In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.

中文翻译：

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基于惯性的频谱切片用于对称二次特征值问题

在所有系数矩阵对称的二次特征值问题（QEP）中，可能存在复杂的特征值。但是，某些应用程序仅需要计算实际特征值。我们提出了一种基于Lanczos的方法，用于计算给定间隔的大型对称QEP中包含的所有实际特征值。该方法使用在不同偏移值处评估的二次多项式的矩阵惯性。这样，对于双曲线问题，可以确保已计算出区间中的所有特征值。我们还将讨论一般的非双曲情形。通过以紧凑​​的张量积表示形式表示计算出的伪兰科斯基础，我们的实现具有存储效率。我们显示了SLEPc库中具有并行实现的计算实验结果。