Sharp growth for cyclic tridiagonal partial pivoting

For Gaussian elimination with partial pivoting on nonsingular real or complex
cyclic tridiagonal matrices of order $n \ge 4$, with nonzeros only on the three
central diagonals and in the corners $(1, n)$ and $(n, 1)$, both corners
nonzero, the manuscript proves that the largest growth factor over all such
matrices and all maximal-modulus pivot ties, measured over every active
Schur-complement entry, is $c_n = F_{n+1} + 1$, where $F_0 = 0$ and $F_1 = 1$;
the first values are $6, 9, 14, 22$. A two-row front argument, which bounds the history of every
column (by Fibonacci numbers in the last two columns), gives the upper bound in the fixed original column
order. A rational construction with both cyclic corners nonzero, built from a
unit lower triangular factor with two subdiagonals, attains it.