The matrix in question is:
$$ A = \begin{bmatrix} 2N-1 & N-1 & 0 & & &\cdots & 0 \\ N-1 & 2N-3 & N-2 & 0 & & \cdots & 0 \\ 0 & N-2 & 2N-5 & \ddots & & & \vdots \\ \vdots & \cdots & \ddots & \ddots & \ddots \\ & & & \ddots & \ddots & 2 & 0\\ \vdots& & & & 2 & 3 & 1\\ 0& \cdots & & & 0 & 1 & 1 \end{bmatrix} $$
What I'm looking for is $U\sqrt{\Sigma}$ and $\sqrt{\Sigma}^{-1}U^T$ where $U$ is an orthogonal matrix, $\Sigma$ is a tridiagonal matrix given by:
$$ U\Sigma^2U^T = A $$
I'm using MATLAB to solve the eigenvalue decomposition. However, the precision is not good enough. Is it possible to exploit the structure of this given matrix to get better accuracy?