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Differential elimination by differential specialization of Sylvester style matrices

Sonia L. Rueda
2016 Advances in Applied Mathematics  
These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials.  ...  This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal  ...  Casal and Scott MacCallum for interesting discussions on these topics, and specially Carlos D'Andrea for helpful comments on this manuscript.  ... 
doi:10.1016/j.aam.2015.07.002 fatcat:k732556cynbxhdkenjpjjmzjve

Differential elimination by differential specialization of Sylvester style matrices

Sonia L. Rueda
2015 ACM Communications in Computer Algebra  
These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials.  ...  This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal  ...  Casal and Scott MacCallum for interesting discussions on these topics, and specially Carlos D'Andrea for helpful comments on this manuscript.  ... 
doi:10.1145/2768577.2768635 fatcat:skxdncfggzasdewztx74mhzlqm

Index reduction of differential algebraic equations by differential algebraic elimination [article]

Xiaolin Qin, Lu Yang, Yong Feng, Bernhard Bachmann, Peter Fritzson
2015 arXiv   pre-print
We propose a new algorithm for index reduction of DAEs and establish the notion of differential algebraic elimination, which can provide the differential algebraic resultant of the enlarged system of original  ...  In this paper, we generalize the idea of differential elimination with Dixon resultant to polynomially nonlinear DAEs.  ...  Rueda [30] presents the differential elimination by differential specialization of Sylvester style matrices to focus on the sparsity with respect to the order of derivation.  ... 
arXiv:1504.04977v1 fatcat:y3lih6anvvcxpnbzq4i7cyhtti

Solving Stable Sylvester Equations via Rational Iterative Schemes

Peter Benner, Enrique S. Quintana-Ortí, Gregorio Quintana-Ortí
2005 Journal of Scientific Computing  
Acknowledgments We would like to express our gratitude to Vasile Sima of the National Institute for Research & Development in Informatics, Bucharest, Romania, for providing some helpful suggestions for  ...  We would obtain the same computational cost had we computed the inverse matrices by means of a Gauss-Jordan elimination procedure and the next matrix in the sequence C k as two matrix products.  ...  , implementation of implicit numerical methods for ordinary differential equations, and block-diagonalization of matrices; see, e.g., [14, 16, 21, 22, 18, 27, 44] to name only a few references.  ... 
doi:10.1007/s10915-005-9007-2 fatcat:ngpyse5ovva3zmhu3k33ydfc54

Numerical Methods for Linear Control Systems [chapter]

D. Boley, B. N. Datta
1997 Systems and Control in the Twenty-First Century  
The Gaussian elimination algorithm without pivoting is efficient, but unstable in general.  ...  Some of the important ones are: controllability and observability problems, the problem of computing the exponential matrix e At , the matrix equations problems: (Lyapunov equations, Sylvester equations  ...  The Sylvester-observer equation is a variation of the Sylvester equation, and the Lyapunov equation is a special case of the algebraic Riccati equation.  ... 
doi:10.1007/978-1-4612-4120-1_4 fatcat:gr5dlxsg5zc55awzxco2klhh7q

Differential resultants and subresultants [chapter]

Marc Chardin
1991 Lecture Notes in Computer Science  
Consider two differential operators If the a i and b j are constants, the condition for the existence of a solution y of L 1 (y) = L 2 (y) = 0 is that the resultant in X of the polynomials (in C[X]) a  ...  We here give a Sylvester style expression for the resultant and the subresultants.  ...  As it was known by Ritt ([R] ), the differential resultant is a polynomial in the a i , b j and their derivatives that tells us-in the general algebraic context of linear differential equations over a  ... 
doi:10.1007/3-540-54458-5_62 fatcat:cadpwzyx3zblhdilxr2gxmiem4

Direct methods for matrix Sylvester and Lyapunov equations

Danny C. Sorensen, Yunkai Zhou
2003 Journal of Applied Mathematics  
We compared all of our schemes with the Matlab Sylvester and Lyapunov solverlyap.m; the results show that our schemes are much more efficient.  ...  We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method forA1X+XA2+D=0and Hammarling's method forAX+XAT+BBT=0withAstable.  ...  This work was supported in part by the National Science Foundation (NSF) Grant CCR-9988393.  ... 
doi:10.1155/s1110757x03212055 fatcat:r4zh6kgbiva45iidwsip3veqye

FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation

Nicholas J. Higham
1988 ACM Transactions on Mathematical Software  
We derive new results concerning the behavior of Hager's method, extend it to complex matrices, and make several algorithmic modifications in order to improve the reliability and efficiency.  ...  The algorithms are based on a convex optimization method for estimating the l-norm of a real matrix devised by Hager [Condition estimates. SIAM J. Sci. Stat. Comput. 5 (1984), 311-3161.  ...  ACKNOWLEDGMENTS It is a pleasure to thank Ian Gladwell for his valuable advice on the development of the algorithms and codes. I also thank Des Higham for suggesting improvements to the manuscript.  ... 
doi:10.1145/50063.214386 fatcat:whd66cmncbfczihlztarpipbaa

Book Review: An Introduction to the Theory of Groups of Finite Order

Arthur Ranum
1909 Bulletin of the American Mathematical Society  
Special mention must be made of his general processes for the construction of contravariants and concomitants, later designated as Uberschiebungen by Gordan and made by him the foundation of the theory  ...  The reader will be struck by the condensed style and compact notation used in the book and by the unusual arrangement of its contents, at least two-thirds of which, I should judge, is in the form of examples  ... 
doi:10.1090/s0002-9904-1909-01747-1 fatcat:he6qx5n5rbgwlesap52ymd4c6a

Numerical and Computational Issues in Linear Control and System Theory [chapter]

A Laub, R Patel, P Van Dooren
2010 The Control Systems Handbook, Second Edition  
It must be emphasized that consideration of large, sparse matrices or matrices with special, exploitable structures may involve significantly different concerns and methodologies than those to be discussed  ...  Open questions remain concerning estimating the condition of Lyapunov and Sylvester equations efficiently and reliably in terms of the coefficient matrices.  ... 
doi:10.1201/b10384-3 fatcat:nr73ocw52jcjpe2te2mjrymora

History of Modern Mathematics

D. E. Smith
1896 Mathematical Gazette  
Differential invariants have been studied by Sylvester, MacMahon, and Hammond.  ...  The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of m -n reduces to  ... 
doi:10.2307/3604091 fatcat:qm7fpamoj5fkpfmbbugomam5oy

Fast Differentiable Matrix Square Root and Inverse Square Root [article]

Yue Song, Nicu Sebe, Wei Wang
2022 arXiv   pre-print
Computing the matrix square root and its inverse in a differentiable manner is important in a variety of computer vision tasks.  ...  and fine-grained recognition, attentive covariance pooling for video recognition, and neural style transfer.  ...  square root satisfies: A 1 2 ∂l ∂A + ∂l ∂A A 1 2 = ∂l ∂A 1 2 (16) As pointed out in [1] , eq. ( 16 ) actually defines the continuoustime Lyapunov equation (BX+XB=C) or a special case of Sylvester equation  ... 
arXiv:2201.12543v1 fatcat:pwycrp44knarhfww3pd7lzjyge

Numerical solution and perturbation theory for generalized Lyapunov equations

Tatjana Stykel
2002 Linear Algebra and its Applications  
We generalize a Bartels-Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side.  ...  Kågström for providing the GUPTRI routine written by J.W. Demmel and himself.  ...  The solution vec(X tq ) can be computed by solving (2.25) via Gaussian elimination with partial pivoting [17] .  ... 
doi:10.1016/s0024-3795(02)00255-0 fatcat:lpgi7fihc5ehnclflqlw2g4zwy

FAST DIFFERENTIABLE MATRIX SQUARE ROOT

Yue Song, Nicu Sebe, Wei Wang
2022 Zenodo  
Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks.  ...  The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function.  ...  ACKNOWLEDGMENTS This work was supported by EU H2020 SPRING No. 871245 and EU H2020 AI4Media No. 951911 projects. We thank Professor Nicholas J. Higham for valuable suggestions.  ... 
doi:10.5281/zenodo.6396092 fatcat:wbgkl2e5cbef3bwrngh3wzffai

Book Review: Lectures on finite precision computations

Hans J. Stetter, Charles Van Loan, Michael Holst, Frank Stenger, Chi-Wang Shu, R. Mattheij, Stephen J. Wright, Thomas F. Coleman, Lars B. Wahlbin
1997 Mathematics of Computation  
The mechanism of finding the interpolation or differentiation matrices is discussed, and examples given for these matrices for different node distributions.  ...  In this setting "proximity" often means the defining matrices differ by a few rows or columns or, more generally, by a low-rank matrix.  ... 
doi:10.1090/s0025-5718-97-00910-1 fatcat:kiiohmhwnrh5rkjl672eqoqnre
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