The authors propose an algorithm to solve numerically the generalized eigenvalue problem \(Tx = \lambda Sx\); \(T\), \(S\) being symmetric tridiagonal matrices. The algorithm is based on finding zeros of the polynomial equation \(\text{det} [T - \lambda S] = 0\); the characteristic polynomial and its derivatives can be evaluated by modified three-term recurrences. This equation is solved by Laguerre's iteration with starting point obtained by a split-merge process. Numerical results and discussion of advantages of the algorithm are also presented.