This work deals with approximate solution of generalized eigenvalue problem with coefficient matrix that is an affine function of d-parameters. The coefficient matrix is assumed to be symmetric positive definite and spectrally equivalent to an average matrix for all parameters in a given set. We develop a Ritz method for rapidly approximating the eigenvalues on the spectral interval of interest $(0,\Lambda)$ for given parameter value. The Ritz subspace is the same for all parameters and it is designed based on the observation that any eigenvector can be split into two components. The first component belongs to a subspace spanned by some eigenvectors of the average matrix. The second component is defined by a correction operator that is a d + 1 dimensional analytic function. We use this structure and build our Ritz subspace from eigenvectors of the average matrix and samples of the correction operator. The samples are evaluated at interpolation points related to a sparse polynomial interpolation method. We show that the resulting Ritz subspace can approximate eigenvectors of the original problem related to the spectral interval of interest with the same accuracy as the sparse polynomial interpolation approximates the correction operator. Bound for Ritz eigenvalue error follows from this and known results. Theoretical results are illustrated by numerical examples. The advantage of our approach is that the analysis treats multiple eigenvalues and eigenvalue crossings that typically have posed technical challenges in similar works.

翻译：本研究探讨系数矩阵为d参数仿射函数的广义特征值问题的近似解法。假设系数矩阵对称正定，且对给定参数集中所有参数均与平均矩阵谱等价。我们发展了一种Ritz方法，用于快速近似给定参数值下感兴趣谱区间$(0,\Lambda)$内的特征值。该Ritz子空间对所有参数保持一致，其设计基于以下观察：任意特征向量均可分解为两个分量。第一分量属于由平均矩阵部分特征向量张成的子空间。第二分量由d+1维解析函数定义的修正算子所确定。利用此结构，我们从平均矩阵的特征向量与修正算子的采样中构建Ritz子空间。采样点选取与稀疏多项式插值方法相关的插值节点。我们证明所得Ritz子空间能以稀疏多项式插值逼近修正算子的相同精度，近似原问题在感兴趣谱区间内的特征向量。基于此结论与已知结果，推导出Ritz特征值误差界。数值算例验证了理论结果。本方法的优势在于：分析过程能处理多重特征值与特征值交叉现象——这些情形在同类研究中通常构成技术挑战。