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Generalized inverses of partitioned matrices in Banachiewicz–Schur form

Jerzy K. Baksalary, George P.H. Styan
2002 Linear Algebra and its Applications  
The problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so-called Banachiewicz-Schur form is reconsidered.  ...  of outer inverses, least-squares generalized inverses, and minimum norm generalized inverses.  ...  Namely, for any fixed generalized inverse G ∈ A{1}, S ∈ C p,q will denote the generalized Schur complement of A in M partitioned as in (1.4), i.e., S = D − CGB, (1.11) and N ∈ C n+q,m+p will stand for  ... 
doi:10.1016/s0024-3795(02)00334-8 fatcat:azpsi2qgjje5rc24pfwergzwfu

An invariant matrix structure in multiantenna communications

A.H. Sayed, W.M. Younis, A. Tarighat
2005 IEEE Signal Processing Letters  
This letter shows that the matrix structure with 2 2 Alamouti sub-blocks remains invariant under several nontrivial matrix operations, including matrix inversion, Schur complementation, Riccati recursion  ...  by using the block matrix inversion formula [4] (2) where denotes the 2 2 Schur complement of with respect to .  ...  Lemma 2 (Invariance Under Schur Complementation): The Schur complement of with respect to any leading block has a similar structure with 2 2 Alamouti sub-blocks.  ... 
doi:10.1109/lsp.2005.856886 fatcat:4y7f4fger5fxpp5k7ovh3veeky

Direct Schur complement method by domain decomposition based on H-matrix approximation

Wolfgang Hackbusch, Boris N. Khoromskij, Ronald Kriemann
2005 Computing and Visualization in Science  
Using the hierarchical (H-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form.  ...  As input, we require the Schur complement matrices corresponding to subdomains and represented in the H-matrix format.  ...  In general, we construct the H-matrix representation of all four matrices involved based on the block structure via standard admissibility criteria.  ... 
doi:10.1007/s00791-005-0008-3 fatcat:aj6j7xklbncpbgvfqutlfnv2nu

Distributed Schur Complement Techniques for General Sparse Linear Systems

Yousef Saad, Maria Sosonkina
1999 SIAM Journal on Scientific Computing  
These techniques utilize the Schur complement system for deriving the preconditioning matrix in a number of ways.  ...  Another preconditioner uses a sparse approximate-inverse technique to obtain certain local approximations of the Schur complement. Comparisons are reported for systems of varying di culty.  ...  Schur complements via approximate inverses Equation (17) describes in general terms an approximate block LU factorization for the global system (15) .  ... 
doi:10.1137/s1064827597328996 fatcat:kix2ztlo6banrjz4w7mxiju5sm

Drazin inverse of partitioned matrices in terms of Banachiewicz–Schur forms

N. Castro-González, M.F. Martínez-Serrano
2010 Linear Algebra and its Applications  
Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S = D − CA D B, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur  ...  Let M = A B C D be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A D . The purpose of this paper is twofold.  ...  In Section 2, we develop conditions under which the Drazin inverse of a partitioned matrix as in (1.3) having group invertible generalized Schur complement, can be expressed in terms of the Banachiewicz-Schur  ... 
doi:10.1016/j.laa.2009.11.024 fatcat:7wfllld5qnbrbe4oyqbfpyvv5a

Fast Algorithms for Displacement and Low-Rank Structured Matrices

Shivkumar Chandrasekaran, Nithin Govindarajan, Abhejit Rajagopal
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures.  ...  First note that T −1 is a Schur complement of M = T I −I 0 . So we could compute the generators of T −1 by running the generalized Schur algorithm on M half-way through.  ...  Though not all matrices with short SSS representations are the inverses of banded matrices, the diagonal representation (4) shows that every SSS representation is the Schur complement of a larger banded  ... 
doi:10.1145/3208976.3209025 dblp:conf/issac/ChandrasekaranG18 fatcat:glbsbjobf5buzp3qthn3ft66f4

Properties of Schur complements in partitioned idempotent matrices

Jerzy K Baksalary, Oskar Maria Baksalary, Tomasz Szulc
2004 Linear Algebra and its Applications  
the minus superscript denotes a generalized inverse of a given matrix.  ...  Related to a complex partitioned matrix P, having A, B, C, and D as its consecutive m × m, m × n, n × m, and n × n submatrices, are generalized Schur complements S = A − BD − C and T = D − CA − B, where  ...  Acknowledgement The authors are very grateful to a referee for helpful comments and suggestions on an earlier version of this paper.  ... 
doi:10.1016/s0024-3795(03)00546-9 fatcat:sqk36eg7rzhnlmvukcofdpicda

A Partition-of-Unity Based Algorithm for Implicit Surface Reconstruction Using Belief Propagation

