scholarly journals Binary Symmetric Matrix Inversion Through Local Complementation

2012 ◽  
Vol 116 (1-4) ◽  
pp. 15-23 ◽  
Author(s):  
Robert Brijder ◽  
Hendrik Jan Hoogeboom
Keyword(s):  
Symmetric Matrix ◽  
Matrix Inversion ◽  
Local Complementation

scholarly journals A Covariance Matrix Inversion Problem arising from the Construction of Phylogenetic Trees

2007 ◽  
Vol 10 ◽  
pp. 119-131
Author(s):  
Tom M. W. Nye ◽  
Brad J. C. Baxter ◽  
Walter R. Gilks
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Symmetric Matrix ◽  
Matrix Inversion ◽  
Covariance Matrices ◽  
Inner Product ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Tree Construction ◽  
Under Construction

AbstractWe describe an efficient algorithm for the inversion of covariance matrices that arise in the context of phylogenetic tree construction. Phylogenetic trees describe the evolutionary relationships between species, and their construction is computationally demanding. Many approaches involve the symmetric matrix of evolutionary distances between species. Regarding these distances as random variables, the corresponding set of variances and covariances form a rank-4 tensor, and the inner-product defined by its inverse can be used to assign statistical scores to candidate trees. We describe a natural set of assumptions for the phylogenetic tree under construction, and show how under these assumptions the covariance tensor for a tree with n leaves can be inverted in O(n2) operations. In addition to presenting the inversion algorithm, we hope this article will open algebraic and computational problems from the field of phylogeny to a wider audience.

QR factorization and algorithm for generalized row (column) symmetric matrix

2013 ◽  
Vol 32 (4) ◽  
pp. 990-993
Author(s):  
Hui-ping YUAN
Keyword(s):  
Symmetric Matrix ◽  
Qr Factorization

Fast Bilinear Algorithms for Symmetric Tensor Contractions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Symmetric Matrix ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Symmetric Tensor ◽  
Partial Sums ◽  
Symmetric Tensors ◽  
Outer Product ◽  
Bilinear Complexity ◽  
Special Cases ◽  
Tensor Contractions

AbstractIn matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

Correction to "A matrix inversion identity"

1971 ◽  
Vol 16 (5) ◽  
pp. 522-522
Author(s):  
T. Fortmann
Keyword(s):  
Matrix Inversion

Hardware Implementation of Floating Point Matrix Inversion Modules on FPGAs

2020 ◽  
Author(s):  
S Chetan ◽  
J Manikandan ◽  
V Lekshmi ◽  
S Sudhakar
Keyword(s):  
Hardware Implementation ◽  
Matrix Inversion ◽  
Floating Point

Certification of Algorithm 58: Matrix inversion

1962 ◽  
Vol 5 (8) ◽  
pp. 438-439
Author(s):  
Richard George
Keyword(s):  
Matrix Inversion
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