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The work of Philip Rabinowitz on numerical integration

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A summary is given of Philip Rabinowitz's contributions to numerical analysis with emphasis on his work on integration.

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References

  1. G. Akrivis and K.-J. Förster, On the definiteness of quadrature formulae of Clenshaw-Curtis type, Computing 33 (1984) 363–366.

    Google Scholar 

  2. L.A. Anderson and W. Gautschi, Optimal weighted Chebyshev-type quadrature formulas, Calcolo 12 (1975) 211–248.

    Google Scholar 

  3. R.E. Barnhill and J.A. Wixom, Quadratures with remainders of minimum norm. I, II, Math. Comp. 21 (1967) 66–75 and 382–387.

    Google Scholar 

  4. K.S. Cole, H.A. Antosiewicz and P. Rabinowitz, Automatic computation of nerve excitation, J. Soc. Ind. Appl. Math. 3 (1955) 153–172. [Correction, ibid. K.S. Cole, H.A. Antosiewicz and P. Rabinowitz, Automatic computation of nerve excitation, J. Soc. Ind. Appl. Math. 6 (1958) 196–197.]

    Article  Google Scholar 

  5. R. Cools and P. Rabinowitz, Monomial cubature rules since “Stroud”: a compilation, J. Comp. Appl. Math. 48 (1993) 309–326.

    Article  Google Scholar 

  6. G. Criscuolo and G. Mastroianni, On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals, Math. Comp. 48 (1987) 725–735.

    Google Scholar 

  7. A.R. Curtis and P. Rabinowitz, On the Gaussian integration of Chebyshev polynomials, Math. Comp. 26 (1972) 207–211.

    Google Scholar 

  8. P. Davis, Errors of numerical approximation for analytic functions, J. Rat. Mech. Anal. 2 (1953) 303–313.

    Google Scholar 

  9. P. Davis and P. Rabinowitz, A multiple purpose orthonormalizing code and its uses, J. ACM 1 (1954) 183–191.

    Article  Google Scholar 

  10. P. Davis and P. Rabinowitz, On the estimation of quadrature errors for analytic functions, Math. Tables Aids Comp. 8 (1954) 193–203.

    Google Scholar 

  11. P.J. Davis and P. Rabinowitz, Abscissas and weights for Gaussian quadratures of high order, J. Res. Nat. Bur. Standards 56 (1956) 35–37.

    Google Scholar 

  12. P.J. Davis and P. Rabinowitz, Some Monte Carlo experiments in computing multiple integrals, Math. Tables Aids Comp. 10 (1956) 1–8.

    Google Scholar 

  13. P. Davis and P. Rabinowitz, Numerical experiments in potential theory using orthonormal functions, J. Washington Acad. Sci. 46 (1956) 12–17.

    Google Scholar 

  14. P. Davis and P. Rabinowitz, Additional abscissas and weights for Gaussian quadratures of high order. Values forn=64, 80, and 96. J. Res. Nat. Bur. Standards 60 (1959) 613–614.

    Google Scholar 

  15. P. Davis and P. Rabinowitz, Advances in orthonormalizing computation, in:Advances in Computers, Vol. 2 (Academic Press, New York, 1961) pp. 55–133.

    Google Scholar 

  16. P.J. Davis and P. Rabinowitz, Some geometrical theorems for abscissas and weights of Gauss type, J. Math. Anal. Appl. 2 (1961) 428–437.

    Article  Google Scholar 

  17. P.J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration, SIAM J. Numer. Anal. B 2 (1965) 367–383.

    Article  Google Scholar 

  18. P.J. Davis and P. Rabinowitz,Numerical Integration (Blaisdell, Waltham, MA, 1967).

    Google Scholar 

  19. P.J. Davis and P. Rabinowitz, On the nonexistence of simplex integration rules for infinite integrals, Math. Comp. 26 (1972) 687–688.

    Google Scholar 

  20. P.J. Davis and P. Rabinowitz,Methods of Numerical Integration (Academic Press, New York, 1975).

    Google Scholar 

  21. P.J. Davis and P. Rabinowitz,Methods of Numerical Integration, 2nd ed. (Academic Press, Orlando, FL, 1984).

    Google Scholar 

  22. D. Elliott and D.F. Paget, Product-integration rules and their convergence, BIT 16 (1976) 32–40.

    Article  Google Scholar 

  23. D. Elliott and D.P. Paget, The convergence of product integration rules, BIT 18 (1978) 137–141.

    Article  Google Scholar 

  24. K.-J. Förster and K. Petras, On estimates for the weights in Gaussian quadrature in the ultraspherical case, Math. Comp. 55 (1990) 243–264.

