113,978 research outputs found
How Ordinary Elimination Became Gaussian Elimination
Newton, in notes that he would rather not have seen published, described a
process for solving simultaneous equations that later authors applied
specifically to linear equations. This method that Euler did not recommend,
that Legendre called "ordinary," and that Gauss called "common" - is now named
after Gauss: "Gaussian" elimination. Gauss's name became associated with
elimination through the adoption, by professional computers, of a specialized
notation that Gauss devised for his own least squares calculations. The
notation allowed elimination to be viewed as a sequence of arithmetic
operations that were repeatedly optimized for hand computing and eventually
were described by matrices.Comment: 56 pages, 21 figures, 1 tabl
On the Identification of Symmetric Quadrature Rules for Finite Element Methods
In this paper we describe a methodology for the identification of symmetric
quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms,
pyramids, and hexahedra. The methodology is free from manual intervention and
is capable of identifying an ensemble of rules with a given strength and a
given number of points. We also present polyquad which is an implementation of
our methodology. Using polyquad we proceed to derive a complete set of
symmetric rules on the aforementioned domains. All rules possess purely
positive weights and have all points inside the domain. Many of the rules
appear to be new, and an improvement over those tabulated in the literature.Comment: 17 pages, 6 figures, 1 tabl
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