Geodesic
Iterative solution of the secular equation
Matematičeskie zametki, Tome 77 (2005) no. 2, pp. 303-310.

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The secular equation with real symmetric positive definite $n\times n$ matrix is transformed into a system of $n^2$ quadratic equations for which it is possible to construct a convergent procedure realizing an iterative solution. An example of the numerical realization of the method for solving the problem of determining the electronic energy spectrum is given. 
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L. S. Chkhartishvili. Iterative solution of the secular equation. Matematičeskie zametki, Tome 77 (2005) no. 2, pp. 303-310. http://geodesic.mathdoc.fr/item/MZM_2005_77_2_a10/

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