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A bound on the minimum rank of solutions to sparse linear matrix equations
2016
2016 American Control Conference (ACC)
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We derive a new upper bound on the minimum rank of matrices belonging to an affine slice of the positive semidefinite cone, when the affine slice is defined according to a system of sparse linear matrix ...
The bound depends on both the number of linear matrix equations and their underlying sparsity pattern. ...
Knowledge of an a priori upper bound on the minimum attainable rank serves to reduce the search space of the algorithms. ...
doi:10.1109/acc.2016.7526693
dblp:conf/amcc/LoucaBB16
fatcat:m4nfy67kk5exjfvhq4eu2lw4zm
Page 4319 of Mathematical Reviews Vol. , Issue 2000f
[page]
2000
Mathematical Reviews
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The author takes under consideration the accelerated overrelax- ation (AOR) method in order to solve a linear system with a symmetric, positive-definite, consistently ordered, sparse coeffi- cient matrix ...
Matrix Anal. Appl. 21 (1999), no. 1, 209-229 (electronic).
Summary: “The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. ...
Page 387 of Mathematical Reviews Vol. , Issue 91A
[page]
1991
Mathematical Reviews
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Summary: “One of the mathematical models in seismic problems is the search for the minimum L, norm solution of an underdeter- mined system of linear equations. ...
These theoretical results may be useful, for example, in case a sequence of linear systems has to be solved, their matrices tend to a rank-deficient matrix and the accuracy lost of the solutions has to ...
Solving sparse linear equations over finite fields
1986
IEEE Transactions on Information Theory
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apply the matrix to a test vector, and the value of k depends on the algorithm. ...
The method of this article is based on the fact that, when a square matrix is repeatedly applied to a vector, the resulting vector sequence is linear recursive. ...
ACKNOWLEDGMENT The author is grateful for the advice of M. Kaminski, R. C. Mullin, A. M. Odlyzko, and an anonymous referee in preparing this article. ...
doi:10.1109/tit.1986.1057137
fatcat:ago7p7ndwfgh5lck5jyk5ox7sy
On the Minimum Distance of Non Binary LDPC Codes
[article]
2009
arXiv
pre-print
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Finally, we have provided a bound on the logarithmic minimum distance of nonbinary codes, using a strategy similar to the girth bound for binary codes. ...
We have carried out a preliminary study on the logarithmic bound on the the minimum distance of non-binary LDPC code ensembles. ...
We consider matrix A to have full rank rank(A) = Tc. We also consider the system (of equations and solutions) constrained such that, each elements of them are non zero. ...
arXiv:0906.2061v1
fatcat:lmrs3bppjfhglm3pdtrkzwkx24
Page 5299 of Mathematical Reviews Vol. , Issue 82m
[page]
1982
Mathematical Reviews
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Author’s summary: “The linear least squares problem Ax=b has a unique solution only if the matrix A has full column rank. ...
A modification of the QR decomposition method of solution of the least squares problem allows a determination of the rank of A’ and a partial interval analysis of the solution vector x. ...
Sherman-morrison-woodbury-formula-based algorithms for the surface smoothing problem
1997
Linear Algebra and its Applications
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a Lyapunov matrix equation or a cascade of two Lyapunov matrix equations. ...
Our solution requires the generation *of a dense capacitance matrix, for which we propose a practical and efficient method. ...
The second term is a minimum-norm solution to a linear system with the matrix 2 }kK .... which can be obtained by solving two cascaded Lyapunov matrix 2 equations. ...
doi:10.1016/s0024-3795(97)80366-7
fatcat:sqnk3zwbnngezn42ftkeksxch4
A Literature Review of Random Greedy Kaczmarz
2022
OALib
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The problem of solving large-scale linear equations widely exists in various research fields. There are many methods to solve the problem. ...
In recent years, the improvement of the algorithm has been a research hotspot in the iterative algorithm. ...
compatible sparse linear equations, and the approximate solution converges to the minimum norm. ...
doi:10.4236/oalib.1108663
fatcat:g7oujqrof5fgrlfqcsdhyrx7ge
Page 1164 of Mathematical Reviews Vol. , Issue 90B
[page]
1990
Mathematical Reviews
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This
method exploits sparsity and rank deficiency of the model ma- trix and is based on orthogonal Givens factorization of a set of sparse columns of the model matrix. ...
The theorem of Prager-Oettli for systems of linear equations is extended to linear complementarity problems. ...
Upper Bounds on the Error of Sparse Vector and Low-Rank Matrix Recovery
[article]
2015
arXiv
pre-print
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Suppose that a solution x to an underdetermined linear system b = Ax is given. x is approximately sparse meaning that it has a few large components compared to other small entries. ...
This result is extended to the case that b is perturbed by noise. Additionally, we generalize the upper bounds to the low-rank matrix recovery problem. ...
Introduction Let x 0 ∈ R m denote a sparse solution of an underdetermined system of linear equations b = Ax (1) in which b ∈ R n and A ∈ R n×m , m > n. ...
arXiv:1504.03195v2
fatcat:4sikbgc4kng7rgyfam5ttfi5qq
A Scheme for Handling Rank-Deficiency in the Solution of Sparse Linear Least Squares Problems
1991
SIAM Journal on Scientific and Statistical Computing
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Introduction In this article we consider the d-irect solution of a linear least squares problem . where A is a sparse M x n matrix, with M 2 n, and b is an m-vector. ...
For large and sparse problems, if the number of new equations is small, it is desirable to be able to update the solution to the first least squares problem using the new equations to obtain the new solutions ...
doi:10.1137/0912062
fatcat:ufyqpgeauraz7dpy62dk4zqzm4
A Unique "Nonnegative" Solution to an Underdetermined System: From Vectors to Matrices
2011
IEEE Transactions on Signal Processing
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This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. ...
A vector solution is the unique solution to an underdetermined linear system only if the measurement matrix has a row-span intersecting the positive orthant. ...
Finding the minimum rank matrix among all matrices satisfying given linear equations is a rank minimization problem. ...
doi:10.1109/tsp.2010.2089624
fatcat:zxaatth5kjhwpmxknnzgat7anm
Conditions for a unique non-negative solution to an underdetermined system
2009
2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
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This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. ...
A vector solution is the unique solution to an underdetermined linear system only if the measurement matrix has a row-span intersecting the positive orthant. ...
Finding the minimum rank matrix among all matrices satisfying given linear equations is a rank minimization problem. ...
doi:10.1109/allerton.2009.5394815
fatcat:euxjfzpb3jac3pr5g43p4lrnca
Preconditioned Iterative Methods for Homotopy Curve Tracking
1992
SIAM Journal on Scientific and Statistical Computing
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The fundamental linear algebra step in these algorithms is the computation of the kernel of the homotopy Jacobian matrix. Problems with large, sparse Jacobian matrices are considered. ...
Homotopy algorithms are a class of methods for solving systems of nonlinear equations that are globally convergent with probability one. ...
Solving large sparse nonlinear systems of equations via homotopy methods involves sparse rectangular linear systems of equations. The sparsity suggests the use of iterative solution methods. ...
doi:10.1137/0913002
fatcat:fmtbwa4lhfhgjhi4snh2gpm7ee
Page 2772 of Mathematical Reviews Vol. , Issue 81G
[page]
1981
Mathematical Reviews
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81¢:05044
The authors consider the problem of solving the positive definite system of linear equations Ax =b where the N x N matrix A is large and sparse. ...
Some analogies between numerical solutions of a system of linear
equations of the form Ax = b, where A is an n-dimensional square
matrix, and numerical solutions of a linear system of ordinary
differential ...
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