For tridiagonals \(T\) replace \(T\) with \(LDL\) (Q1591178)

The author discusses two of the ideas needed to compute eigenvectors that are orthogonal without making use of the Gram-Schmidt procedure when some of the eigenvalues are tightly clustered. In the first of the new schemes, the radical new goal is to compute an approximate eigenvector for a given approximate eigenvalue with a relative residual property. In the second scheme, due to the relative gaps in the spectrum the origin is shifted and the triangular factorization is used. In the development, the recently discovered differential stationary QD algorithms are used. Examples of both ideas, using four by four systems, are given.