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Let $n\ge 1$. Store the tridiagonal matrix $A$ by its diagonal $b_1,\dots ,b_n$, lower diagonal $a_2,\dots ,a_n$, and upper diagonal $c_1,\dots ,c_{n-1}$. The linear system $Ax=d$ iswith the nonexistent endpoint terms omitted. Initialize $p_1=b_1$ and $g_1=d_1$. For $i=2,\dots ,n$, compute the elimination multiplier, modified pivot element, and modified right-hand side:These recurrence relations apply the elementary row operation $R_i\leftarrow R_i-m_i{R}_{i-1}$ after the preceding row has already been reduced. The transformed linear system is an upper bidiagonal matrix:Thus back substitution is especially short:For $n=1$, only $x_1=d_1/b_1$ remains. Check each pivot element before its first division, including $p_n$ before back substitution.

Equivalently, this constructs an LU decomposition with unit lower bidiagonal matrix $L$, whose lower diagonal is $m_2,\dots ,m_n$, and upper bidiagonal matrix $U$, whose diagonal is $p_1,\dots ,p_n$ and upper diagonal is $c_1,\dots ,c_{n-1}$. The $g$ sweep is forward substitution in $Lg=d$. There is no fill-in outside these bands when no rows are interchanged. LAPACK's DGTSV source uses these same coefficient and right-hand-side updates in its branch without row interchange.^[ref]

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]