Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T07:29:19.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear algebra algorithms as dynamical systems

Published online by Cambridge University Press:  25 April 2008

Moody T. Chu
Moody T. Chu
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205, USA E-mail: chu@math.ncsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Any logical procedure that is used to reason or to infer either deductively or inductively, so as to draw conclusions or make decisions, can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state develops into another state by following a certain specific rule. Such an action is generally formalized as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyse, and modify realization processes for various applications.

The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in submanifolds of matrices are cast in fairly general frameworks, of which special cases have been found to afford unified and fundamental insights into the structure and behaviour of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

Type
Research Article
Information
Acta Numerica , Volume 17 , May 2008 , pp. 1 - 86
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Absil, P.-A. and Kurdyka, K. (2006), ‘On the stable equilibrium points of gradient systems’, Systems Control Lett. 55, 573–577.CrossRefGoogle Scholar
Absil, P.-A., Mahony, R. and Andrews, B. (2005), ‘Convergence of the iterates of descent methods for analytic cost functions’, SIAM J. Optim. 16, 531–547 (electronic).CrossRefGoogle Scholar
Akhiezer, N. I. (1965), The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York. Translated by Kemmer, N..Google Scholar
Allgower, E. and Georg, K. (1980), ‘Simplicial and continuation methods for approximating fixed points and solutions to systems of equations’, SIAM Rev. 22, 28–85.CrossRefGoogle Scholar
Allgower, E. L. and Georg, K. (2003), Introduction to Numerical Continuation Methods, Vol. 45 of Classics in Applied Mathematics, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Antoulas, A. C. (2005), Approximation of Large-Scale Dynamical Systems, Vol. 6 of Advances in Design and Control, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Aptekarev, A. I., Branquinho, A. and Marcellán, F. (1997), ‘Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation’, J. Comput. Appl. Math. 78, 139–160.CrossRefGoogle Scholar
Arbenz, P. and Golub, G. H. (1995), ‘Matrix shapes invariant under the symmetric QR algorithm’, Numer. Linear Algebra Appl. 2, 87–93.CrossRefGoogle Scholar
Arnold, V. I. (1988), Geometrical Methods in the Theory of Ordinary Differential Equations, Vol. 250 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), second edn, Springer, New York. Translated from the Russian by Szücs, Joseph (Szűcs, József M.).Google Scholar
Ashlock, D. A., Driessel, K. R. and Hentzel, I. R. (1997), On matrix structures invariant under Toda-like isospectral flows, in Proc. Fifth Conference of the International Linear Algebra Society, Atlanta 1995, Vol. 254, pp. 29–48.Google Scholar
Baker, A. (2002), Matrix Groups: An Introduction to Lie Group Theory, Springer Undergraduate Mathematics Series, Springer, London.CrossRefGoogle Scholar
Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. and Van der Vorst, H. (1994), Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Beelen, T. and Van Dooren, P. (1990), Computational aspects of the Jordan canonical form, in Reliable Numerical Computation, Oxford University Press, New York, pp. 57–72.CrossRefGoogle Scholar
Benner, P. and Kressner, D. (2006), ‘Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II’, ACM Trans. Math. Software 32, 352–373.CrossRefGoogle Scholar
Benner, P., Byers, R., Mehrmann, V. and Xu, H. (2002), ‘Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils’, SIAM J. Matrix Anal. Appl. 24, 165–190 (electronic).CrossRefGoogle Scholar
Benner, P., Kressner, D. and Mehrmann, V. (2005), Skew-Hamiltonian and Hamiltonian eigenvalue problems: theory, algorithms and applications, in Proc. Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, pp. 3–39.CrossRefGoogle Scholar
Benner, P., Mehrmann, V. and Xu, H. (1998), ‘A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils’, Numer. Math. 78, 329–358.CrossRefGoogle Scholar
