The Significant Order of Symmetric Tridiagonal Matrices
Thuy T. T. Bui, Gary L. Hunter · IMA Journal of Applied Mathematics · 1974
It is demonstrated that an eigenvalue of a symmetric tridiagonal matrix as calculated by the Sturm sequence bisection procedure, is determined in general by a leading principal partition of the whole matrix. The order of this leading principal partition is called the significant order of the matrix. The significant order depends upon the specific eigenvalue being calculated, and upon the working precision of the computer. This numerical phenomenon may be employed to accelerate the bisection procedure. In applications it either indicates the adequacy or otherwise of the working precision for finite matrices, or in the case of theoretically infinite matrices, the effective truncation order for the precision employed.