Calculation involving determinant of a matrix Suppose I have the following Toeplitz symmetric matrix
\begin{align}
M=\begin{bmatrix}
1 & c & c & x \\
c & 1 & c & c \\
c & c & 1 & c \\
x & c & c & 1 
\end{bmatrix}
\end{align}
I want to write an algorithm that takes $c$ as input and calculates the range of $x$ for which matrix $M$ is positive semidefinite.
Currently, I do Gaussian elimination by hand and reduce the problem to checking the determinant of a $2 \times 2$ matrix. But how do I automate the process so I can write a function that takes $c$ and $n$ as inputs, where $n$ is the dimension of $M$, and returns the range of $x$. Thanks!

Essentially, you want to compute the diameter of a $1$-dimensional spectrahedron. This diameter can be found by solving two semidefinite programs
$$\begin{array}{ll} \text{minimize} & \pm x\\ \text{subject to} & \begin{bmatrix} 1 & c & c & x \\
c & 1 & c & c \\
c & c & 1 & c \\
x & c & c & 1 
\end{bmatrix} \succeq \mathrm O_4\end{array}$$

spectrahedra linear-matrix-inequality semidefinite-programming
