The eigenvalues of a tridiagonal matrix are bounded by the maximum and minimum roots of a sequence of functions that form a chain sequence.

Specifically, given a general tridiagonal matrix

$$ A_n= \begin{pmatrix} a_{1} & b_{1} \\\ c_{1} & a_{2} & b_{2} \\\ & c_{2} & a_{3} & \ddots \\\ & & \ddots & \ddots & b_{n-1} \\\ & & & c_{n-1} & a_{n} \end{pmatrix} $$

The eigenvalues of $A_n$ belong to the real interval $(A,B)$ if and only if:

1. $ \forall 1 \leq k < n $: $ A < a_k < B $
2. $ \left\{\frac{b_k c_k}{(A-a_k)(A-a_{k+1})}\right\}^{n-1}_1 $ and $ \left\{\frac{b_k c_k}{(B-a_k)(B-a_{k+1})}\right\}^{n-1}_1 $ are both chain sequences

For more information, see theorem 1 in paper. Further, they reformulate an equivalent condition in theorem 2, which is of more practical use.

For chain sequences, Wikipedia is a good place to start, but I also recommend this paper.

The eigenvalues of a tridiagonal matrix are bounded by the maximum and minimum roots of a sequence of functions that form a chain sequence.

Specifically, given a general tridiagonal matrix

$$ A_n= \begin{pmatrix} a_{1} & b_{1} \\\ c_{1} & a_{2} & b_{2} \\\ & c_{2} & a_{3} & \ddots \\\ & & \ddots & \ddots & b_{n-1} \\\ & & & c_{n-1} & a_{n} \end{pmatrix} $$

The eigenvalues of $A_n$ belong to the real interval $(A,B)$ if and only if:

1. $ \forall 1 \leq k < n $: $ A < a_k < B $
2. $ \left\{\frac{b_k c_k}{(A-a_k)(A-a_{k+1})}\right\}^{n-1}_1 $ and $ \left\{\frac{b_k c_k}{(B-a_k)(B-a_{k+1})}\right\}^{n-1}_1 $ are both chain sequences

For more information, see theorem 1 in the paper "Bound on the Extreme Zeros of Orthogonal Polynomials" (see references below). Further, they reformulate an equivalent condition in theorem 2, which is of more practical use.

For chain sequences, Wikipedia is a good place to start, but I also recommend the paper "Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices" for convergence properties of parameter sequences, which can help determine whether a given sequence satisfies the properties to be a chain sequence.

References:

[1] Ismail, Mourad E. H.; Li, Xin, Bound on the extreme zeros of orthogonal polynomials, Proc. Am. Math. Soc. 115, No. 1, 131-140 (1992). ZBL0744.33005.

[2] Szwarc, Ryszard, Chain sequences, orthogonal polynomials, and Jacobi matrices, J. Approximation Theory 92, No. 1, 59-73 (1998). ZBL0892.42014.

The eigenvalues of a tridiagonal matrix are bounded by the maximum and minimum roots of a sequence of functions that form a chain sequence.

Specifically, given a general tridiagonal matrix

$$ A_n= \begin{pmatrix} a_{1} & b_{1} \\\ c_{1} & a_{2} & b_{2} \\\ & c_{2} & a_{3} & \ddots \\\ & & \ddots & \ddots & b_{n-1} \\\ & & & c_{n-1} & a_{n} \end{pmatrix} $$

The eigenvalues of $A_n$ belong to the real interval $(A,B)$ if and only if:

1. $ \forall 1 \leq k < n $: $ A < a_k < B $
2. $ \left\{\frac{b_k c_k}{(A-a_k)(A-a_{k+1})}\right\}^{n-1}_1 $ and $ \left\{\frac{b_k c_k}{(B-a_k)(B-a_{k+1})}\right\}^{n-1}_1 $ are both chain sequences

For more information, see theorem 1 in paper. Further, they reformulate an equivalent condition in theorem 2, which is of more practical use.

For chain sequences, Wikipedia is a good place to start, but I also recommend this paper.

The eigenvalues of a tridiagonal matrix are bounded by the maximum and minimum roots of a sequence of functions that form a chain sequence.

Specifically, given a general tridiagonal matrix

$$ A_n= \begin{pmatrix} a_{1} & b_{1} \\\ c_{1} & a_{2} & b_{2} \\\ & c_{2} & a_{3} & \ddots \\\ & & \ddots & \ddots & b_{n-1} \\\ & & & c_{n-1} & a_{n} \end{pmatrix} $$

The eigenvalues of $A_n$ belong to the real interval $(A,B)$ if and only if:

1. $ \forall 1 \leq k < n $: $ A < a_k < B $
2. $ \left\{\frac{b_k c_k}{(A-a_k)(A-a_{k+1})}\right\}^{n-1}_1 $ and $ \left\{\frac{b_k c_k}{(B-a_k)(B-a_{k+1})}\right\}^{n-1}_1 $ are both chain sequences

For more information, see theorem 1 in paper. Further, they reformulate an equivalent condition in theorem 2, which is of more practical use.

For chain sequences, Wikipedia is a good place to start, but I also recommend this paper.