Yi-Ling Chen, Shang-Hong Lai
2007 IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07)  
To achieve multi-scale reconstruction, we propose a novel progressive reconstruction algorithm which utilizes the Schur complement formula to reduce the computational cost of iteratively updating the radial  ...  The variational implicit patches are then combined together to form the overall surface via a set of blending functions, which is also referred to as the partition-of-unity method.  ...  Schur complement formula gives the closed-form expression for the inverse of a partitioned matrix.  ... 
doi:10.1109/smi.2007.3 dblp:conf/smi/ChenL07 fatcat:qeqwlc45uvftnkohufofobzmve

Preconditioning iterative MLFMA solutions of integral equations

Levent Gurel, Tahir Malas, Ozgur Ergul
2010 2010 URSI International Symposium on Electromagnetic Theory  
In this paper, we present our efforts to devise effective preconditioners for MLFMA solutions of difficult electromagnetics problems involving both conductors and dielectrics.  ...  The multilevel fast multipole algorithm (MLFMA) is a powerful method that enables iterative solutions of electromagnetics problems with low complexity.  ...  It is possible to provide fast and yet successful approximations to solutions of such partitioned matrix systems using Schur complement reduction.  ... 
doi:10.1109/ursi-emts.2010.5637219 fatcat:5stnk2ubvfftnkqomltcznelfa

Schur complement solver for Quantum Monte-Carlo simulations of strongly interacting fermions

Maksim Ulybyshev, Nils Kintscher, Karsten Kahl, Pavel Buividovich
2019 Computer Physics Communications  
We present a non-iterative solver based on the Schur complement method for sparse linear systems of special form which appear in Quantum Monte-Carlo (QMC) simulations of strongly interacting fermions on  ...  While the number of floating-point operations for this solver scales as the cube of the number of lattice sites, for practically relevant lattice sizes it is still significantly faster than iterative solvers  ...  The path integral representation of the partition function (4) starts from the Suzuki-Trotter decomposition of the partition function Tr e −βĤ ≈ Tr e −∆τĤ (2) e −∆τĤ (4) e −∆τĤ (2) e −∆τĤ (4) ...  ... 
doi:10.1016/j.cpc.2018.10.023 fatcat:auxxk2yg6vfr7mh2uvmlhvrfwi

parGeMSLR: A Parallel Multilevel Schur Complement Low-Rank Preconditioning and Solution Package for General Sparse Matrices [article]

Tianshi Xu, Vassilis Kalantzis, Ruipeng Li, Yuanzhe Xi, Geoffrey Dillon, Yousef Saad
2022 arXiv   pre-print
From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner as the sum between the matrix inverse of the interface coupling matrix and a low-rank correction term  ...  To reduce the cost associated with the computation of the approximate inverse matrices, parGeMSLR exploits a multilevel partitioning of the algebraic domain.  ...  the gap between the first term and the actual Schur complement matrix inverse associated with that level.  ... 
arXiv:2205.03224v1 fatcat:tvagn3qu4jaolmufswcachofcq

Page 1806 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews  
Suppose T is a skew tableau of shape a/u whose entries are the complement (in [”]) of {j,---, j,}, and U is a skew tableau of shape a/y whose entries are the complement of {i},---,i,}.  ...  As a corollary, they obtain a square relation between the skew Q- function of shape A/y and the skew Schur function of shape 1/7, with an appropriate interpretation, where A, u are partitions with distinct  ... 

Page 1806 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
As a corollary, they obtain a square relation between the skew Q- function of shape A/ and the skew Schur function of shape 4/7, with an appropriate interpretation, where A, u are partitions with distinct  ...  Suppose T is a skew tableau of shape a/u whose entries are the complement (in [”]) of {j1,---, j,}, and U is a skew tableau of shape a/y whose entries are the complement of {ij,---,i,}.  ... 

A hierarchical preconditioner for wave problems in quasilinear complexity [article]

Boris Bonev, Jan S. Hesthaven
2021 arXiv   pre-print
We exploit the property that Schur complements arising in such problems can be well approximated by hierarchical matrices.  ...  An approximate factorization can be computed matrix-free and in a (quasi-)linear number of operations.  ...  Summary of the algorithm. We have now discussed all steps necessary to compute the HSS representation of the Schur complement (3.5) via random sampling.  ... 
arXiv:2105.07791v1 fatcat:f3jbal7wazhupfqoajm3lsns6y

Page 129 of Mathematical Reviews Vol. , Issue 90A [page]

1990 Mathematical Reviews  
The author also studies the connections between InA and InB'AB and the inertia of quadratic forms defined by partitioned matrices and forms involving the generalized Schur complement of A in (s 4 (defined  ...  ) + (d,d,e —2d), where 1 is the symbol of usual orthogonal complement, R(A) is the range of A, d = dim(ASNS*+), e = dim(Ker(A)) — dim[Ker(A) 5], A* is the generalized inverse of A (Theorem 3.1).  ... 
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