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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Some inequalities for continued fractions
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by R. M. Dudley PDF
Math. Comp. 49 (1987), 585-593 Request permission


For some continued fractions $Q = {b_0} + {a_1}/({b_1} + \cdots )$ with mth convergent ${Q_m}$, it is shown that relative errors are monotone in some arguments. If all the entries ${a_j}$ and ${b_j}$ in Q are positive, then the relative error $|{Q_m}/Q - 1|$ is bounded by $|{Q_m}/{Q_{m + 1}} - 1|$, which is nonincreasing in the partial denominator ${b_j}$ for each $j \geqslant 0$, as is $|{Q_m}/Q - 1|$ for $j \leqslant m + 1$. If ${b_j} \geqslant 1$ for all $j \geqslant 1$, ${b_0} \geqslant 0$, and ${a_j} = {( - 1)^{j + 1}}{c_j}$ where ${c_j} \geqslant 0$ and for j even, ${c_j} < 1$, then $|{Q_m}/Q - 1|$ is bounded by $|{Q_m}/{Q_{m + 2}} - 1|$, and both are nonincreasing in ${b_j}$ for even $j \leqslant m + 2$. These facts apply to continued fractions of Euler, Gauss and Laplace used in computing Poisson, binomial and normal probabilities, respectively, giving monotonicity of relative errors as functions of the variables in suitable ranges.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 49 (1987), 585-593
  • MSC: Primary 40A15; Secondary 33A20, 65D20
  • DOI:
  • MathSciNet review: 906191