Let LDLt be the triangular factorization of a real symmetric n\Theta n tridiagonal matrix so that L is a unit lower bidiagonal matrix, D is diagonal. Let (*; v) be an eigenpair, * 6 = 0, with the property that both * and v are determined to high relative accuracy by the parameters in L and D. Suppose also that the relative gap between * and its nearest neighbor _ in the spectrum exceeds 1=n; nj * \Gamma _j? j*j. This paper presents a new O(n) algorithm and a proof that, in the presence of round-off error, the algorithm computes an approximate eigenvector ^v that is accurate to working precision: j sin &quot;(v; ^v)j = O(n&quot;), where &quot; is the round-off unit. It follows that ^v is numerically orthogonal to all the other eigenvectors. This result forms part of a program to compute numerically orthogonal eigenvectors without resorting to the Gram-Schmidt process. The contents of this paper provide a high-level description and theoretical justification for LAPACK (version 3.0) subroutine DLAR1V
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