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The $ T\sb{+m}$ translformation

Author: Roland F. Streit
Journal: Math. Comp. 30 (1976), 505-511
MSC: Primary 65B10
MathSciNet review: 0421028
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Abstract: This paper discusses a nonlinear sequence-to-sequence transformation, known as the $ {T_{ + m}}$ transform, which is used to accelerate the convergence of an infinite series. A brief history of the transform is given; a number of theorems are established which enable one to make effective use of the transform, and several examples are presented to illustrate this effectiveness.

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  • [1] A. C. AITKEN, "On Bernoulli's numerical solution of algebraic equations," Proc. Roy. Soc. Edinburgh, v. 46, 1926, pp. 289-305.
  • [2] D. SHANKS & T. S. WALTON, The Use of Rational Functions as Approximate Solutions of Certain Trajectory Problems, Naval Ordnance Laboratory Memorandum #9524, White Oak, Md., 1948.
  • [3] D. R. HARTREE, "Notes on iterative processes," Proc. Cambridge Philos. Soc., v. 45, 1949, pp. 230-236. MR 10, 574. MR 0029268 (10:574a)
  • [4] G. ISAKSON, "A method for accelerating the convergence of an iteration procedure," J. Aeronaut. Sci., v. 16, 1949, p. 443. MR 11, 57. MR 0030814 (11:57c)
  • [5] FOREST R. MOULTON, Introduction to Celestial Mechanics, MacMillan, New York, 1916, p. 364.
  • [6] P. A. SAMUELSON, "A convergent iterative process," J. Math. and Phys., v. 24, 1945, pp. 131-134. MR 7, 337. MR 0014827 (7:337j)
  • [7] D. SHANKS, An Analogy Between Transients and Mathematical Sequences and Some Non-Linear Sequence-to-Sequence Transforms Suggested by It. I, Naval Ordnance Laboratory Memorandum #9994, White Oak, Md., 1949.
  • [8] D. SHANKS, "Non-linear transformations of divergent and slowly convergent sequences," J. Math. and Phys., v. 34, 1955, pp. 1-42. MR 16, 961. MR 0068901 (16:961e)
  • [9] S. LUBKIN, "A method of summing infinite series," J. Res. Nat. Bur. Standards, v. 48, 1952, pp. 228-254. MR 14, 500. MR 0051576 (14:500g)
  • [10] H. L. GRAY & W. D. CLARK, "On a class of nonlinear transformations and their applications to the evaluation of infinite series," J. Res. Nat. Bur. Standards Sect. B, v. 73B, 1969, pp. 251-274. MR 42 #2206. MR 0267304 (42:2206)
  • [11] T. J. I'A. BROMWICH, An Introduction to the Theory of Infinite Series, MacMillan, London, 1908.

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Article copyright: © Copyright 1976 American Mathematical Society

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