    Google Scholar 

  25. J.H. Freilich and P. Rabinowitz, Asymptotic approximation by polynomials in theL 1 norm, J. Approx. Theory 8 (1973) 304–314.

    Article  Google Scholar 

  26. W. Gautschi, Numerical quadrature in the presence of a singularity, SIAM J. Numer. Anal. 4 (1967) 357–362.

    Article  Google Scholar 

  27. W. Gautschi, Construction of Gauss-Christoffel quadrature formulas, Math. Comp. 22 (1968) 251–270.

    Google Scholar 

  28. W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comp. 3 (1982) 289–317.

    Article  Google Scholar 

  29. W. Gautschi, Questions of numerical condition related to polynomials, in:Studies in Numerical Analysis, ed. G.H. Golub, Studies in Mathematics vol. 24 (The Mathematical Association of America, 1984) pp. 140–177.

  30. W. Gautschi, Remainder estimates for analytic functions, in:Numerical Integration: Recent Developments, Software and Applications, eds. O. Espelid and A. Genz, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 357 (Kluwer, Dordrecht, 1992) pp. 133–145.

    Google Scholar 

  31. W. Gautschi and G. Monegato, On optimal Chebyshev-type quadratures, Numer. Math. 28 (1977) 59–67.

    Article  Google Scholar 

  32. W. Gautschi and H. Yanagiwara, On Chebyshev-type quadratures, Math. Comp. 28 (1974) 125–134.

    Google Scholar 

  33. W. Gautschi and S.E. Notaris, An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions, Math. Comp. 51 (1988) 231–248.

    Google Scholar 

  34. P.C. Hammer and A.H. Stroud, Numerical evaluation of multiple integrals. II, Math. Tables Aids Comp. 12 (1958) 272–280.

    Google Scholar 

  35. A.S. Kronrod,Nodes and Weights for Quadrature Formulae. Sixteen-place Tables (Russian) (Izdat. “Nauka”, Moscow, 1964). [English transl.: Consultants Bureau, New York, 1965.]

    Google Scholar 

  36. A.N. Lowan, N. Davids and A. Levenson, Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss' mechanical quadrature formula, Bull. Amer. Math. Soc. 48 (1942) 739–743.

    Google Scholar 

  37. D.S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Math. Comp. 43 (1984) 219–242.

    Google Scholar 

  38. D.S. Lubinsky and P. Rabinowitz, Hermite and Hermite-Fejér interpolation and associated product integration rules on the real line: TheL 1 theory, Canad. J. Math. 44 (1992) 561–590.

    Google Scholar 

  39. F. Mantel and P. Rabinowitz, The application of integer programming to the computation of fully symmetric integration formulas in two and three dimensions, SIAM J. Numer. Anal. 14 (1977) 391–425.

    Article  Google Scholar 

  40. P.G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18, no. 213 (1979).

  41. P. Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984) 669–698.

    Google Scholar 

  42. D. Nicholson, P. Rabinowitz, N. Richter and D. Zeilberger, On the error in the numerical integration of Chebyshev polynomials, Math. Comp. 25 (1971) 79–86.

    Google Scholar 

  43. J. Nowinski and P. Rabinowitz, The method of the kernel function in the theory of elastic plates, J. Appl. Math. Phys. 13 (1962) 26–42.

    Article  Google Scholar 

  44. T.N.L. Patterson, The optimum addition of points to quadrature formulae, Math. Comp. 22 (1968) 847–856. [Errata, ibid. T.N.L. Patterson, The optimum addition of points to quadrature formulae, Math. Comp. 23, 892.]

    Google Scholar 

  45. T.N.L. Patterson, On some Gauss and Lobatto based integration formulae, Math. Comp. 22 (1968) 877–881.

    Google Scholar 

  46. K. Petras, Gaussian quadrature formulae — second Peano kernels, nodes, weights and Bessel functions, Calcolo 30 (1993) 1–27.

    Google Scholar 

  47. K. Petras, Gaussian integration of Chebyshev polynomials and analytic functions,Proc. on Special Functions (dedicated to Luigi Gatteschi on his seventieth birthday), Ann. Numer. Math. 3 (1995), to appear.

  48. R. Piessens, Modified Clenshaw-Curtis integration and applications to numerical computation of integral transforms, in:Numerical Integration: Recent Developments, Software and Applications, eds. P. Keast and G. Fairweather, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 203 (Reidel, Dordrecht, 1987) pp. 35–51.

    Google Scholar 

  49. P. Rabinowitz, Abscissas and weights for Lobatto quadrature of high order, Math. Comp. 14 (1960) 47–52.

    Google Scholar 

  50. P. Rabinowitz, Numerical experiments in conformal mapping by the method of orthonormal polynomials, J. ACM 13 (1966) 296–303.