Bhaya, A. and Kaszkurewicz, E. (2006), Control Perspectives on Numerical Algorithms and Matrix Problems, Vol. 10 of Advances in Design and Control, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Blanes, S., Casas, F., Oteo, J. A. and Ros, J. (1998), ‘Magnus and Fer expansions for matrix differential equations: The convergence problem’, J. Phys. A 31, 259–268.CrossRefGoogle Scholar
Bloch, A. M. and Iserles, A. (2006), ‘On an isospectral Lie-Poisson system and its Lie algebra’, Found. Comput. Math. 6, 121–144.CrossRefGoogle Scholar
Bloch, A. M., Brockett, R. W. and Ratiu, T. S. (1992), ‘Completely integrable gradient flows’, Comm. Math. Phys. 147, 57–74.CrossRefGoogle Scholar
Bramble, J. H. (1993), Multigrid Methods, Vol. 294 of Pitman Research Notes in Mathematics Series, Longman, Harlow.Google Scholar
Briggs, W. L. (1987), A Multigrid Tutorial, SIAM, Philadelphia, PA.Google Scholar
Brockett, R. W. (1991), ‘Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems’, Linear Algebra Appl. 146, 79–91.CrossRefGoogle Scholar
Brockett, R. W. (1993), Differential geometry and the design of gradient algorithms, in Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles 1990, Vol. 54 of Proc. Sympos. Pure Math., AMS, Providence, RI, pp. 69–92.CrossRefGoogle Scholar
Bunse-Gerstner, A., Byers, R., Mehrmann, V. and Nichols, N. K. (1991), ‘Numerical computation of an analytic singular value decomposition of a matrix valued function’, Numer. Math. 60, 1–39.CrossRefGoogle Scholar
Calvo, M. P., Iserles, A. and Zanna, A. (1997), ‘Numerical solution of isospectral flows’, Math. Comp. 66, 1461–1486.CrossRefGoogle Scholar
Celledoni, E. and Iserles, A. (2000), ‘Approximating the exponential from a Lie algebra to a Lie group’, Math. Comp. 69, 1457–1480.CrossRefGoogle Scholar
Chill, R. (2003), ‘On the Lojasiewicz–Simon gradient inequality’, J. Funct. Anal. 201, 572–601.CrossRefGoogle Scholar
Chu, D., Liu, X. and Mehrmann, V. (2007), ‘A numerical method for computing the Hamiltonian Schur form’, Numer. Math. 105, 375–412.CrossRefGoogle Scholar
Chu, M. (1986 a), ‘A continuous approximation to the generalized Schur decomposition’, Linear Algebra Appl. 78, 119–132.CrossRefGoogle Scholar
Chu, M. T. (1986 b), ‘A differential equation approach to the singular value decomposition of bidiagonal matrices’, Linear Algebra Appl. 80, 71–79.CrossRefGoogle Scholar
Chu, M. T. (1988), ‘On the continuous realization of iterative processes’, SIAM Rev. 30, 375–387.CrossRefGoogle Scholar
Chu, M. T. (1991), ‘A continuous Jacobi-like approach to the simultaneous reduction of real matrices’, Linear Algebra Appl. 147, 75–96.CrossRefGoogle Scholar
Chu, M. T. (1995), ‘Constructing a Hermitian matrix from its diagonal entries and eigenvalues’, SIAM J. Matrix Anal. Appl. 16, 207–217.CrossRefGoogle Scholar
Chu, M. T. and Buono, N. Del (2008 a), ‘Total decoupling of general quadratic pencils I: Theory’, J. Sound Vibration 309, 96–111.CrossRefGoogle Scholar
Chu, M. T. and Buono, N. Del (2008 b), ‘Total decoupling of general quadratic pencils II: Structure preserving isospectral flows’, J. Sound Vibration 309, 112–128.CrossRefGoogle Scholar
Chu, M. T. and Driessel, K. R. (1990), ‘The projected gradient method for least squares matrix approximations with spectral constraints’, SIAM J. Numer. Anal. 27, 1050–1060.CrossRefGoogle Scholar
Chu, M. T. and Funderlic, R. E. (2002), ‘The centroid decomposition: Relationships between discrete variational decompositions and SVDs’, SIAM J. Matrix Anal. Appl. 23, 1025–1044 (electronic).CrossRefGoogle Scholar
Chu, M. T. and Golub, G. H. (2002), Structured inverse eigenvalue problems, in Acta Numerica, Vol. 11, Cambridge University Press, pp. 1–71.Google Scholar
Chu, M. T. and Golub, G. H. (2005), Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York.CrossRefGoogle Scholar
Chu, M. T. and Guo, Q. (1998), ‘A numerical method for the inverse stochastic spectrum problem’, SIAM J. Matrix Anal. Appl. 19, 1027–1039 (electronic).CrossRefGoogle Scholar
Chu, M. T. and Norris, L. K. (1988), ‘Isospectral flows and abstract matrix factorizations’, SIAM J. Numer. Anal. 25, 1383–1391.CrossRefGoogle Scholar
