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Mathematics of Computation

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Stability of rounded off inverses under iteration

Author: Harold G. Diamond
Journal: Math. Comp. 32 (1978), 227-232
MSC: Primary 65G05
MathSciNet review: 0461879
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Abstract: Let f be a monotone and strictly convex (or concave) function on a real interval and let g be the inverse function. Let $ I(x) = x$. For $ \phi $ a real valued function and N a positive integer let $ {\phi _N}(x)$ denote the rounding of $ \phi (x)$ to N significant figures. Let $ h = {g_N} \circ {f_N}$ , the composition of $ {f_N}$ and $ {g_N}$. It is shown that

$\displaystyle h \circ h \circ {I_N} = h \circ h \circ h \circ {I_N},$

and that equality can fail for fewer iterations.

References [Enhancements On Off] (What's this?)

  • [1] C. T. FIKE, Computer Evaluation of Mathematical Functions, Prentice-Hall, Englewood Cliffs, N. J., 1968. MR 0235700 (38:4003)
  • [2] JOHN F. REISER & DONALD E. KNUTH, "Evading the drift in floating point addition," Information Processing Lett., v. 3, 1975, pp. 84-87.
  • [3] DAVID W. MATULA, "The base conversion theorem," Proc. Amer. Math. Soc., v. 19, 1968, pp. 716-723. MR 0234908 (38:3222)

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Article copyright: © Copyright 1978 American Mathematical Society

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