Abstract

Continuous matrix factorizations show great promise in a number of contexts. In this chapter we survey results on continuous matrix factorizations paying particular attention to smooth matrix factorizations of fundamental matrix solutions of linear differential equations and differential-algebraic equations with special emphasis on smooth QR and smooth SVD.

Keywords

Lyapunov Exponent Singular Value Decomposition Matrix Function Exponential Dichotomy Fundamental Matrix Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Adrianova, L.Ya.: Introduction to Linear Systems of Differential Equations (Transl. from the Russian by Peter Zhevandrov). Translations of Mathematical Monographs, vol. 146, x+204pp. American Mathematical Society, Providence (1995)Google Scholar
  2. 2.
    Badawy, M., Van Vleck, E.: Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space. Linear Algebra Appl. 437(1), 37–59 (2012)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.: Lyapunov exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: theory, and Part 2: numerical applications. Meccanica 15, 9–20, 21–30 (1980)Google Scholar
  4. 4.
    Bohl, P.: Über Differentialungleichungen. J. F. d. Reine Und Angew. Math. 144, 284–313 (1913)Google Scholar
  5. 5.
    Breda, D., Van Vleck, E.: Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations. Numer. Math. 126(2), 225–257 (2014)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N.K.: Numerical computation of an analytic singular value decomposition of a matrix valued function. Numer. Math. 60, 1, 1–39 (1991)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bylov, B.F., Izobov, N.A.: Necessary and sufficient conditions for stability of characteristic exponents of a linear system. Differ. Uravn. 5, 1794–1903 (1969)MATHMathSciNetGoogle Scholar
  8. 8.
    Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows. Math. Comput. 66(220), 1461–1486 (1997)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Champneys, A.R., Kuznetsov, Yu.A., Sandstede, B.: A numerical toolbox for homoclinic bifurcation analysis. Int. J. Bifur. Chaos Appl. Sci. Eng. 6(5), 867–887 (1996)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Chu, M.T.: On the continuous realization of iterative processes. SIAM Rev. 30, 375–387 (1988)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629, ii+98 pp. Springer, Berlin/New York (1978)Google Scholar
  12. 12.
    Demmel, J.W., Dieci, L., Friedman, M.J.: Computing connecting orbits via an improved algorithm for continuing invariant subspaces. SIAM J. Sci. Comput. 22(1), 81–94 (2000)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Dieci, L., Eirola, T.: On smooth decompositions of matrices. SIAM J. Matrix Anal. Appl. 20(3), 800–819 (1999) (electronic)Google Scholar
  14. 14.
    Dieci, L., Elia, C.: The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects. J. Differ. Eqn. 230(2), 502–531 (2006)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Dieci, L., Elia, C., Van Vleck, E.: Exponential dichotomy on the real line: SVD and QR methods. J. Differ. Eqn. 248(2), 287–308 (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Dieci, L., Elia, C., Van Vleck, E.: Detecting exponential dichotomy on the real line: SVD and QR algorithms. BIT 51(3), 555–579 (2011)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Dieci, L., Friedman, M.J.: Continuation of invariant subspaces. Numer. Linear Algebra Appl. 8(5), 317–327 (2001)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dieci, L., Jolly, M., Rosa, R., Van Vleck, E.: Error on approximation of Lyapunov exponents on inertial manifolds: the Kuramoto-Sivashinsky equation. J. Discret. Contin. Dyn. Syst. Ser. B 9(3–4), 555–580 (2008)MATHGoogle Scholar
  19. 19.
    Dieci, L., Jolly, M.S., Van Vleck, E.S.: Numerical techniques for approximating Lyapunov exponents and their implementation. ASME J. Comput. Nonlinear Dyn. 6, 011003–1–7 (2011)Google Scholar
  20. 20.
    Dieci, L., Russell, R.D., Van Vleck, E.S.: Unitary integrators and applications to continuous orthonormalization techniques. SIAM J. Numer. Anal. 31(1), 261–281 (1994)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Dieci, L., Russell, R.D., Van Vleck, E.S.: On the computation of Lyapunov exponents for continuous dynamical systems. SIAM J. Numer. Anal. 34(1), 402–423 (1997)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Dieci, L., Van Vleck, E.S.: Computation of a few Lyapunov exponents for continuous and discrete dynamical systems. Numerical methods for ordinary differential equations (Atlanta, 1994). Appl. Numer. Math. 17(3), 275–291 (1995)Google Scholar
  23. 23.
    Dieci, L., Van Vleck, E.S.: Computation of orthonormal factors for fundamental solution matrices. Numer. Math. 83(4), 599–620 (1999)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Dieci, L., Van Vleck, E.S.: Lyapunov spectral intervals: theory and computation. SIAM J. Numer. Anal. 40(2), 516–542 (2002) (electronic)Google Scholar
  25. 25.
    Dieci, L., Van Vleck, E.S.: On the error in computing Lyapunov exponents by QR methods. Numer. Math. 101(4), 619–642 (2005)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Dieci, L., Van Vleck, E.S.: Perturbation theory for approximation of Lyapunov exponents by QR methods. J. Dyn. Differ. Eqn. 18(3), 815–840 (2006)CrossRefMATHGoogle Scholar
