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Integration formulas and schemes based on $ g$-splines


Authors: George D. Andria, George D. Byrne and David R. Hill
Journal: Math. Comp. 27 (1973), 831-838
MSC: Primary 65D30; Secondary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1973-0339460-5
MathSciNet review: 0339460
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Abstract: Numerical integration formulas of interpolatory type are generated by the integration of g-splines. These formulas, which are best in the sense of Sard, are used to construct predictor-corrector and block implicit schemes. The schemes are then compared with Adams-Bashforth-Adams-Moulton and Rosser schemes for a particular set of prototype problems. Moreover, an improved error bound for linear multistep formulas based on g-splines and a comparison of $ {L^2}$ norms of Peano kernels for Adams and natural spline formulas are given.


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  • [1] L. V. Ahlfors, Complex Analysis, An Introduction to the Theory of Analytic Functions of One Variable, McGraw-Hill, New York, 1953. MR 14, 857. MR 510197 (80c:30001)
  • [2] J. H. Ahlberg & E. N. Nilson, "The approximation of linear functionals," SIAM J. Numer. Anal., v. 3, 1966, pp. 173-182. MR 36 #589. MR 0217500 (36:589)
  • [3] G. D. Andria, G. D. Byrne & D. R. Hill, "Natural spline block implicit methods," Nordisk. Tidskr. Informationsbehandling (BIT), v. 13, 1973. (To appear.) MR 0323110 (48:1468)
  • [4] I. Babuška, M. Práger & E. Vitásek, Numerical Solution of Differential Equations, Státní Nakladatelství Technické Literatury, Prague, 1964; English transl., Interscience, New York, 1966. MR 36 #6149; #6150. MR 0223100 (36:6149)
  • [5] G. D. Byrne & D. N. H. Chi, "Linear multistep formulas based on g-splines," SIAM J. Numer. Anal., v. 9, 1972, pp. 316-324. MR 0311111 (46:10207)
  • [6] E. D. Callender, "Single step methods and low order splines for solution of ordinary differential equations," SIAM J. Numer. Anal., v. 8, 1971, pp. 61-66. MR 0315897 (47:4446)
  • [7] F. Ceschino & J. Kuntzman, Numerical Solution of Initial Value Problems, Dunod, Paris, 1963; English transl., Prentice-Hall, Englewood Cliffs, N.J., 1966. MR 28 #2639; MR 33 #3465. MR 0195262 (33:3465)
  • [8] P. E. Chase, "Stability properties of predictor-corrector methods for ordinary differential equations," J. Assoc. Comput. Mach., v. 9, 1962, pp. 457-468. MR 29 #738. MR 0163436 (29:738)
  • [9] C. S. Duris, Optimal Quadrature Formulas Using Generalized Inverses. Part II: Sard "Best" Formulas, Math. Rep. 70-71, Dept. of Mathematics, Drexel University, Philadelphia, 1970.
  • [10] T. N. E. Greville, "Spline functions, interpolation, and numerical quadrature," Mathematical Methods for Digital Computers. Vol. II, A. Ralston and H. S. Wilf (Editors), Wiley, New York, 1968, pp. 156-168.
  • [11] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 24 #B1772. MR 0135729 (24:B1772)
  • [12] T. E. Hull & A. L. Creemer, "Efficiency of predictor-corrector procedures," J. Assoc. Comput. Mach., v. 10, 1963, pp. 291-301. MR 27 #4367. MR 0154419 (27:4367)
  • [13] T. E. Hull, W. H. Enright, B. M. Feller & A. E. Sedgwick, Comparing Numerical Methods for Ordinary Differential Equations, Technical Report #29, Dept. of Computer Science, University of Toronto, 1971.
  • [14] B. L. Hulme, One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems, Technical Memo. SC-TM-710127, Sandia Laboratories, Albuquerque, N.M., 1971; Math. Comp., v. 26, 1972, pp. 415-426. MR 0321301 (47:9834)
  • [15] B. L. Hulme, "Piecewise polynomial Taylor methods for initial value problems," Numer. Math., v. 17, 1971, pp. 367-381. MR 0298953 (45:8002)
  • [16] L. Lapidus & J. H. Seinfeld, Numerical Solution of Ordinary Differential Equations, Math. in Sci. and Engineering, vol. 74, Academic Press, New York, 1971. MR 43 #7073. MR 0281355 (43:7073)
  • [17] F. R. Loscalzo, An Introduction to the Application of Spline Functions to Initial Value Problems (Proc. Sem., Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1968), Academic Press, New York, 1969, pp. 37-64. MR 39 #2334. MR 0240989 (39:2334)
  • [18] F. R. Loscalzo, Numerical Solution of Ordinary Differential Equations by Spline Functions (SPLINDIFF), Technical Summary Report 842, Mathematics Research Center, U. S. Army, University of Wisconsin, Madison, Wis., 1968.
  • [19] F. R. Loscalzo & I. J. Schoenberg, On the Use of Spline Functions for Solutions of Ordinary Differential Equations, Technical Summary Report 723, Mathematics Research Center, U.S. Army, University of Wisconsin, Madison, Wis., 1967.
  • [20] F. R. Loscalzo & T. D. Talbot, "Spline function approximation for solutions of ordinary differential equations," Bull. Amer. Math. Soc., v. 73, 1967, pp. 438-442. MR 35 #1218. MR 0210325 (35:1218)
  • [21] F. R. Loscalzo & T. D. Talbot, "Spline function approximations for solutions of ordinary differential equations," SIAM J. Numer. Anal., v. 4, 1967, pp. 433-445. MR 36 #4808. MR 0221756 (36:4808)
  • [22] J. B. Rosser, "A Runge-Kutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417-452. MR 36 #2325. MR 0219242 (36:2325)
  • [23] A. Sard, Linear Approximation, Math. Surveys, no. 9, Amer. Math. Soc., Providence, R.I., 1963. MR 28 #1429. MR 0158203 (28:1429)
  • [24] I. J. Schoenberg, "On best approximation of linear operators," Nederl. Akad. Wetensch. Proc. Ser. A, v. 67 = Indag. Math., v. 26, 1964, pp. 155-163. MR 28 #4284. MR 0161075 (28:4284)
  • [25] I. J. Schoenberg, "On the Ahlberg-Nilson extension of spline interpolation: the g-splines and their optimal properties," J. Math. Anal. Appl., v. 21, 1968, pp. 207-231. MR 36 #6849. MR 0223802 (36:6849)
  • [26] L. F. Shampine & H. A. Watts, "Block implicit one-step methods," Math. Comp., v. 23, 1969, pp. 731-740. MR 41 #9445. MR 0264854 (41:9445)

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DOI: https://doi.org/10.1090/S0025-5718-1973-0339460-5
Article copyright: © Copyright 1973 American Mathematical Society

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