SPECTRAL METHODS AND QUADRATURES

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Resumo

Classical interpolation quadratures and, in particular, Gaussian quadratures are considered in the context of spectral methods, i.e., methods for solving boundary value problems for linear ODE by expanding them into series over orthogonal (and not only) polynomials. Fourier transforms are shown to play a key role here and allow calculating the required quadratures quite easily. Explicit formulas are given for some quadratures, and their efficiency is compared for high-accuracy computation of integrals. A simple Maple procedure for the Clenshaw–Curtis quadrature is given, and its application to computing the integral yielding the function of the sum of divisors of a natural number is considered.

Sobre autores

V. Varin

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Email: varin@keldysh.ru
Moscow, Russia

Bibliografia

  1. Bailey D.H. et. all. A Comparison of Three High-Precision Quadrature Schemes // Experimental Mathematics. 2005. V. 14. №3. P. 317–329.
  2. Варин В.П. Аппроксимация дифференциальных операторов с учетом граничных условий // Ж. вычисл. матем. и матем. физ. 2023. Т. 63. №8. С. 1251–1271.
  3. Варин В.П. Спектральные методы решения дифференциальных и функциональных уравнений // Ж. вычисл. матем. и матем. физ. 2024. Т. 64. №5. С. 713–728.
  4. Варин В.П. Специальные решения уравнения Шази // Ж. вычисл. матем. и матем. физ. 2017. Т. 57. №2. С. 210–236.
  5. Krylov V.I. Approximate calculation of integrals. New-York, London: Macmillan, 1962.
  6. Wilf H.S. Mathematics for the physical sciences. New-York: Wiley, 1962.
  7. Golub G.H., Welsch J.H. Calculation of Gauss Quadrature Rules // Mathematics of Computation. 1969. V. 23. №106. P. 221–230.
  8. Gantmacher F.R. Application of the Theory of Matrices. New-York: Chelsea Press, 1960.
  9. Bogaert I. Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights // SIAM Journal on Scientific Computing. 2014. V. 36. P. A1008–A1026.
  10. Fejér L. Mechanische Quadraturen mit positiven Cotesschen Zahlen // Math. Z. 1933. V. 37. P. 287–309.
  11. Clenshaw C.W., Curtis A.R. A method for numerical integration on an automatic computer // Numerische Mathematik. 1960. V. 2. P. 197–205.
  12. Варин В.П. Рациональные коэффициенты ортогональных разложений некоторых функций // Ж. вычисл. матем. и матем. физ. 2025. Т. 65. №7. С. 1211–1224.
  13. Варин В.П. Спектральные методы полиномиальной интерполяции и аппроксимации // Ж. вычисл. матем. и матем. физ. 2025. Т. 65. №2. С. 221–232.
  14. Ahmad Z. Definitely an Integral // Mathematical Monthly. 2002. V. 109. №7. P. 670–671.
  15. Shampine L.F. et. al. Solving Boundary Value Problems for Ordinary Differential Equations in Matlab with bvp4c // (https://www.mathworks.com/matlabcentral/fileexchange/3819-tutorial-on-solving-bvps-with-bvp4c)
  16. Numerical Algorithms Group Inc. NAG FORTRAN 77 Library Manual, Mark 17. (Oxford, UK, 1996).
  17. Lagarias J.C. An elementary problem equivalent to the Riemann hypothesis // Mathematical Monthly. 2002. V. 109. №6. P. 534–543.
  18. Sloane online encyclopedia of integer sequences, (http://oeis.org).

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