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Differential elimination by differential specialization of Sylvester style matrices

2016
*
Advances in Applied Mathematics
*

These are determinants

doi:10.1016/j.aam.2015.07.002
fatcat:k732556cynbxhdkenjpjjmzjve
*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. ...##
###
Differential elimination by differential specialization of Sylvester style matrices

2015
*
ACM Communications in Computer Algebra
*

These are determinants

doi:10.1145/2768577.2768635
fatcat:skxdncfggzasdewztx74mhzlqm
*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. ...##
###
Index reduction of differential algebraic equations by differential algebraic elimination
[article]

2015
*
arXiv
*
pre-print

We propose a new algorithm for index reduction

arXiv:1504.04977v1
fatcat:y3lih6anvvcxpnbzq4i7cyhtti
*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. ...##
###
Solving Stable Sylvester Equations via Rational Iterative Schemes

2005
*
Journal of Scientific Computing
*

Acknowledgments We would like to express our gratitude to Vasile Sima

doi:10.1007/s10915-005-9007-2
fatcat:ngpyse5ovva3zmhu3k33ydfc54
*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. ...##
###
Numerical Methods for Linear Control Systems
[chapter]

1997
*
Systems and Control in the Twenty-First Century
*

The Gaussian

doi:10.1007/978-1-4612-4120-1_4
fatcat:gr5dlxsg5zc55awzxco2klhh7q
*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. ...##
###
Differential resultants and subresultants
[chapter]

1991
*
Lecture Notes in Computer Science
*

Consider two

doi:10.1007/3-540-54458-5_62
fatcat:cadpwzyx3zblhdilxr2gxmiem4
*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 ...##
###
Direct methods for matrix Sylvester and Lyapunov equations

2003
*
Journal of Applied Mathematics
*

We compared all

doi:10.1155/s1110757x03212055
fatcat:r4zh6kgbiva45iidwsip3veqye
*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. ...##
###
FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation

1988
*
ACM Transactions on Mathematical Software
*

We derive new results concerning the behavior

doi:10.1145/50063.214386
fatcat:whd66cmncbfczihlztarpipbaa
*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. ...##
###
Book Review: An Introduction to the Theory of Groups of Finite Order

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 ...

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

2010
*
The Control Systems Handbook, Second Edition
*

It must be emphasized that consideration

doi:10.1201/b10384-3
fatcat:nr73ocw52jcjpe2te2mjrymora
*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*. ...##
###
History of Modern Mathematics

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 ...

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

2022
*
arXiv
*
pre-print

Computing the matrix square root and its inverse in a

arXiv:2201.12543v1
fatcat:pwycrp44knarhfww3pd7lzjyge
*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 ...##
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Numerical solution and perturbation theory for generalized Lyapunov equations

2002
*
Linear Algebra and its Applications
*

We generalize a Bartels-Stewart method and a Hammarling method to compute a partial solution

doi:10.1016/s0024-3795(02)00255-0
fatcat:lpgi7fihc5ehnclflqlw2g4zwy
*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] . ...##
###
FAST DIFFERENTIABLE MATRIX SQUARE ROOT

2022
*
Zenodo
*

Computing the matrix square root or its inverse in a

doi:10.5281/zenodo.6396092
fatcat:wbgkl2e5cbef3bwrngh3wzffai
*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. ...##
###
Book Review: Lectures on finite precision computations

1997
*
Mathematics of Computation
*

The mechanism

doi:10.1090/s0025-5718-97-00910-1
fatcat:kiiohmhwnrh5rkjl672eqoqnre
*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. ...
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