If is the computed solution to a tridiagonal system Ax b obtained by Gaussian elimination, what is the "best" bound available for the error x and how can it be computed efficiently? This question is answered using backward error analysis, perturbation theory, and properties of the LU factorization of A. For three practically important classes of tridiagonal matrix, those that are symmetric posi...

Infrared Non-Destructive Testing (INDT) is known as an effective and rapid method for nondestructive inspection. It can detect a broad range of near-surface structuring flaws in metallic and composite components. Those flaws are modeled as a smooth contour centered at peaks of stored thermal energy, termed Regions of Interest (ROI). Dedicated methodologies must detect the presence of those ROIs...

We present a smoothed analysis of Gaussian elimination, both with partial pivoting and without pivoting. Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the ...