A Quasi-Separable Approach to Solve the Symmetric Definite Tridiagonal Generalized Eigenvalue Problem

@article{Vandebril2009AQA,
  title={A Quasi-Separable Approach to Solve the Symmetric Definite Tridiagonal Generalized Eigenvalue Problem},
  author={Raf Vandebril and Gene H. Golub and Marc Van Barel},
  journal={SIAM J. Matrix Analysis Applications},
  year={2009},
  volume={31},
  pages={154-174}
}
We present a new fast algorithm for solving the generalized eigenvalue problem $T\mathbf{x}=\lambda S\mathbf{x}$, in which both $T$ and $S$ are real symmetric tridiagonal matrices and $S$ is positive definite. A method for solving this problem is to compute a Cholesky factorization $S=LL^T$ and solve the equivalent symmetric standard eigenvalue problem $L^{-1}TL^{-T}(L^T\mathbf{x})=\lambda(L^T\mathbf{x})$. We prove that the matrix $L^{-1}TL^{-T}$ is quasi-separable; that is, all submatrices… CONTINUE READING

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