Some limiting cases of the transformation
Authors:
H. L. Gray and W. R. Schucany
Journal:
Math. Comp. 23 (1969), 849859
MSC:
Primary 65.55; Secondary 41.00
MathSciNet review:
0260172
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Abstract: In this paper some new nonlinear transformations are introduced. They arise from considering the limit of the transformation as a particular parameter approaches its limiting value. The primary purpose of these transformations is to increase the rate of convergence of an improper integral. However, by introduction of an iteration method it is shown that they may also be used to produce approximating functions for the tail of an improper integral. Several examples are included.
 [1]
Samuel
Lubkin, A method of summing infinite series, J. Research Nat.
Bur. Standards 48 (1952), 228–254. MR 0051576
(14,500g)
 [2]
Daniel
Shanks, Nonlinear transformations of divergent and slowly
convergent sequences, J. Math. and Phys. 34 (1955),
1–42. MR
0068901 (16,961e)
 [3]
P.
Wynn, On a device for computing the
𝑒_{𝑚}(𝑆_{𝑛}) tranformation, Math. Tables Aids Comput. 10 (1956), 91–96. MR 0084056
(18,801e), http://dx.doi.org/10.1090/S00255718195600840566
 [4]
P.
Wynn, The epsilon algorithm and operational
formulas of numerical analysis, Math. Comp.
15 (1961),
151–158. MR 0158513
(28 #1736), http://dx.doi.org/10.1090/S0025571819610158513X
 [5]
P.
Wynn, On the convergence and stability of the epsilon
algorithm, SIAM J. Numer. Anal. 3 (1966), no. 1,
91–122. MR
0207180 (34 #6996)
 [6]
H.
L. Gray and T.
A. Atchison, Nonlinear transformations related to the evaluation of
improper integrals. I, SIAM J. Numer. Anal. 4 (1967),
363–371. MR 0223096
(36 #6145)
 [7]
Wyman
G. Fair and Yudell
L. Luke, Rational approximations to the
incomplete elliptic integrals of the first and second kinds, Math. Comp. 21 (1967), 418–422. MR 0222348
(36 #5400), http://dx.doi.org/10.1090/S00255718196702223485
 [8]
T.
A. Atchison and H.
L. Gray, Nonlinear transformations related to the evaluation of
improper integrals. II, SIAM J. Numer. Anal. 5
(1968), 451–459. MR 0229367
(37 #4941)
 [9]
W. R. Schucany & H. L. Gray, ``A new approximation related to the error function,'' Math. Comp., v. 22, 1968, pp. 201202.
 [10]
H. L. Gray & W. R. Schucany, ``On the evaluation of distribution functions,'' J. Amer. Statist Assoc., v. 63, 1968, pp. 715720.
 [11]
H.
L. Gray and T.
A. Atchison, The generalized
𝐺transform, Math. Comp. 22 (1968), 595–605. MR 0229368
(37 #4942), http://dx.doi.org/10.1090/S00255718196802293686
 [12]
H.
L. Gray, R.
W. Thompson, and G.
V. McWilliams, A new approximation for the chisquare
integral, Math. Comp. 23 (1969), 85–89. MR 0238470
(38 #6746), http://dx.doi.org/10.1090/S00255718196902384705
 [1]
 S. Lubkin, ``A method of summing infinite series,'' J. Res. Nat. Bur. Standards, v. 48, 1952, pp. 228254. MR 14, 500. MR 0051576 (14:500g)
 [2]
 D. Shanks, ``Nonlinear transformations of divergent and slowly convergent sequences,'' J. Math. Phys., v. 34, 1955, pp. 142. MR 16, 961. MR 0068901 (16:961e)
 [3]
 P. Wynn, ``On a device for computing the transformation,'' MTAC, v. 10, 1956, pp. 9196. MR 18, 801. MR 0084056 (18:801e)
 [4]
 P. Wynn, ``The epsilon algorithm and operational formulas of numerical analysis,'' Math. Comp., v. 15, 1961, pp. 151158. MR 28 #1736. MR 0158513 (28:1736)
 [5]
 P. Wynn, ``On the convergence and stability of the epsilon algorithm,'' SIAM J. Numer. Anal., v. 3, 1966, no. 1, pp. 91122. MR 34 #6996. MR 0207180 (34:6996)
 [6]
 H. L. Gray & T. A. Atchison, ``Nonlinear transformations related to the evaluation of improper integrals. I,'' SIAM J. Numer. Anal., v. 4, 1967, pp. 363371. MR 36 #6145. MR 0223096 (36:6145)
 [7]
 W. G. Fair & Y. L. Luke, ``Rational approximations to the incomplete elliptic integrals of the first and second kinds,'' Math. Comp., v. 21, 1967, pp. 418422. MR 36 #5400. MR 0222348 (36:5400)
 [8]
 T. A. Atchison & H. L. Gray, ``Nonlinear transformations related to the evaluation of improper integrals. II,'' SIAM J. Numer. Anal., v. 5, 1968, pp. 451459. MR 37 #4941. MR 0229367 (37:4941)
 [9]
 W. R. Schucany & H. L. Gray, ``A new approximation related to the error function,'' Math. Comp., v. 22, 1968, pp. 201202.
 [10]
 H. L. Gray & W. R. Schucany, ``On the evaluation of distribution functions,'' J. Amer. Statist Assoc., v. 63, 1968, pp. 715720.
 [11]
 H. L. Gray & T. A. Atchison, ``The generalized transform,'' Math. Comp., v. 22, 1968, pp. 595605. MR 37 #4942. MR 0229368 (37:4942)
 [12]
 H. L. Gray, R. W. Thompson & G. V. McWilliams, ``A new approximation for the chisquare integral,'' Math. Comp., v. 23, 1969, pp. 8589. MR 0238470 (38:6746)
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DOI:
http://dx.doi.org/10.1090/S0025571819690260172X
PII:
S 00255718(1969)0260172X
Article copyright:
© Copyright 1969
American Mathematical Society
