Notes on linear systems with positive definite tri-diagonal coefficient matrices

In this article the author shows that under certain conditions a three-term recurrence for a tri-diagonal matrix becomes a two-term recurrence. Using this new recurrence, the possibility of the LU factorization of any tri-diagonal matrix is now easy to investigate. The positive definiteness of any real symmetric tri-diagonal matrix of any positive definite tri-diagonal is given. Some numerical examples are also given.