arXiv subjects

Manuel Radons

Publications and source records attributed to Manuel Radons.

20 records · Page 2Linked to original sources

arXiv2016

Direct solution of piecewise linear systems

Let $S$ be a real $n\times n$ matrix, $z,\hat c\in \mathbb R^n$, and $| z|$ the componentwise modulus of $z$. Then the piecewise linear equation system $$z-S| z| = \hat c$$ is called an \textit{absolute value equation} (AVE). It has been proven to be equivalent to the general \textit{linear complementarity problem}, which means that it is NP hard in general. We will show that for several system classes the AVE essentially retains the good natured solvability properties of regular linear systems. I.e., it can be solved directly by a slightly modified Gaussian elimination that we call the signed Gaussian elimination. For dense matrices $S$ this algorithm has the same operations count as the classical Gaussian elimination with symmetric pivoting. For tridiagonal systems in $n$ variables its computational cost is roughly that of sorting $n$ floating point numbers. The sharpness of the proposed restrictions on $S$ will be established.

math.OC↗

arXiv2016

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

A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz 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 matrices are dense and there exist well known explicit formulas by which they can be calculated in $\mathcal O(n^2)$. In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If $\vert a\vert > 2\vert b\vert$, that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of $n$ for sufficiently large $n$. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses $\mathcal O(1)$ floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver.

math.NA↗