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I have an application that is somewhat similar to the situation of computing Gaussian quadrature nodes and weights: simply put, I need to compute the eigenvalues and the last two (normalized) components of the eigenvectors of a symmetric tridiagonal matrix.

I am aware of the Golub-Welsch scheme that uses the implicit QL/QR algorithm for Gaussian quadrature rule generation; the difference between it and my situation is that one uses the first component of the eigenvectors for Golub-Welsch.

I can modify the relevant LAPACK routines for my specific situation, but I was wondering if there were any new developments in this area. From my limited searching, it does not seem that MRRR or divide-and-conquer allow for computing only selected components.

Literature pointers would be extremely appreciated.

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    $\begingroup$ It sounds like an interesting application. Perhaps you could clarify what is meant by "normalized" in connection with the last two components. One thinks of permuting the rows and columns of your matrix to make the corresponding last two components become the first two components. In any case it might help to know more about the application. $\endgroup$ – hardmath Aug 24 '18 at 16:03

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