Journal Article10.1145/360767.360783
Tridiagonalization by permutations
Norman E. Gibbs,William G. Poole +1 more
10
TL;DR: A graph-theoretic algorithm which examines an arbitrary n × n matrix and determines whether or not it can be permuted into tridiagonal form is given and gives the explicit tridiagon form.
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Abstract: Tridiagonalizing a matrix by similarity transformations is an important computational tool in numerical linear algebra. Consider the class of sparse matrices which can be tridiagonalized using only row and corresponding column permutations. The advantages of using such a transformation include the absence of round-off errors and improved computation time when compared with standard transformations. A graph-theoretic algorithm which examines an arbitrary n × n matrix and determines whether or not it can be permuted into tridiagonal form is given. The algorithm requires no arithmetic while the number of comparisons, the number of assignments, and the number of increments are linear in n. This compares very favorably with standard transformation methods. If the matrix is permutable into tridiagonal form, the algorithm gives the explicit tridiagonal form. Otherwise, early rejection will occur.
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Citations
Computational methods of linear algebra
D. K. Faddeev,V. N. Faddeeva +1 more
TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.
689
The bandwidth problem for graphs and matrices—a survey
TL;DR: This survey describes all the results known to the authors as of approximately August 1981 and describes the effect on bandwidth of local operations such as refinement and contraction of graphs, bounds on bandwidth in terms of other graph invariants, the bandwidth of special classes of graph, and approximate bandwidth algorithms for graphs and matrices.
330
Complexity results for bandwidth minimization
TL;DR: In this article, a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form (bandwidth 2) whenever it is possible to do so using simultaneous row and column permutations is presented.
The profile of the Cartesian product of graphs
David Kuo,Jing-Ho Yan +1 more
TL;DR: If the vertices of a graph G can be ordered to satisfy a special neighborhood property, then so can the graph GxQ"n, which can be used to determine the profile of Q"n and K"mxQ" n.
1
On tradeoffs between width- and fill-like graph parameters
Dariusz Dereniowski,Adam Stański +1 more
TL;DR: It is proved that for an arbitrary graph G and an integer t ∈ {1, …, pw(G) + 1}, there exists an interval supergraph G′ of G such that for its clique number it holds ω(G′)≤(1+2t) p(G)+1 and the number of its edges is bounded by |E (G′)| ≤ (t + 2)p(G).
References
An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
TL;DR: In this article, a systematic method for finding the latent roots and principal axes of a matrix, without reducing the order of the matrix, has been proposed, which is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of minimized iterations.
Numerical Computation of the Characteristic Values of a Real Symmetric Matrix
Wallace Givens
- 01 Mar 1954
TL;DR: In this paper, the authors discuss the use of computers to determine the characteristic values of a symmetric matrix and deal with the accumulation of round-off errors, which is a common problem in symmetric matrices.
Householder's tridiagonalization of a symmetric matrix
TL;DR: In this article, an improved version of Householder's algorithm for tridiagonalization of a real symmetric matrix was discussed. But the most efficient form of the procedure depends on the method used to solve the eigenproblem of the derived tridiagon matrix.
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