Zero-pivot failure of tridiagonal elimination ($\det \left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right)=-1$)

An invertible matrix need not admit the unpivoted Thomas algorithm. For example,Here $\det A=-1$ and $Ax=d$, but $p_1=0$ makes $m_2=1/0$ undefined. Interchanging the two equations instead gives the upper triangular matrix system $x_1+x_2=3$, $x_2=2$, solved immediately by back substitution. Even a nonzero first diagonal does not guarantee all later pivot elements: $\left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 1\end{array}\right)$ has determinant $-1$, yet $p_2=1-1=0$.

For a general tridiagonal matrix, use a solver with partial pivoting when these structural guarantees are unavailable or a dangerously small pivot element occurs. Adjacent row interchanges can introduce a second upper diagonal, so a pivoted solver must track that additional fill-in; it cannot blindly continue the unpivoted recurrence relations. LAPACK's DGTSV explicitly performs partial pivoting, stores this extra upper diagonal, and includes it in back substitution. Its fixed-width storage and bounded work per row retain $O(n)$ time for one right-hand side and $O(n)$ storage.^[ref]