The textured, iterative approximation algorithms are a class fast linear equation solvers [1], [2] and differ from the classical iterative algorithms fundamentally in their approximations of system matrices. The textured approach uses different approximations of a system matrix in a round-robin fashion while the classical approaches use a single fixed approximation. It therefore has a better approximation of system matrix and a potentially faster speed. In this note we prove that the convergent speed of the textured iterative algorithms for linear equations with a class of tridiagonal system matrices is strictly faster than the corresponding classical iterative algorithms. We also give the spectral radii of the textured iterative and classical algorithms for this class of linear equations. These results provide some insights and theoretical supports for the textured iterative algorithms.