Are there special methods to get exact eigenvalues of a tridiagonal matrix?
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Yes. For general tridiagonal matrices, see The Numerical Recipes, Chapter 11, or Golub-Van Loan. For symmetric tridiagonal matrices, you can do better, see Coakley/Rochlin's paper.
Coakley, Ed S.; Rokhlin, Vladimir, A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices, Appl. Comput. Harmon. Anal. 34, No. 3, 379-414 (2013). ZBL1264.65051.
Golub, Gene; Van Loan, Charles F., Matrix computations., Baltimore, MD: The Johns Hopkins Univ. Press. xxvii, 694 p. (1996). ZBL0865.65009.