Chu, M. T. and Trendafilov, N. T. (2001), ‘The orthogonally constrained regression revisited’, J. Comput. Graph. Statist. 10, 746–771.CrossRefGoogle Scholar
Chu, M. T. and Xu, S.-F. (2005), ‘On computing minimal realizable spectral radii of non-negative matrices’, Numer. Linear Algebra Appl. 12, 77–86.CrossRefGoogle Scholar
Chu, M., Buono, N. Del, Lopez, L. and Politi, T. (2005), ‘On the low-rank approximation of data on the unit sphere’, SIAM J. Matrix Anal. Appl. 27, 46–60 (electronic).CrossRefGoogle Scholar
Cox, T. F. and Cox, M. A. A. (1994), Multidimensional Scaling, Vol. 59 of Monographs on Statistics and Applied Probability, Chapman & Hall, London.Google Scholar
Curtis, M. L. (1984), Matrix Groups, second edn, Universitext, Springer, New York.CrossRefGoogle Scholar
Daniel, J. W. (1967), ‘The conjugate gradient method for linear and nonlinear operator equations’, SIAM J. Numer. Anal. 4, 10–26.CrossRefGoogle Scholar
Date, E., Kashiwara, M., Jimbo, M. and Miwa, T. (1983), Transformation groups for soliton equations, in Nonlinear Integrable Systems: Classical Theory and Quantum Theory, Kyoto 1981, World Scientific, Singapore, pp. 39–119.Google Scholar
Deift, P., Demmel, J., Li, L. C. and Tomei, C. (1991), ‘The bidiagonal singular value decomposition and Hamiltonian mechanics’, SIAM J. Numer. Anal. 28, 1463–1516.CrossRefGoogle Scholar
Deift, P., Nanda, T. and Tomei, C. (1983), ‘Ordinary differential equations and the symmetric eigenvalue problem’, SIAM J. Numer. Anal. 20, 1–22.CrossRefGoogle Scholar
Buono, N. Del and Lopez, L. (2002), ‘Geometric integration on manifold of square oblique rotation matrices’, SIAM J. Matrix Anal. Appl. 23, 974–989 (electronic).CrossRefGoogle Scholar
Della-Dora, J. (1975), ‘Numerical linear algorithms and group theory’, Linear Algebra Appl. 10, 267–283.CrossRefGoogle Scholar
Demmel, J. and Kahan, W. (1990), ‘Accurate singular values of bidiagonal matrices’, SIAM J. Sci. Statist. Comput. 11, 873–912.CrossRefGoogle Scholar
Demmel, J., Gu, M., Eisenstat, S., Slapničar, I., Veselić, K. and Drmač, Z. (1999), ‘Computing the singular value decomposition with high relative accuracy’, Linear Algebra Appl. 299, 21–80.CrossRefGoogle Scholar
Devaney, R. L. (1992), A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley Studies in Nonlinearity, Addison-Wesley, Reading, MA.Google Scholar
Dieci, L., Russell, R. D. and Van Vleck, E. S. (1994), ‘Unitary integrators and applications to continuous orthonormalization techniques’, SIAM J. Numer. Anal. 31, 261–281.CrossRefGoogle Scholar
Edelman, A., Arias, T. A. and Smith, S. T. (1999), ‘The geometry of algorithms with orthogonality constraints’, SIAM J. Matrix Anal. Appl. 20, 303–353 (electronic).CrossRefGoogle Scholar
Elaydi, S. (2005), An Introduction to Difference Equations, Undergraduate Texts in Mathematics, third edn, Springer, New York.Google Scholar
Engø, K. (2003), ‘Partitioned Runge–Kutta methods in Lie-group setting’, BIT 43, 21–39.CrossRefGoogle Scholar
Faßbender, H., Mackey, D. S., Mackey, N. and Xu, H. (1999), ‘Hamiltonian square roots of skew-Hamiltonian matrices’, Linear Algebra Appl. 287, 125–159.CrossRefGoogle Scholar
Faybusovich, L. (1991), ‘Hamiltonian structure of dynamical systems which solve linear programming problems’, Phys. D 53, 217–232.CrossRefGoogle Scholar
Fernando, K. V. and Parlett, B. N. (1994), ‘Accurate singular values and differential qd algorithms’, Numer. Math. 67, 191–229.CrossRefGoogle Scholar
Francis, J. G. F. (1961/1962), ‘The QR transformation: A unitary analogue to the LR transformation I’, Comput. J. 4, 265–271.CrossRefGoogle Scholar
Freund, R. W. and Nachtigal, N. M. (1991), ‘QMR: A quasi-minimal residual method for non-Hermitian linear systems’, Numer. Math. 60, 315–339.CrossRefGoogle Scholar
Galor, O. (2005), ‘Discrete dynamical systems’, GE, Growth, Math Methods, Econ-WPA. Available at http://ideas.repec.org/p/wpa/wuwpge/0504001.html.Google Scholar
Garcia, C. B. and Gould, F. J. (1980), ‘Relations between several path following algorithms and local and global Newton methods’, SIAM Rev. 22, 263–274.CrossRefGoogle Scholar
Garvey, S. D., Friswell, M. I. and Prells, U. (2002 a), ‘Co-ordinate transformations for second order systems I: General transformations’, J. Sound Vibration 258, 885–909.CrossRefGoogle Scholar