  27. 27.
    Dieci, L., Van Vleck, E.S.: Lyapunov and Sacker-Sell spectral intervals. J. Dyn. Differ. Eqn. 19(2), 265–293 (2007)CrossRefMATHGoogle Scholar
  28. 28.
    Dieci, L., Van Vleck, E.S.: On the error in QR integration. SIAM J. Numer. Anal. 46(3), 1166–1189 (2008)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Diliberto, S.P.: On systems of ordinary differential equations. In: Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematical Studies, vol. 20, pp. 1–38. Princeton University Press, Princeton (1950)Google Scholar
  30. 30.
    Holtz, O., Mehrmann, V., Schneider, H.: Matrices that commute with their derivative. On a letter from Schur to Wielandt. Linear Algebra Appl. 438(5), 2574–2590 (2013)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Grundlehren der Mathematischen Wissenschaften. Band 132, xxi+619 pp. Springer, Berlin/New York (1976)Google Scholar
  32. 32.
    Kressner, D.: The periodic QR algorithm is a disguised QR algorithm. Linear Algebra Appl. 417(2–3), 423–433 (2006)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Kressner, D.: A periodic Krylov-Schur algorithm for large matrix products. Numer. Math. 103(3), 461–483 (2006)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Textbooks in Mathematics, viii+377 pp. European Mathematical Society (EMS), Zürich (2006)Google Scholar
  35. 35.
    Leimkuhler, B.J., Van Vleck, E.S.: Orthosymplectic integration of linear Hamiltonian systems. Numer. Math. 77(2), 269–282 (1997)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Linh, V.H., Mehrmann, V.: Lyapunov, Bohl and Sacker-Sell spectral intervals for differential-algebraic equations. J. Dyn. Differ. Eqn. 21(1), 153–194 (2009)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Linh, V.H., Mehrmann, V.: Spectral analysis for linear differential-algebraic equations. In: 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden, 2011. Discrete and Continuous Dynamical Systems Supplement, vol. II, pp. 991–1000. ISBN:978-1-60133-008-6; 1-60133-008-1Google Scholar
  38. 38.
    Linh, V.H., Mehrmann, V.: Approximation of spectral intervals and leading directions for differential-algebraic equation via smooth singular value decompositions. SIAM J. Numer. Anal. 49(5), 1810–1835 (2011)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Linh, V.H., Mehrmann, V.: Spectra and leading directions for linear DAEs. In: Control and Optimization with Differential-Algebraic Constraints. Advances in Design and Control, vol. 23, pp. 59–78. SIAM, Philadelphia (2012)Google Scholar
  40. 40.
    Linh, V.H.: Mehrmann, V., Van Vleck, E.S.: QR methods and error analysis for computing Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations. Adv. Comput. Math. 35(2–4), 281–322 (2011)Google Scholar
  41. 41.
    Lyapunov, A.: Problém géneral de la stabilité du mouvement. Int. J. Control 53, 531–773 (1992)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Millionshchikov, V.M.: Structurally stable properties of linear systems of differential equations. Differ. Uravn. 5, 1775–1784 (1969)MATHGoogle Scholar
  43. 43.
    Millionshchikov V.M.: Systems with integral division are everywhere dense in the set of all linear systems of differential equations. Differ. Uravn. 5, 1167–1170 (1969)MATHGoogle Scholar
  44. 44.
    Oliveira, S., Stewart, D.E.: exponential splittings of products of matrices and accurately computing singular values of long products. In: Proceedings of the International Workshop on Accurate Solution of Eigenvalue Problems, University Park, 1998. Linear Algebra Applications, vol. 3091–3, pp. 175–190 (2000)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197 (1998)Google Scholar
  46. 46.
    Palmer, K.J.: The structurally stable systems on the half-line are those with exponential dichotomy. J. Differ. Eqn. 33, 16–25 (1979)CrossRefMATHGoogle Scholar
  47. 47.
    Palmer, K.J.: Exponential dichotomy, integral separation and diagonalizability of linear sys temsof ordinary differential equations. J. Differ. Eqn. 43, 184–203 (1982)CrossRefMATHGoogle Scholar
  48. 48.
    Palmer, K.J.: Exponential separation, exponential dichotomy and spectral theory for linear s ystems of ordinary differential equations. J. Differ. Eqn. 46, 324–345 (1982)CrossRefMATHGoogle Scholar
  49. 49.
    Perron, O.: Die Ordnungszahlen Linearer Differentialgleichungssysteme. Math. Zeits. 31, 748–766 (1930)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Rheinboldt, W.C.: On the computation of multidimensional solution manifolds of parametrized equations. Numer. Math. 53(1–2), 165–181 (1988)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Eqn. 7, 320–358 (1978)CrossRefMathSciNetGoogle Scholar
  52. 52.
    Stewart, D.E.: A new algorithm for the SVD of a long product of matrices and the stability of products. Electron. Trans. Numer. Anal. 5, 29–47 (1997) (electronic)Google Scholar
  53. 53.
    Van Vleck, E.S. On the error in the product QR decomposition. SIAM J. Matrix Anal. Appl. 31(4), 1775–1791 (2009/2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA

Personalised recommendations