S. Rong, "A Fast Method for Time Delay,
Doppler Shift and Doppler Rate Estimation", International Conference on
Radar (CIE-06), Shanghai, China, October 2006, pp. 1-4.
[18] O. Rabaste and T. Chonavel, "Estimation of Multipath Channels With
Long Impulse Response at Low SNR via an MCMC Method", IEEE
Trans. Sig. Proc., vol. 55, pp. 1312-1325, 2007.
[19] R. J. Baxley and G. T. Zhou, "Computational Complexity Analysis of
FFT Pruning - A Markov Modeling Approch", Digital Signal processing
Workshop, Proc. IEEE 12th Digital Signal Processing Workshop, 2006,
pp. 535-539.
[20] N. Macon and A. Spitzbart, "Inverses of vandermonde matrices", The
American Mathematical Monthly, vol. 65, pp. 95-100, 1958.
[21] A. Klinger, "The vandermonde matrix", The American Mathematical
Monthly, vol. 74(5), pp. 571-574, 1967.
[1] M. B. Allen and E. L. Isaacson, Numerical Analysis for Applied Science,
Wiley-Interscience, 1998.
[2] B. Fornberg and P. G. Ciarlet and A. Iserles, A Practical Guide to
Pseudospectral Methods, Cambridge University Press, 1998.
[3] J. P. Berrut and L. N. Trefethen, "Barycentric Lagrange interpolation,
SIAM rev.", pp. 501-517, 2004.
[4] H. Jeffreys and B. S. Jeffreys, Weierstrass-s Theorem on Approximation
by Polynomials, P 14.08 in Methods of Mathematical Physics, 3rd ed.
England: Cambridge University Press, 1988.
[5] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, N.Y.:
Academic Press, 1975.
[6] C. W. Clenshaw A. R. Curtis, "A method for numerical integration on an
automatic computer", Numer. Math., vol. 2, pp. 197-205, 1960.
[7] L. N. Trefethen, "Is Gauss Quadrature Better Than Clenshaw-Curtis ?",
SIAM Rev., Society for Industrial and Applied Mathematics, vol. 50(1),
pp. 67-87, 2008.
[8] P. K. Kythe and M. R. Schäferkotter, Handbook of Computational
Methods for Integration, CRC Press, 2005.
[9] G. H. Golub and J. H. Welsch, "Calculation of gauss quadrature rules",
Math. Comp., vol. 23, pp. 221-230, 1969.
[10] W. M. Gentleman, "Implementing clenshaw-curtis quadrature, i- computing
the cosine transformation", Communications of the ACM, vol.
15(5), pp. 337-342, 1972.
[11] V. Levanon, Radar Principles, New York: John Wiley & Sons, 1988.
[12] J. Graves and P. M. Prenter, "Numerical iterative filters applied to first
kind Fredholm integral equation", Numer. Math., vol. 30, pp. 281-299,
1978.
[13] K. C. Aas and K. A. Duell and C. T. Mullis, "Synthesis of extremal
wavelet-generating filters using Gaussian quadrature", IEEE Transaction
on signal processing, vol. 43(5), pp. 1045-1057, 1995.
[14] B. R. Mahafza, Radar Systems Analysis ans Design Using Matlab, New
York: Chapman & Hall/CRC, 2000.
[15] P. M. Woodward, Probability and information Teory, with Application
to Radar, 2nd ed. Oxford: Pergamon Press, 1964.
[16] A. Papoulis, Signal Analysis, New York: McGraw-Hill, 1977.
[17] T. Shan and R. Tao, and R. S. Rong, "A Fast Method for Time Delay,
Doppler Shift and Doppler Rate Estimation", International Conference on
Radar (CIE-06), Shanghai, China, October 2006, pp. 1-4.
[18] O. Rabaste and T. Chonavel, "Estimation of Multipath Channels With
Long Impulse Response at Low SNR via an MCMC Method", IEEE
Trans. Sig. Proc., vol. 55, pp. 1312-1325, 2007.
[19] R. J. Baxley and G. T. Zhou, "Computational Complexity Analysis of
FFT Pruning - A Markov Modeling Approch", Digital Signal processing
Workshop, Proc. IEEE 12th Digital Signal Processing Workshop, 2006,
pp. 535-539.
[20] N. Macon and A. Spitzbart, "Inverses of vandermonde matrices", The
American Mathematical Monthly, vol. 65, pp. 95-100, 1958.
[21] A. Klinger, "The vandermonde matrix", The American Mathematical
Monthly, vol. 74(5), pp. 571-574, 1967.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:49175", author = "Khalid Minaoui and Thierry Chonavel and Benayad Nsiri and Driss Aboutajdine", title = "Quadrature Formula for Sampled Functions", abstract = "This paper deals with efficient quadrature formulas involving functions that are observed only at fixed sampling points. The approach that we develop is derived from efficient continuous quadrature formulas, such as Gauss-Legendre or Clenshaw-Curtis quadrature. We select nodes at sampling positions that are as close as possible to those of the associated classical quadrature and we update quadrature weights accordingly. We supply the theoretical quadrature error formula for this new approach. We show on examples the potential gain of this approach.
", keywords = "Gauss-Legendre, Clenshaw-Curtis, quadrature, Peano kernel, irregular sampling.", volume = "4", number = "1", pages = "5-7", }
