Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:


$ \begin{pmatrix}
1 & a\_{1} & 0 & ... & 0 \\\ 
a\_{1} & 1 & a\_{2} & & ... \\\
0 & a\_{2} & 1 & ... & 0 \\\ 
... & & ... & & a\_{n-1} \\\ 
0 & ... & 0 & a\_{n-1} & 1
\end{pmatrix}, $


where all $0< a\_i < 1$ for $i = 1\ldots n-1$ but the matrix is not necessarily diagonally dominant. I am interested in finding a tight upper-bound for the largest eigenvalue of this matrix.


Unfortunately, [Eigenvalues of Symmetric Tridiagonal Matrices](https://mathoverflow.net/questions/131527/eigenvalues-of-symmetric-tridiagonal-matrices) doesn't have the answer that I am looking for.



Upper Bounds on the Largest Eigenvalue of Jacobi Matrices
The entries are nonnegative, so the dominant eigenvector has
all entries positive, and its eigenvalue is an increasing function of the $a\_i$.
If each $a\_i = 1$ then that eigenvalue is $1 + 2 \cos\frac\pi{n+1}$
if I did this right; since you allow only $a\_i < 1$, this bound
$1 + 2 \cos\frac\pi{n+1}$ is not attained, but it is still the supremum of
eigenvalues of such matrices.


