I am looking for analytical expressions for the the singular values of a Toeplitz matrix. If possible for a general Toeplitz matrix but I would also take results for a tridiagonal Toeplitz matrix
\begin{equation} A = \begin{pmatrix} a & b \\ c & a & b \\ & \ddots & \ddots & \ddots \\ & & c & a\end{pmatrix} \end{equation}
While I have references for the eigenvalues of such a matrix, e.g. the book by Smith, I am struggling to find expressions for the singular values.
I am aware that if $c=b$ and $A$ is real and symmetric, the singular values are basically the eigenvalues (absolute values of the eigenvalues to be precise). But what if that is not the case?
Smith, G. D., Numerical solution of partial differential equations. Finite difference methods. 2nd ed, Oxford Applied Mathematics and Computing Science Series. Oxford: Clarendon Press. XII, 304 p. hbk: \textsterling 9.00; pbk: \textsterling 4.95 (1978). ZBL0389.65040.