    Article  Google Scholar 

  51. P. Rabinowitz, Calculations of the conductivity of a medium containing cylindrical inclusions by the method of orthogonalized particular solutions, J. Appl. Phys. 37 (1966) 557–560.

    Article  Google Scholar 

  52. P. Rabinowitz, Gaussian integration in the presence of a singularity, SIAM J. Numer. Anal. 4 (1967) 191–201.

    Article  Google Scholar 

  53. P. Rabinowitz, Applications of linear programming to numerical analysis, SIAM Rev. 10 (1968) 121–159.

    Article  Google Scholar 

  54. P. Rabinowitz, Practical error coefficients for estimating quadrature errors for analytic functions, Commun. ACM 11 (1968) 45–46.

    Article  Google Scholar 

  55. P. Rabinowitz, Rough and ready error estimates in Gaussian integration of analytic functions, Commun. ACM 12 (1969) 268–270.

    Article  Google Scholar 

  56. P. Rabinowitz, Mathematical programming and approximation, in:Approximation Theory, ed. A. Talbot (Academic Press, London, 1970) pp. 217–231.

    Google Scholar 

  57. P. Rabinowitz (ed.),Numerical Methods for Nonlinear Algebraic Equations (Gordon and Breach, London, 1970).

    Google Scholar 

  58. P. Rabinowitz, A short bibliography on solution of systems of nonlinear algebraic equations, in [57Numerical Methods for Nonlinear Algebraic Equations, pp. 195–199].

  59. P. Rabinowitz, Ignoring the singularity in numerical integration, in:Topics in Numerical Analysis, ed. J.J.H. Miller (Academic Press, London, 1977) pp. 361–368.

    Google Scholar 

  60. P. Rabinowitz, The numerical evaluation of Cauchy principal value integrals,Proc. 4th Symp. on Numerical Mathematics, University of Natal, Durban, South Africa (1978) pp. 54–82.

    Google Scholar 

  61. P. Rabinowitz, The exact degree of precision of generalized Gauss-Kronrod integration rules, Math. Comp. 35 (1980) 1275–1283.

    Google Scholar 

  62. P. Rabinowitz, Generalized composite integration rules in the presence of a singularity, Calcolo 20 (1983) 231–238.

    Google Scholar 

  63. P. Rabinowitz, Gauss-Kronrod integration rules for Cauchy principal value integrals, Math. Comp. 41 (1983) 63–78 [Corrigenda, ibid P. Rabinowitz, Gauss-Kronrod integration rules for Cauchy principal value integrals, Math. Comp. 45 91985 277; 50 (1988) 655.]

    Google Scholar 

  64. P. Rabinowitz, Rates of convergence of Gauss, Lobatto, and Radau integration rules for singular integrals, Math. Comp. 47 (1986) 625–638.

    Google Scholar 

  65. P. Rabinowitz, The convergence of interpolatory product integration rules, BIT 26 (1986) 131–134.

    Article  Google Scholar 

  66. P. Rabinowitz, On the convergence of interpolatory product integration rules based on Gauss, Radau and Lobatto points, Israel J. Math. 56 (1986) 66–74.

    Google Scholar 

  67. P. Rabinowitz, On the definiteness of Gauss-Kronrod integration rules, Math. Comp. 46 (1986) 225–227.

    Google Scholar 

  68. P. Rabinowitz, A stable Gauss-Kronrod algorithm for Cauchy principal-value integrals, Comp. Math. Appl. Part B 12 (1986) 1249–1254.

    Article  Google Scholar 

  69. P. Rabinowitz, Numerical integration in the presence of an interior singularity, J. Comp. Appl. Math. 17 (1987) 31–41.

    Article  Google Scholar 

  70. P. Rabinowitz, The convergence of noninterpolatory product integration rules, in:Numerical Integration: Recent Developments, Software and Applications, eds. P. Keast and G. Fairweather, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 203 (Reidel, Dordrecht, 1987) pp. 1–16.

    Google Scholar 

  71. P. Rabinowitz, Convergence results for piecewise linear quadratures for Cauchy principal value integrals, Math. Comp. 51 (1988) 741–747.

    Google Scholar 

  72. P. Rabinowitz, Product integration based on Hermite-Fejér interpolation, J. Comp. Appl. Math. 28 (1989) 85–101.

    Article  Google Scholar 

  73. P. Rabinowitz, On an interpolatory product rule for evaluating Cauchy principal value integrals, BIT 29 (1989) 347–355.

    Article  Google Scholar 

  74. P. Rabinowitz, Numerical integration based on approximating splines, J. Comp. Appl. Math. 33 (1990) 73–83.

    Article  Google Scholar 

  75. P. Rabinowitz, Numerical evaluation of Cauchy principal value integrals with singular integrands, Math. Comp. 55 (1990) 265–276.

    Google Scholar 

  76. P. Rabinowitz, Generalized noninterpolatory rules for Cauchy principal value integrals, Math. Comp. 54 (1990) 217–279.

    Google Scholar 

  77. P. Rabinowitz, Uniform convergence of Cauchy principal value integrals of interpolating splines, Israel Math. Conf. Proc. 4 (1991) 225–231.

    Google Scholar 

  78. P. Rabinowitz, Extrapolation methods in numerical integration, Numer. Algor. 3 (1992) 17–28.

    Article  Google Scholar 

  79. P. Rabinowitz, S. Elhay and J. Kautsky, Empirical mathematics: the first Patterson extension of Gauss-Kronrod rules, Int. J. Comp. Math. 36 (1990) 119–129.

    Google Scholar 

  80. P. Rabinowitz and L. Gori,L 1-norm convergence of Hermite-Fejér interpolation based on the Laguerre and Hermite abscissas, Rend. Mat. Appl. (7) 14 (1994) 159–176.

    Google Scholar 

  81. P. Rabinowitz, J. Kautsky, S. Elhay and J.C. Butcher, On sequences of imbedded integration rules, in:Numerical Integration: Recent Developments, Software and Applications, eds. P. Keast and G. Fairweather, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 203 (Reidel, Dordrecht, 1987) pp. 113–139.

    Google Scholar 

  82. P. Rabinowitz and D.S. Lubinsky, Noninterpolatory integration rules for Cauchy principal value integrals, Math. Comp. 53 (1989) 279–295.

    Google Scholar 

  83. P. Rabinowitz and N. Richter, Perfectly symmetric two-dimensional integration formulas with minimal number of points, Math. Comp. 23 (1969) 765–779.

    Google Scholar 

  84. P. Rabinowitz and N. Richter, New error coefficients for estimating quadrature errors for analytic functions, Math. Comp. 24 (1970) 561–570.

    Google Scholar 

  85. P. Rabinowitz and N. Richter, Asymptotic properties of minimal integration rules, Math. Comp. 24 (1970) 593–609.

    Google Scholar 

  86. P. Rabinowitz and N. Richter, Chebyshev-type integration rules of minimum norm, Math. Comp. 24 (1970) 831–845.

    Google Scholar 

  87. P. Rabinowitz and I.H. Sloan, Product integration in the presence of a singularity, SIAM J. Numer. Anal. 21 (1984) 149–166.

    Article  Google Scholar 

  88. P. Rabinowitz and W.E. Smith, Interpolatory product integration for Riemann-integrable functions, J. Austral. Math. Soc. Ser. B 29 (1987) 195–202.

    Google Scholar 

  89. P. Rabinowitz and W.E. Smith, Interpolatory product integration in the presence of singularities:L 2 theory, J. Comp. Appl. Math. 39 (1992) 79–87.

    Article  Google Scholar 

  90. P. Rabinowitz and W.E. Smith, Interpolatory product integration in the presence of singularities:L p theory, in:Numerical Integration: Recent Developments, Software and Applications, eds. T.O. Espelid and A. Genz, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 357 (Kluwer, Dordrecht, 1992) pp. 93–109.

    Google Scholar 

  91. P. Rabinowitz and G. Weiss, Tables of abscissas and weights for numerical evaluation of integrals of the form\(\int_0^\infty {e^{ - x} x^n f(x)dx} \), Math. Tables Aids Comp. 13 (1959) 285–294.

    Google Scholar 

  92. A. Ralston and P. Rabinowitz,A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).

    Google Scholar 

  93. T. Schira, Ableitungsfreie Fehlerabschätzungen bei numerischer Integration holomorpher Funktionen, Dissertation, University of Karlsruhe (1994).

  94. I.H. Sloan and W.E. Smith, Properties of interpolatory product integration rules, SIAM J. Numer. Anal. 19 (1982) 427–442.

    Article  Google Scholar 

  95. W.E. Smith and I.H. Sloan, Product-integration rules based on the zeros of Jacobi polynomials, SIAM J. Numer. Anal. 17 (1980) 1–13.

    Article  Google Scholar 

  96. A.H. Stroud,Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, NJ, 1971).

    Google Scholar 

  97. G. Szegö, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1935) 501–513. [Collected Papers (ed. R. Askey), Vol. 2, 545–557.]

    Article  Google Scholar 

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Gautschi, W. The work of Philip Rabinowitz on numerical integration. Numer Algor 9, 199–222 (1995). https://doi.org/10.1007/BF02141588

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