Journal ArticleDOI
On Block Elimination for Sparse Linear Systems
TLDR
The surprising result is obtained that it may be beneficial to compute an unsymmetric factorization of a symmetric matrix under certain sparsity conditions.Abstract:
We consider the solution of linear equations involving a sparse coefficient matrix having a triangular factorization. In addition to the usual triangular factorization, we consider block factorizations, where the diagonal blocks of the factors are not necessarily triangular. We show that under certain sparsity conditions, these alternate factorizations may require fewer arithmetic operations and less storage. In particular, we obtain the surprising result that it may be beneficial to compute an unsymmetric factorization of a symmetric matrix.read more
Citations
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Journal ArticleDOI
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.
Journal ArticleDOI
Two Fast Algorithms for Sparse Matrices: Multiplication and Permuted Transposition
TL;DR: An O(M) algorithm is produced to solve A x = b where M is the number of multiplications needed to factor A into L U and the concept of an unordered merge plays a key role in obtaining this algorithm.
Journal ArticleDOI
A survey of sparse matrix research
TL;DR: This paper surveys the state of the art in sparse matrix research in January 1976, and discusses the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms.
Journal ArticleDOI
A survey of direct methods for sparse linear systems
TL;DR: The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems.
Journal ArticleDOI
An automatic one-way dissection algorithm for irregular finite element problems
TL;DR: A formal definition of a nested dissection ordering of the graph of a general sparse symmetric matrix A is given and some preliminary results which provide a direct relationship between these orders and theorems are introduced.
References
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Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Book
The finite element method in engineering science
TL;DR: In this paper, the authors describe how people search numerous times for their favorite books like this the finite element method in engineering science, but end up in malicious downloads, and instead they cope with some infectious bugs inside their computer.
Journal ArticleDOI
Nested Dissection of a Regular Finite Element Mesh
TL;DR: This paper presents an unusual numbering of the mesh (unknowns) and shows that if the authors avoid operating on zeros, the $LDL^T $ factorization of A can be computed using the same standard algorithm in $O(n^3 )$ arithmetic operations.
Book ChapterDOI
A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations
TL;DR: By following the formulation of elimination as a combinatorial process a considerable insight into the elimination process by studying the evolution of the cycle structure and the vertex-separator, or cut-set, structure of a graph under elimination can be gained.