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Solving sparse rational linear systems

Wayne Eberly, Mark Giesbrecht, Pascal Giorgi, Arne Storjohann, Gilles Villard
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
We propose a new algorithm to find a rational solution to a sparse system of linear equations over the integers.  ...  It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections.  ...  Introducing blocks to solve integer sparse linear systems is then an advantage since it allows us to use such fast dense routines.  ... 
doi:10.1145/1145768.1145785 dblp:conf/issac/EberlyGGSV06 fatcat:7rlftjlgxrhsnlywmkry36ynby

Solving Very Sparse Rational Systems of Equations

William Cook, Daniel E. Steffy
2011 ACM Transactions on Mathematical Software  
Efficient methods for solving linear-programming problems in exact precision rely on the solution of sparse systems of linear equations over the rational numbers.  ...  We consider a test set of instances arising from exact-precision linear programming and use this test set to compare the performance of several techniques designed for symbolic sparse linear-system solving  ...  INTRODUCTION Solving systems of linear equations is a fundamental mathematical problem.  ... 
doi:10.1145/1916461.1916463 fatcat:6tvw7gxubzed7ges55t6ows6iu

Interior Eigensolver for Sparse Hermitian Definite Matrices Based on Zolotarev's Functions [article]

Yingzhou Li, Haizhao Yang
2020 arXiv   pre-print
Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the  ...  This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil (A,B).  ...  for solving the shifted linear system (A − σB)x = b using sparse direct solvers.  ... 
arXiv:1701.08935v5 fatcat:jnvnnezmezhljaq5d4cojvaov4

Page 2212 of Mathematical Reviews Vol. 56, Issue 6 [page]

1978 Mathematical Reviews  
Each aims to maintain both stability and sparseness and allows further systems with the same matrix or another matrix having the same sparsity pattern to be solved economically.  ...  In the deferred back solution method two systems with coefficient matrix G are solved and one system with coeffi- cient matrix A=B—RG~'C.  ... 


A. Bultheel, M. Van Barel
1997 Journal of Computational and Applied Mathematics  
The partial realization problem in linear system theory can be seen as Pad6 approximation at infinity.  ...  In linear system theory, there is a tradition of generalizing the scalar problem to the vector and matrix case, to deal with multi-input multi-output systems.  ...  The same recurrences appear in the iterative methods for solving (large sparse) linear systems and (generalized) eigenvalue problems in linear algebra. See for example Brezinski's paper.  ... 
doi:10.1016/s0377-0427(96)00119-7 fatcat:irxv5u7s5veylnvviowpipjjby

An algorithm to solve integer linear systems exactly using numerical methods

Zhendong Wan
2006 Journal of symbolic computation  
Success of this algorithm on a linear equation requires that the linear system must be sufficiently well-conditioned for the numeric linear algebra method being used to compute a solution with sufficient  ...  In this paper, we present a new algorithm for the exact solutions of linear systems with integer coefficients using numerical methods.  ...  Experimentation on dense linear systems Rational solver for sparse integer linear systems The main idea is to apply Algorithm 1 by replacing the numeric solver with a successful sparse linear system  ... 
doi:10.1016/j.jsc.2005.11.001 fatcat:6bfxnfkonfbt3bbsxos7fh6kjq

Page 2706 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
Summary: “This paper introduces tensor methods for solving large sparse systems of nonlinear equations.  ...  We 65 NUMERICAL ANALYSIS 2706 develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations.  ... 

Page 2601 of Mathematical Reviews Vol. , Issue 85f [page]

1985 Mathematical Reviews  
A code for solving sparse, unsymmetric systems of linear equa- tions using a frontal method designed by the author is discussed from several points of view.  ...  S. 85f:65039 Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core. SIAM J. Sci. Statist. Comput. 5 (1984), no. 2, 270-280.  ... 

A determinant-free method to simulate the parameters of large Gaussian fields

Louis Ellam, Heiko Strathmann, Mark Girolami, Iain Murray
2017 Stat  
We develop a Markov chain Monte Carlo sampling scheme for the auxiliary model that requires no more than the application of inverse-matrix-square-roots and the solution of linear systems.  ...  These operations can be performed at large scales with rational approximations.  ...  In this setting, we make use of the sparse linear system in (10), which can be solved using a banded linear system solver.  ... 
doi:10.1002/sta4.153 fatcat:7jvz2kssqne2lasppybtbrmjea

Page 3238 of Mathematical Reviews Vol. , Issue 81H [page]

1981 Mathematical Reviews  
For solving linear systems of equations the authors develop a two-stage gradient descent method.  ...  System Sci. 4 (1970), 473-491; MR 42 #7088] to the block tridiagonal systems of linear equations.  ... 

Macromodeling of Multiport Systems Using a Fast Implementation of the Vector Fitting Method

D. Deschrijver, M. Mrozowski, T. Dhaene, D. De Zutter
2008 IEEE Microwave and Wireless Components Letters  
Broadband macromodeling of large multiport systems by vector fitting can be time consuming and resource demanding when all elements of the system matrix share a common set of poles.  ...  Index Terms-Broadband macromodeling, numerical techniques, system identification, vector fitting (VF).  ...  Once the final poles are found, the residues of the transfer function can be solved as a linear problem by setting in (1) equal to 1.  ... 
doi:10.1109/lmwc.2008.922585 fatcat:wu2kpjexcbb2rfe6viomplxj2e

Geometric Modeling and Calibration of Planar Multi-Projector Displays Using Rational Bezier Patches

Ezekiel S. Bhasker, Aditi Majumder
2007 2007 IEEE Conference on Computer Vision and Pattern Recognition  
It can be further used to develop an efficient and accurate geometric calibration method with a sparse sampling of the function.  ...  In this paper, we present a new closed-form model that relates projectors to cameras in planar multi-projector displays, using rational Bezier patches.  ...  The four correspondences result in a system of linear equations that can be solved to recovery the homography matrix.  ... 
doi:10.1109/cvpr.2007.383466 dblp:conf/cvpr/BhaskerM07 fatcat:z7ndelrngzfnzevtt6ux6ozqui

Page 8126 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
|[Razzaghi, Mohsen| Rational Chebyshev tau method for solving Volterra’s population model. (English summary) Appl. Math. Comput. 149 (2004), no. 3, 893-900.  ...  Summary: “An approximate method for solving Volterra’s popu- lation model for population growth of a species in a closed system is proposed.  ... 

Sparse Polynomial Interpolation in Nonstandard Bases

Y. N. Lakshman, B. David Saunders
1995 SIAM journal on computing (Print)  
In this paper, we consider the problem of interpolating univariate polynomials over a eld of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis.  ...  Our algorithms may be regarded as generalizations of Ben-Or and Tiwari's (1988) algorithm (based on the BCH decoding algorithm) for interpolating polynomials that are sparse in the standard basis.  ...  jj = A i;j : 2 From the above lemmas we see that we can compute the coe cients of (z) by solving the linear system of equations given by A~ = ?  ... 
doi:10.1137/s0097539792237784 fatcat:7uvqeqb7uveglkhh5hspfgoofa

Page 2642 of Mathematical Reviews Vol. , Issue 87e [page]

1987 Mathematical Reviews  
Over the past decade, several efficient iterative methods have been developed to solve large sparse (nonsymmetric) systems of linear algebraic equations.  ...  Frequently, a substantial portion of the total computational-work and storage required to solve stiff IVPs is devoted to solving these linear alge- braic systems, particularly if the systems are large.  ... 
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