Perturbation theory and error analysis for the Cauchy formula

In this work, we analyze the numerical behavior of the classical Cauchy identity $$ \sum _{\lambda } s_\lambda (a_1,\dots ,a_n)s_\lambda (x_1,\dots ,x_m) = \prod _{j=1}^n \prod _{i=1}^m \frac{1}{1 - a_j x_i}, $$ by developing perturbation and running error analyses. We show that relative perturbations in the nodes $$x_i$$ and coefficients $$a_j$$ only induce small relative changes in the output provided some relative gaps are sufficiently large. We also propose an algorithm computing a posteriori relative error bound with low computational overhead. Finally, we derive truncation error bounds for the Schur expansion of the formula. Numerical experiments confirm the sharpness of the theoretical results and illustrate the effectiveness of the proposed bounds in practice.