This paper focuses on computing the eigenvalues of the generalized collocation matrix of the rational Said–Ball basis, also called as the quasi-rational Said–Ball–Vandermonde (q-RSBV) matrix, with high relative accuracy. To achieve this, we propose explicit expressions for the minors of the q-RSBV matrix and develop a high-precision algorithm to compute these parameters. Additionally, we present perturbation theory and error analysis to further analyze the accuracy of our approach. Finally, we provide some numerical examples to demonstrate the high relative accuracy of our algorithms.

中文翻译：

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计算拟有理 Said-Ball-Vandermonde 矩阵的特征值

本文重点计算有理 Said-Ball 基的广义配置矩阵（也称为准有理 Said-Ball-Vandermonde (q-RSBV) 矩阵）的特征值，具有较高的相对精度。为了实现这一目标，我们提出了 q-RSBV 矩阵次数的显式表达式，并开发了一种高精度算法来计算这些参数。此外，我们提出了扰动理论和误差分析，以进一步分析我们方法的准确性。最后，我们提供了一些数值示例来证明我们的算法具有较高的相对精度。