Abstract: Suppose that one knows a very accurate approximation (+ to an eigenvalue A of a
symmetric tridiagonal matrix T. A good way to approximate the eigenvector x is to
discard an appropriate equation, say the rth, from the system (T – aI)x = 0 and
then to solve the resulting underdetermined system in any of several stable ways.
However the output x can be completely inaccurate if T is chosen poorly, and in the
absence of a quick and reliable way to choose r, this method has lain neglected for
over 35 years. Experts in boundary value problems have known about the special
structure of the inverse of a tridiagonal matrix since the 1960s and their double
triangular factorization technique (down and up) gives directly the redundancy of
each equation and so reveals the set of good choices for r. The relation of double
factorization to the eigenvector algorithm of Godunov and his collaborators is described.
The results extend to band matrices and to zero entries in eigenvectors, and
have uses beyond eigenvector computation.
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Citation
- Fernando’s Solution to Wilkinson’s Problem: An Application of Double Factorization (pdf, software)
B. Parlett, I. Dhillon.
Linear Algebra and its Applications, pp. 247-279, 1997.
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