= Safe pivots for tridiagonal elimination
{title2=$p_i\ne0\quad(1\leq i\leq n)$}
The displayed <Thomas algorithm> requires every modified <pivot element> to be nonzero in exact <arithmetic>. Nonzero original diagonal entries alone are insufficient: later <pivot elements> can vanish after elimination. Two useful sufficient contexts are a real <symmetric positive-definite matrix>, and <strict diagonal dominance> by rows:
$$
|b_i|>|a_i|+|c_i|\quad(1\leq i\leq n),\qquad a_1=c_n=0.
$$
For the <symmetric positive-definite matrix> case, elimination leaves positive-definite remaining blocks by the <Schur complement>, so each $p_i>0$. Equivalently, the initial <principal minors> $\Delta_i=\det A_{1:i,1:i}$ are positive, and $p_i=\Delta_i/\Delta_{i-1}$ with $\Delta_0=1$.
For <strict diagonal dominance>, a short induction shows $|p_i|>|c_i|$. The first row gives $|p_1|=|b_1|>|c_1|$. Assuming the preceding bound, the <triangle inequality> gives
$$
|p_i|\geq|b_i|-\frac{|a_i|\,|c_{i-1}|}{|p_{i-1}|}\geq|b_i|-|a_i|>|c_i|.
$$
In particular $p_n\ne0$. These are structural guarantees against a zero <pivot element>, not universal bounds on forward error in <floating-point arithmetic>. A poorly conditioned <linear system> can still amplify <roundoff error>; <numerical stability> and the <condition number> must be distinguished.
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