214 research outputs found
Symmetric indefinite triangular factorization revealing the rank profile matrix
We present a novel recursive algorithm for reducing a symmetric matrix to a
triangular factorization which reveals the rank profile matrix. That is, the
algorithm computes a factorization where is a permutation matrix,
is lower triangular with a unit diagonal and is
symmetric block diagonal with and antidiagonal
blocks. The novel algorithm requires arithmetic
operations. Furthermore, experimental results demonstrate that our algorithm
can even be slightly more than twice as fast as the state of the art
unsymmetric Gaussian elimination in most cases, that is it achieves
approximately the same computational speed. By adapting the pivoting strategy
developed in the unsymmetric case, we show how to recover the rank profile
matrix from the permutation matrix and the support of the block-diagonal
matrix. There is an obstruction in characteristic for revealing the rank
profile matrix which requires to relax the shape of the block diagonal by
allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient.
This relaxed decomposition can then be transformed into a standard
decomposition at a
negligible cost
The antitriangular factorisation of saddle point matrices
Mastronardi and Van Dooren recently introduced the block antitriangular ("Batman") decomposition for symmetric indefinite matrices. Here we show the simplification of this factorisation for saddle point matrices and demonstrate how it represents the common nullspace method. We show the relation of this factorisation to constraint preconditioning and how it transforms but preserves the block diagonal structure of block diagonal preconditioning
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The state-of-the-art of preconditioners for sparse linear least-squares problems
In recent years, a variety of preconditioners have been proposed for use in solving large sparse linear least-squares problems. These include simple diagonal preconditioning, preconditioners based on incomplete factorizations and stationary inner iterations used with Krylov subspace methods. In this study, we briefly review preconditioners for which software has been made available and then present a numerical evaluation of them using performance profiles and a large set of problems arising from practical applications. Comparisons are made with state-of-the-art sparse direct methods
On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a
2x2-block-matrix by its transpose using 5 block products (3 recursive calls and
2 general products) over C or any finite field.We use geometric considerations
on the space of bilinear forms describing 2x2 matrix products to obtain this
algorithm and we show how to reduce the number of involved additions.The
resulting algorithm for arbitrary dimensions is a reduction of multiplication
of a matrix by its transpose to general matrix product, improving by a constant
factor previously known reductions.Finally we propose schedules with low memory
footprint that support a fast and memory efficient practical implementation
over a finite field.To conclude, we show how to use our result in LDLT
factorization.Comment: ISSAC 2020, Jul 2020, Kalamata, Greec
A New Preconditioning Approachfor an Interior Point–Proximal Method of Multipliers for Linear and Convex Quadratic Programming
In this paper, we address the efficient numerical solution of linear and
quadratic programming problems, often of large scale. With this aim, we devise
an infeasible interior point method, blended with the proximal method of
multipliers, which in turn results in a primal-dual regularized interior point
method. Application of this method gives rise to a sequence of increasingly
ill-conditioned linear systems which cannot always be solved by factorization
methods, due to memory and CPU time restrictions. We propose a novel
preconditioning strategy which is based on a suitable sparsification of the
normal equations matrix in the linear case, and also constitutes the foundation
of a block-diagonal preconditioner to accelerate MINRES for linear systems
arising from the solution of general quadratic programming problems. Numerical
results for a range of test problems demonstrate the robustness of the proposed
preconditioning strategy, together with its ability to solve linear systems of
very large dimension
On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a block-matrix by its transpose over C or any finite field using 5 recursive products. We use geometric considerations on the space of bilinear forms describing 2×2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose space and time efficient schedules that enable us to provide fast practical implementations for higher-dimensional matrix products
Efficient Algorithms for Solving Structured Eigenvalue Problems Arising in the Description of Electronic Excitations
Matrices arising in linear-response time-dependent density functional theory and many-body perturbation theory, in particular in the Bethe-Salpeter approach, show a 2 × 2 block structure. The motivation to devise new algorithms, instead of using general purpose eigenvalue solvers, comes from the need to solve large problems on high performance computers. This requires parallelizable and communication-avoiding algorithms and implementations. We point out various novel directions for diagonalizing structured matrices. These include the solution of skew-symmetric eigenvalue problems in ELPA, as well as structure preserving spectral divide-and-conquer schemes employing generalized polar decompostions
Simultaneous and Two-step Reconciliation of Systems of Time Series.
The reconciliation of systems of time series subject to both temporal and contemporaneous constraints can be solved in such a way that the temporal profiles of the original series be preserved “at the best” (movement preservation principle). Thanks to the sparsity of the linear system to be solved, a feasible procedure can be developed to solve simultaneously the problem. A two-step strategy might be more suitable in the case of large systems: firstly, each series is aligned to the corresponding temporal constraints according to a movement preservation principle; secondly, all series are reconciled within each low-frequency period according to the given constraints. This work compares the results of simultaneous and two-step approaches for medium/large datasets from real-life and discusses conditions under which the two-step procedure can be a valid alternative to the simultaneous one
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