On an inverse formula of a tridiagonal matrix (Q2914865)

In this article, the author considers the inverse \(Z_n=(z_{ij})\) of \(Y_n^{-1}\) of a general \(n\times n\) tridiagonal matrix NEWLINE\[NEWLINEY_n=(y_{ij})=\begin{pmatrix} \alpha_1 & \gamma_1 & 0 & \cdots & 0 \\ \beta_2 & \alpha_2 & \gamma_2 & \ddots & \vdots \\ 0 & \beta_3 & \ddots & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & \gamma_{n-1} \\ 0 & \cdots & 0 & \beta_n & \alpha_n \end{pmatrix}.NEWLINE\]NEWLINE By a transformation of the well-known Usmani's formula, he obtains the followingNEWLINENEWLINE\textbf{Theorem 1}: For a tridiagonal matrix \(Y_n\), define sequences \(f_l\) and \(g_l\) by NEWLINE\[NEWLINE f_l=\frac{-\gamma_l}{f_{l-1}\beta_l+\alpha_l}, g_l=\frac{-\beta_l}{g_{l+1}\gamma_l+\alpha_l} NEWLINE\]NEWLINE with \(f_0=g_{n+1}=0\). Further define NEWLINE\[NEWLINE p_{ki}=\begin{cases} \Pi_{l=k}^{i-1}f_l, & \text{if } k<i \\ 1, & \text{if } k=i \\ \Pi_{l=i+1}^kg_l & \text{if } k>i \\ \end{cases}. NEWLINE\]NEWLINE Then the \((i,j)\)-th element of \(Z_n\) is NEWLINE\[NEWLINEz_{ij}=\frac{p_{ij}}{\beta_jf_{j-1}+\alpha_j+\gamma_jg_{j+1}}.NEWLINE\]NEWLINE The author claims that, compared to Usmani's formula, the new formulation is (i) still quite simple, (ii) more stable in numerical calculation (which must be dealt with caution when zero minors are involved, as discussed in Section 5), (iii) better for study the limit form (which is discussed in Section 4).