Stability under perturbations of $\pi$-fraction approximants
DOI:
https://doi.org/10.3842/umzh.v78i9-10.9921Keywords:
неперервний дріб, $\pi$--дріб, апроксиманта, стійкість до збурень, відносна похибка, число обумовленостіAbstract
UDC 517.518: 519.65
We define the condition number of element perturbations for an approximant of a continued fraction. On this basis, we investigate the stability of $\pi$-fraction approximants under coefficient perturbations. A formula for the relative error of the approximant is obtained in terms of the relative errors of fraction's coefficients. The differentiability of the function of relative error at zero is established. A formula for the condition number of the coefficient perturbations of the $\pi$-fraction approximant is derived, and a condition for its stability under perturbations is obtained based on the finiteness of this number. It is proved that the set of nonnegative values of the variable is the set of stability under the coefficient perturbations. An estimate for the condition number is obtained and, under additional restrictions imposed on the coefficients, its independence of the order of approximant is established.
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