Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Reductions of operator pencils

Author: Olivier Verdier
Journal: Math. Comp. 83 (2014), 189-214
MSC (2010): Primary 15A21, 15A22, 34A30, 47A10, 65L80
Published electronically: June 27, 2013
MathSciNet review: 3120586
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study problems associated with an operator pencil, i.e., a pair of operators on Banach spaces. Two natural problems to consider are linear constrained differential equations and the description of the generalized spectrum. The main tool to tackle either of those problems is the reduction of the pencil. There are two kinds of natural reduction operations associated to a pencil, which are conjugate to each other.

Our main result is that those two kinds of reductions commute, under some mild assumptions that we investigate thoroughly.

Each reduction exhibits moreover a pivot operator. The invertibility of all the pivot operators of all possible successive reductions corresponds to the notion of regular pencil in the finite dimensional case, and to the inf-sup condition for saddle point problems on Hilbert spaces.

Finally, we show how to use the reduction and the pivot operators to describe the generalized spectrum of the pencil.

References [Enhancements On Off] (What's this?)

  • [1] A. G. Baskakov, Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators, Sovrem. Mat. Fundam. Napravl. 9 (2004), 3-151 (electronic) (Russian); English transl., J. Math. Sci. (N. Y.) 137 (2006), no. 4, 4885-5036. MR 2123307 (2005j:47005),
  • [2] A. G. Baskakov, Linear relations as generators of semigroups of operators, Mat. Zametki 84 (2008), no. 2, 175-192 (Russian, with Russian summary); English transl., Math. Notes 84 (2008), no. 1-2, 166-183. MR 2475046 (2010c:47101),
  • [3] A. G. Baskakov and K. I. Chernyshov, Spectral analysis of linear relations, and degenerate semigroups of operators, Mat. Sb. 193 (2002), no. 11, 3-42 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 11-12, 1573-1610. MR 1937028 (2004k:47001),
  • [4] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [5] S. L. Campbell and W. Marszalek, The index of an infinite-dimensional implicit system, Math. Comput. Model. Dyn. Syst. 5 (1999), no. 1, 18-42. MR 1689804 (2000h:35026),
  • [6] K. Debrabant and K. Strehmel, Convergence of Runge-Kutta methods applied to linear partial differential-algebraic equations, Appl. Numer. Math. 53 (2005), no. 2-4, 213-229. MR 2128523 (2005k:65125),
  • [7] Angelo Favini and Atsushi Yagi, Degenerate differential equations in Banach spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 215, Marcel Dekker Inc., New York, 1999. MR 1654663 (99i:34079)
  • [8] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649 (21 #6372c)
  • [9] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Theory and algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
  • [10] Marcus Hausdorf and Werner M. Seiler, On the numerical analysis of overdetermined linear partial differential systems, Symbolic and numerical scientific computation (Hagenberg, 2001) Winkler, Frank, et al., (eds.), Lecture Notes in Comput. Sci., vol. 2630, Springer, Berlin, 2003, pp. 152-167. MR 2043704 (2005a:35210),
  • [11] Roswitha März, Numerical methods for differential algebraic equations, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 141-198. MR 1165725 (93e:65096)
  • [12] Patrick J. Rabier and Werner C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, J. Differential Equations 109 (1994), no. 1, 110-146. MR 1272402 (96a:34004),
  • [13] Sebastian Reich, Beitrag zur theorie der algebrodifferentialgleichungen, Ph.D. thesis, TU Dresden, 1990.
  • [14] Gregory J. Reid, Ping Lin, and Allan D. Wittkopf, Differential elimination-completion algorithms for DAE and PDAE, Stud. Appl. Math. 106 (2001), no. 1, 1-45. MR 1805484 (2002b:65124),
  • [15] Bernd Simeon, Radu Serban, and Linda R. Petzold, A model of macroscale deformation and microvibration in skeletal muscle tissue, M2AN Math. Model. Numer. Anal. 43 (2009), no. 4, 805-823. MR 2542878 (2010h:74018),
  • [16] B. Thaller and S. Thaller, Factorization of degenerate Cauchy problems: the linear case, J. Operator Theory 36 (1996), no. 1, 121-146. MR 1417190 (98a:34062)
  • [17] Caren Tischendorf, Coupled systems of differential algebraic and partial differential equations in circuit and device simulation, Ph.D. thesis, Humboldt University of Berlin, 2003, Habilitation Thesis.
  • [18] P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra Appl. 27 (1979), 103-140. MR 545726 (80g:65042),
  • [19] Olivier Verdier, Differential equations with constraints, Doctoral theses in mathematical sciences, University of Lund, June 2009, URL
  • [20] -, Reduction and normal forms of matrix pencils, URL
  • [21] J. H. Wilkinson, Linear differential equations and Kronecker's canonical form, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978), Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York, 1978, pp. 231-265. MR 519065 (80e:65077)
  • [22] Kai Tak Wong, The eigenvalue problem $ \lambda Tx+Sx$, J. Differential Equations 16 (1974), 270-280. MR 0349711 (50 #2204)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 15A21, 15A22, 34A30, 47A10, 65L80

Retrieve articles in all journals with MSC (2010): 15A21, 15A22, 34A30, 47A10, 65L80

Additional Information

Olivier Verdier
Affiliation: Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway

Received by editor(s): May 27, 2011
Received by editor(s) in revised form: May 22, 2012
Published electronically: June 27, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society