Stability to perturbations of $g$-fractions
Abstract
This paper investigates the stability of $g$--fractions to perturbations. Recurrence formulas for the relative errors of the approximant tails of the $g$--fraction are established, based on which a~formula for the relative error of the approximant is obtained, describing its accumulation in the sequence of approximants. Sufficient conditions for the stability of the $g$--fraction are established { using the majorant estimation technique}. These conditions are formulated in terms of a majorant continued fraction that bounds the magnitude of the relative error of the approximant. It is shown that stability is guaranteed if the relative errors of the coefficients and the variable of the fraction are bounded, and the numerical series formed from the parameters of the majorant diverges. The theoretical results are applied to find stability sets in the complex plane { using the method of value sets for the approximant tails}. In particular, it is proven that under certain conditions on the coefficients, the $g$--fraction is stable in the closed unit disk. The obtained error estimates can be used for accuracy control in practical problems that use $g$--fractions.
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