The spectral transformation Lanczos method for the sparse symmetric definite generalized eigenvalue problem for matrices $A$ and $B$ is an iterative method that addresses the case of semidefinite or ill conditioned $B$ using a shifted and inverted formulation of the problem. This paper proposes the same approach for dense problems and shows that with a shift chosen in accordance with certain constraints, the algorithm can conditionally ensure that every computed shifted and inverted eigenvalue is close to the exact shifted and inverted eigenvalue of a pair of matrices close to $A$ and $B$. Under the same assumptions on the shift, the analysis of the algorithm for the shifted and inverted problem leads to useful error bounds for the original problem, including a bound that shows how a single shift that is of moderate size in a scaled sense can be chosen so that every computed generalized eigenvalue corresponds to a generalized eigenvalue of a pair of matrices close to $A$ and $B$. The computed generalized eigenvectors give a relative residual that depends on the distance between the corresponding generalized eigenvalue and the shift. If the shift is of moderate size, then relative residuals are small for generalized eigenvalues that are not much larger than the shift. Larger shifts give small relative residuals for generalized eigenvalues that are not much larger or smaller than the shift.

翻译：针对矩阵$A$和$B$的稀疏对称正定广义特征值问题的谱变换Lanczos方法，是一种通过采用移位求逆形式来处理$B$为半正定或病态情况的迭代方法。本文将该方法推广至稠密问题，并证明在满足特定约束条件下选取移位参数时，该算法能够保证每个计算得到的移位求逆特征值都接近于一对接近$A$和$B$的矩阵的精确移位求逆特征值。在相同移位假设下，对移位求逆问题的算法分析可推导出原始问题的有效误差界，其中包括一个关键界限：通过选取在缩放意义下大小适中的单一移位参数，可使每个计算得到的广义特征值都对应一对接近$A$和$B$的矩阵的广义特征值。计算得到的广义特征向量产生的相对残差取决于对应广义特征值与移位参数的距离。若移位参数大小适中，则对于不大于移位参数太多的广义特征值，其相对残差较小；而更大的移位参数可使广义特征值在既不太大于也不太小于移位参数时获得较小的相对残差。