The location of real eigenvalues provides critical insights into the stability and resonance properties of physical systems. This paper presents a hybrid symbolic numeric approach for certified real eigenvalue localization. Our method combines Gershgorin disk analysis with Hermite matrix certification to compute certified intervals that enclose the real eigenvalues. These intervals can be further refined through bisectionlike procedures to achieve the desired precision. The proposed approach delivers reliable interval certifications while preserving computational efficiency. The effectiveness of the framework is demonstrated through a concise, fully worked computational example.

翻译：实特征值的位置为理解物理系统的稳定性和共振特性提供了关键信息。本文提出了一种混合符号数值方法，用于实现认证的实特征值定位。我们的方法将Gershgorin圆盘分析与Hermite矩阵认证相结合，以计算包含实特征值的认证区间。这些区间可以通过类二分法程序进一步细化，以达到所需的精度。所提出的方法在保持计算效率的同时，提供了可靠的区间认证。通过一个简洁、完整的计算实例，验证了该框架的有效性。