A Way to Find the Most Redundant Equation in a Tridiagonal System,
CALIFORNIA UNIV BERKELEY CENTER FOR PURE AND APPLIED MATHEMATICS
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Suppose that one knows a very accurate approximation delta to an eigenvalue lambda of a symmetric tridiagonal matrix T. A good way to approximate the eigenvector is to discard an appropriate equation, say the rth, from the system T- delta Ix 0 and then to solve the resulting underdetermined system in any of several stable ways. However the output can be completely inaccurate if r 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. We show how double triangular factorization down and up, which is closely related to twisted factorization, gives us directly the redundancy of each equation and so reveals the set of good choices for r. The results extend to band matrices and the applications go beyond eigenvector computation to determinant evaluation and solution of well conditioned systems.
- Numerical Mathematics