Sturm sequence

Sturm sequence

[′stərm ‚sēkwəns]
(mathematics)
For a polynomial p (x), this is the sequence of functions ƒ0(x), ƒ1(x),…, where ƒ0(x) = p (x), ƒ1(x) = p ′(x), and ƒn (x) is the negative remainder that occurs by finding the greatest common divisor of ƒn-2(x) and ƒn-1(x) via the euclidean algorithm.
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The application of this method to Hermitian matrices is essentially based on the Sturm sequence property, which means that for any given real number [lambda], the number of sign changes in the sequence of the characteristic polynomials of the principal leading submatrices of an N x N Hermitian matrix A equals the number of eigenvalues which are less than that [lambda].
To this end, we use the Sturm sequence property of the polynomials (2.6) and the bisection method described in Section 2.5.
Compute the the ratios of consecutive pairs of Sturm sequence of polynomials (2.6).
Find the number of sign changes in the Sturm sequence of polynomials (2.6) which is in fact equal to the number of negative [D.sub.k]([lambda]), k = 1, ..., N, that have been just obtained since these are ratios of pairs of consecutive Sturm polynomials.
The function v = find Nu([lambda], generators) in the algorithm below gathers the operations in Theorem 3.1 for finding the number v of sign changes in the Sturm sequence. This is a simplified--but still well working--version of the algorithm in [1].