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A well known numerical task is the inversion of large symmetric tridiagonalToeplitz matrices, i.e., matrices whose entries equal $a$ on the diagonal and$b$ on the extra diagonals ($a, b\in \mathbb R$). The inverses of such matricesare dense and there exist well known explicit formulas by which they can becalculated in $\mathcal O(n^2)$. In this note we present a simplification ofthe problem that has proven to be rather useful in everyday practice: If $\verta\vert > 2\vert b\vert$, that is, if the matrix is strictly diagonallydominant, its inverse is a band matrix to working precision and the bandwidthis independent of $n$ for sufficiently large $n$. Employing this observation,we construct a linear time algorithm for an explicit tridiagonal inversion thatonly uses $\mathcal O(1)$ floating point operations. On the basis of thissimplified inversion algorithm we outline the cornerstones for an efficientparallelizable approximative equation solver.

$\mathcal O(n)$ working precision inverses for symmetric tridiagonal Toeplitz matrices with $\mathcal O(1)$ floating point calculations