The translformation

Author:
Roland F. Streit

Journal:
Math. Comp. **30** (1976), 505-511

MSC:
Primary 65B10

MathSciNet review:
0421028

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper discusses a nonlinear sequence-to-sequence transformation, known as the 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.

**[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.**45**(1949), 230–236. MR**0029268****[4]**Gabriel Isakson,*A method for accelerating the convergence of an iteration procedure*, J. Aeronaut. Sci.**16**(1949), 443. MR**0030814****[5]**FOREST R. MOULTON,*Introduction to Celestial Mechanics*, MacMillan, New York, 1916, p. 364.**[6]**Paul A. Samuelson,*A convergent iterative process*, J. Math. Phys. Mass. Inst. Tech.**24**(1945), 131–134. MR**0014827****[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]**Daniel Shanks,*Non-linear transformations of divergent and slowly convergent sequences*, J. Math. and Phys.**34**(1955), 1–42. MR**0068901****[9]**Samuel Lubkin,*A method of summing infinite series*, J. Research Nat. Bur. Standards**48**(1952), 228–254. MR**0051576****[10]**H. L. Gray and 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**73B**(1969), 251–274. MR**0267304****[11]**T. J. I'A. BROMWICH,*An Introduction to the Theory of Infinite Series*, MacMillan, London, 1908.

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0421028-6

Article copyright:
© Copyright 1976
American Mathematical Society