Some inequalities for continued fractions

Author:
R. M. Dudley

Journal:
Math. Comp. **49** (1987), 585-593

MSC:
Primary 40A15; Secondary 33A20, 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906191-X

MathSciNet review:
906191

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Abstract | References | Similar Articles | Additional Information

Abstract: For some continued fractions $Q = {b_0} + {a_1}/({b_1} + \cdots )$ with *m*th 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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Keywords:
Alternating continued fractions,
monotonicity of errors

Article copyright:
© Copyright 1987
American Mathematical Society