Gaussian elimination with partial pivoting is performed routinely, millions times per day around the world, but partial pivoting (that is, row interchange of an input matrix) is communication intensive and has become the bottleneck of the elimination algorithm in the present day computer environment, in both cases of matrices of large and small size. Gaussian elimination with no pivoting as wel...

The deformation of applicable surfaces such as sheets of paper satisfies the differential geometric constraints of isometry (lengths and areas are conserved) and vanishing Gaussian curvature. We show that these constraints lead to a closed set of equations that allow recovery of the full geometric structure from a single image of the surface and knowledge of its undeformed shape. We show that t...

When solving systems of equations by using matrices, many teachers present a Gauss-Jordan elimination approach to row reducing matrices that can involve painfully tedious operations with fractions (which I will call the traditional method). In this essay, I present an alternative method to row reduce matrices that does not introduce additional fractions until the very last steps. The students i...

The history of Model Theory can be traced back to the work of Charles Sanders Peirce and Ernst Schröder, when semantics started playing a role in Logic. But the first outcome dates from 1915. It appears in the paper Über Möglichkeiten im Relativkalkül (Math. Ann. 76, 445-470) by Leopold Löwenheim. The period from 1915 to 1935 is, in words of R.L. Vaught, extraordinary. The method of elimination...

Let B ∈Mn(C) be a row diagonally dominant matrix, i.e., σi|bii| = n ∑ j=1 j 6=i |bij |, i = 1, . . . , n, where 0 ≤ σi < 1, i = 1, . . . , n, with σ = max1≤i≤n σi. We show that no pivoting is necessary when Gaussian elimination is applied to A = B−1. Moreover, the growth factor for A does not exceed 1 + σ. The same results are true with row diagonal dominance being replaced by column diagonal d...

1 Problem The interaction energy of an electric charge q and a " point " electric dipole with moment p of fixed magnitude, both at rest, is, in Gaussian units,

Neville elimination is an elimination procedure alternative to Gaussian elimination. It is very useful when dealing with totally positive matrices, for which nice stability results are known. Here we include examples, most of them test matrices used in MATLAB which are not totally positive matrices, where Neville elimination outperforms Gaussian elimination. 2010 Elsevier Inc. All rights reserved.