Gaussian elimination is not optimal, revisited
نویسندگان
چکیده
منابع مشابه
How Accurate is Gaussian Elimination?∗
J.H. Wilkinson put Gaussian elimination (GE) on a sound numerical footing in the 1960s when he showed that with partial pivoting the method is stable in the sense of yielding a small backward error. He also derived bounds proportional to the condition number κ(A) for the forward error ‖x − x̂‖, where x̂ is the computed solution to Ax = b. More recent work has furthered our understanding of GE, la...
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Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V (D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs (x, z) and (y, z) are both present in D. The competition number k(G) for an undirected graph G is the least number r such that there exists an acyclic digraph F on |V (G)| + r vertice...
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As the standard method for solving systems of linear equations, Gaussian elimination (GE) is one of the most important and ubiquitous numerical algorithms. However, its successful use relies on understanding its numerical stability properties and how to organize its computations for efficient execution on modern computers. We give an overview of GE, ranging from theory to computation. We explai...
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Newton, in an unauthorized textbook, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — that Newton did not want to publish, that Euler did not recommend, that Legendre called “ordinary,” and that Gauss called “common” — is now named after Gauss: “Gaussian” elimination. (One suspects, he would not be amused.) Gauss’s...
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We consider the solution of a linear system Ax = b on a distributed memory machine when the matrix A has full rank and is large, sparse and nonsymmetric. We use our Cartesian nested dissection algorithm to compute a ll-reducingcolumn ordering of the matrix. We develop algorithms that use the associated separator tree to estimate the structure of the factor and to distribute and perform numeric ...
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ژورنال
عنوان ژورنال: Journal of Logical and Algebraic Methods in Programming
سال: 2016
ISSN: 2352-2208
DOI: 10.1016/j.jlamp.2016.06.003