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Solving sparse rational linear systems

2006
*
Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06
*

We propose a new algorithm to find a

doi:10.1145/1145768.1145785
dblp:conf/issac/EberlyGGSV06
fatcat:7rlftjlgxrhsnlywmkry36ynby
*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. ...##
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Solving Very Sparse Rational Systems of Equations

2011
*
ACM Transactions on Mathematical Software
*

Efficient methods for

doi:10.1145/1916461.1916463
fatcat:6tvw7gxubzed7ges55t6ows6iu
*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. ...##
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Interior Eigensolver for Sparse Hermitian Definite Matrices Based on Zolotarev's Functions
[article]

2020
*
arXiv
*
pre-print

Compared to the state-of-the-art algorithm FEAST, the proposed

arXiv:1701.08935v5
fatcat:jnvnnezmezhljaq5d4cojvaov4
*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. ...##
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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. ...##
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Preface

1997
*
Journal of Computational and Applied Mathematics
*

The partial realization problem in

doi:10.1016/s0377-0427(96)00119-7
fatcat:irxv5u7s5veylnvviowpipjjby
*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. ...##
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An algorithm to solve integer linear systems exactly using numerical methods

2006
*
Journal of symbolic computation
*

Success of this algorithm on a

doi:10.1016/j.jsc.2005.11.001
fatcat:6bfxnfkonfbt3bbsxos7fh6kjq
*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*...##
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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. ...##
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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. ...##
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A determinant-free method to simulate the parameters of large Gaussian fields

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

doi:10.1002/sta4.153
fatcat:7jvz2kssqne2lasppybtbrmjea
*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. ...##
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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. ...##
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Macromodeling of Multiport Systems Using a Fast Implementation of the Vector Fitting Method

2008
*
IEEE Microwave and Wireless Components Letters
*

Broadband macromodeling of large multiport

doi:10.1109/lmwc.2008.922585
fatcat:wu2kpjexcbb2rfe6viomplxj2e
*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. ...##
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Geometric Modeling and Calibration of Planar Multi-Projector Displays Using Rational Bezier Patches

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

doi:10.1109/cvpr.2007.383466
dblp:conf/cvpr/BhaskerM07
fatcat:z7ndelrngzfnzevtt6ux6ozqui
*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. ...##
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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. ...##
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Sparse Polynomial Interpolation in Nonstandard Bases

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

doi:10.1137/s0097539792237784
fatcat:7uvqeqb7uveglkhh5hspfgoofa
*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~ = ? ...##
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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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