Topic
Tridiagonal matrix algorithm
About: Tridiagonal matrix algorithm is a research topic. Over the lifetime, 1070 publications have been published within this topic receiving 21084 citations.
Papers published on a yearly basis
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Papers
TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
Abstract: t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical operations (all logarithms in this paper are for base 2, thus tog 7 ~ 2.8; the usual method requires approximately 2n 3 arithmetical operations). The algorithm induces algorithms for invert ing a matr ix of order n, solving a system of n linear equations in n unknowns, comput ing a determinant of order n etc. all requiring less than const n l°g 7 arithmetical operations. This fact should be compared with the result of KLYUYEV and KOKOVKINSHCHERBAK [1 ] tha t Gaussian elimination for solving a system of l inearequations is optimal if one restricts oneself to operations upon rows and columns as a whole. We also note tha t WlNOGRAD [21 modifies the usual algorithms for matr ix multiplication and inversion and for solving systems of linear equations, trading roughly half of the multiplications for additions and subtractions. I t is a pleasure to thank D. BRILLINGER for inspiring discussions about the present subject and ST. COOK and B. PARLETT for encouraging me to write this paper. 2. We define algorithms e~, ~ which mult iply matrices of order m2 ~, by induction on k: ~ , 0 is the usual algorithm, for matr ix multiplication (requiring m a multiplications and m 2 ( m t) additions), e~,k already being known, define ~ , ~ +t as follows: If A, B are matrices of order m 2 k ~ to be multiplied, write
2,978 citations
TL;DR: A graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations is considered, and it is conjecture that the problem of finding a minimum ordering is NP-complete.
Abstract: We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to calculate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with special properties. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in $O(n + e)$ time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in $O(ne)$ time. We conjecture that the problem of finding a minimum ordering is NP-complete
1,457 citations
28 Mar 1983
TL;DR: In this article, the authors give upper and lower bounds for the degrees of the elements of a Gr6bner base, which are based on projective algebraic geometry and the choice of the ordering appears to be critical.
Abstract: In the past few years, two very different methods have been developed for solving systems of algebraic equations : the method of Gr6bner bases or standard bases [Buc I, Buc 2, Tri, P.Y] and the one which I presented in Eurosam 79 [Laz 2, Laz 3] based on gaussian elimination in some matrices. Although they look very different, they are, in fact, very similar, at least if we restrict ourselves to the first step of my method. On the other hand Gr6bner base algorithms are very close to the tangent cone algorithm of Mora [Mor]. All of these algorithms are related to Gaussian elimination. In the first part of this paper, we try to develop all these relations and to show that this leads to improvements in some of these algorithms. In the second part we give upper and lower bounds for the degrees of the elements of a Gr6bner base. These bounds are based on projective algebraic geometry. The choice of the ordering appears to be critical : lexicographical orderings give Gr6bner bases of high degree, while reverse lexicographical orderings lead to low degrees.
496 citations
TL;DR: The method is shown to be stable and for a large class of matrices it is, asymptotically, faster by an order of magnitude than theQR method.
Abstract: A method is given for calculating the eigenvalues of a symmetric tridiagonal matrix The method is shown to be stable and for a large class of matrices it is, asymptotically, faster by an order of magnitude than theQR method
460 citations
TL;DR: This paper presents an efficient technique for performing a spatially inhomogeneous edge-preserving image smoothing, called fast global smoother, focusing on sparse Laplacian matrices consisting of a data term and a prior term that approximate the solution of the memory- and computation-intensive large linear system by solving a sequence of 1D subsystems.
Abstract: This paper presents an efficient technique for performing a spatially inhomogeneous edge-preserving image smoothing, called fast global smoother. Focusing on sparse Laplacian matrices consisting of a data term and a prior term (typically defined using four or eight neighbors for 2D image), our approach efficiently solves such global objective functions. In particular, we approximate the solution of the memory-and computation-intensive large linear system, defined over a d-dimensional spatial domain, by solving a sequence of 1D subsystems. Our separable implementation enables applying a linear-time tridiagonal matrix algorithm to solve d three-point Laplacian matrices iteratively. Our approach combines the best of two paradigms, i.e., efficient edge-preserving filters and optimization-based smoothing. Our method has a comparable runtime to the fast edge-preserving filters, but its global optimization formulation overcomes many limitations of the local filtering approaches. Our method also achieves high-quality results as the state-of-the-art optimization-based techniques, but runs ∼10-30 times faster. Besides, considering the flexibility in defining an objective function, we further propose generalized fast algorithms that perform Lγ norm smoothing (0 < γ < 2) and support an aggregated (robust) data term for handling imprecise data constraints. We demonstrate the effectiveness and efficiency of our techniques in a range of image processing and computer graphics applications.
385 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |