Multiple quadrature with central differences on one line

Author:
Herbert E. Salzer

Journal:
Math. Comp. **16** (1962), 244-248

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1962-0145655-9

MathSciNet review:
0145655

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Abstract: The coefficients in the *n*-fold quadrature formulas for the stepwise integration of (1) , at intervals of *h*, namely, for *n* even, (2) , for *n* odd, (3) , are tabulated exactly for *n* = 1(1)6, *m* = 1(1)10. They were calculated from the well-known symbolic formulas (4) , (5) and (6) . For calculating , replace *n* by *n* - *r* in (2) and (3). Use of (2) and (3) avoids the solution of (1) by simultaneous lower-order systems for , as well as mid-interval tabular arguments, requires only even-order differences, on a single line, and provides great accuracy due to rapid decrease of as *m* increases. However, the integration may be slowed down by the need to estimate and refine iteratively the later values of required in . Reference to earlier collected formulas of Legendre, Oppolzer, Thiele, Lindow, Salzer, Milne and Buckingham, reveals that Thiele and Buckingham come closest to (2), (3), as their works contain schemes that involve just tabular arguments throughout. For *n* odd, they give formulas that are based upon the series in for instead of as in the present arrangement.

**[1]**A. M. Legendre,*Traité des Fonctions Elliptiques*, v. 2, Paris, 1826, Chapter 3, p. 41-60 (For errors, see*MTAC*, v. 5, 1951, p. 27).**[2]**T. R. Oppolzer,*Lehrbuch zur Bahnbestimmung der Cometen und Planeten*, v. 2, W. Engelmann, Leipzig, 1880, p. 35, 53-54, 545, 596.**[3]**T. N. Thiele,*Interpolationsrechnung*, B. G. Teubner, Leipzig, 1909, p. 95-97. (Some misprints are noted in*Math. Comp.*, v. 15, 1961, p. 321.)**[4]**M. Lindow,*Numerische Infinitesimalrechnung*, F. Dümmler, Berlin and Bonn, 1928, p. 170-171.**[5]**H. E. Salzer, ``Coefficients for mid-interval numerical integration with central differences,''*Phil. Mag.*, ser. 7, v. 36, 1945, p. 216-218. MR**0013924 (7:219b)****[6]**W. E. Milne,*Numerical Calculus*, Princeton, 1949, p. 196-197.**[7]**H. E. Salzer, ``Coefficients for repeated integration with central differences,''*J. Math. Phys.*, v. 28, 1949, p. 54-61. MR**0029284 (10:576b)****[8]**R. A. Buckingham,*Numerical Methods*, Pitman Publishing Corp., New York and London, 1957, p. 150-154. (For errors, see*Math. Comp.*, v. 15, 1961, p. 319.).

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DOI:
https://doi.org/10.1090/S0025-5718-1962-0145655-9

Article copyright:
© Copyright 1962
American Mathematical Society