Garvey, S. D., Friswell, M. I. and Prells, U. (2002 b), ‘Co-ordinate transformations for second order systems II: Elementary structure-preserving transformations’, J. Sound Vibration 258, 911–930.CrossRefGoogle Scholar
Gear, C. W. (1981), ‘Numerical solution of ordinary differential equations: Is there anything left to do?’, SIAM Rev. 23, 10–24.CrossRefGoogle Scholar
Gohberg, I., Lancaster, P. and Rodman, L. (1982), Matrix Polynomials, Academic Press, New York.Google Scholar
Goldberg, D. (1991), ‘What every computer scientist should know about floating point arithmetic’, ACM Computing Surveys 23, 5–48.CrossRefGoogle Scholar
Golub, G. and Kahan, W. (1965), ‘Calculating the singular values and pseudo-inverse of a matrix’, J. Soc. Indust. Appl. Math. Ser. B, Numer. Anal. 2, 205–224.CrossRefGoogle Scholar
Golub, G. H. and Van Loan, C. F. (1996), Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, third edn, Johns Hopkins University Press, Baltimore, MD.Google Scholar
Golub, G. H. and Wilkinson, J. H. (1976), ‘Ill-conditioned eigensystems and the computation of the Jordan canonical form’, SIAM Rev. 18, 578–619.CrossRefGoogle Scholar
Greenbaum, A. (1997), Iterative Methods for Solving Linear Systems, Vol. 17 of Frontiers in Applied Mathematics, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Guckenheimer, J. (2002), Numerical analysis of dynamical systems, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, pp. 345–390.Google Scholar
Hageman, L. A. and Young, D. M. (1981), Applied Iterative Methods, Academic Press, New York. Also, unabridged republication of the 1981 original: Dover, Mineola, NY (2004).Google Scholar
Hairer, E. and Wanner, G. (1996), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Vol. 14 of Springer Series in Computational Mathematics, second edn, Springer, Berlin.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, second edn, Springer, Berlin.Google Scholar
Hairer, E., Nørsett, S. P. and Wanner, G. (1993), Solving ordinary differential equations I: Nonstiff Problems, Vol. 8 of Springer Series in Computational Mathematics, second edn, Springer, Berlin.Google Scholar
Hauser, R. and Nedić, J. (2007), ‘On the relationship between the convergence rates of iterative and continuous processes’, SIAM J. Optim. 18, 52–64 (electronic).CrossRefGoogle Scholar
Helmke, U. and Moore, J. B. (1994), Optimization and Dynamical Systems, Communications and Control Engineering Series, Springer, London.CrossRefGoogle Scholar
Helmke, U. and Wirth, F. (2000), Controllability of the shifted inverse power iteration: the case of real shifts, in International Conference on Differential Equations, Berlin 1999, Vols 1, 2, World Scientific, River Edge, NJ, pp. 859–864.Google Scholar
Helmke, U. and Wirth, F. (2001), ‘On controllability of the real shifted inverse power iteration’, Systems Control Lett. 43, 9–23.CrossRefGoogle Scholar
Helmke, U., Moore, J. B. and Perkins, J. E. (1994), ‘Dynamical systems that compute balanced realizations and the singular value decomposition’, SIAM J. Matrix Anal. Appl. 15, 733–754.CrossRefGoogle Scholar
Hestenes, M. R. and Stiefel, E. (1952), ‘Methods of conjugate gradients for solving linear systems’, J. Research Nat. Bur. Standards 49, 409–436.CrossRefGoogle Scholar
Higham, N. J. (1989), Matrix nearness problems and applications, in Applications of Matrix Theory, Bradford 1988, Vol. 22 of Inst. Math. Appl. Conf. Ser., New Ser., Oxford University Press, New York, pp. 1–27.Google Scholar
Hirota, R., Tsujimoto, S. and Imai, T. (1993), Difference scheme of soliton equations, in Future Directions of Nonlinear Dynamics in Physical and Biological Systems, Lyngby 1992, Vol. 312 of NATO Adv. Sci. Inst. Ser. B, Phys., Plenum, New York, pp. 7–15.CrossRefGoogle Scholar
Hirsch, M. W. and Smale, S. (1979), ‘On algorithms for solving f(x) = 0’, Comm. Pure Appl. Math. 32, 281–313.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (1990), Matrix Analysis, Cambridge University Press, Cambridge.Google Scholar
Horst, P. (1965), Factor Analysis of Data Matrices, Holt, Rinehart and Winston, New York.Google Scholar
Howe, R. (1983), ‘Very basic Lie theory’, Amer. Math. Monthly 90, 600–623. Correction to ‘Very basic Lie theory’, 91 (1984), 247.CrossRefGoogle Scholar
Iserles, A. (2002), ‘On the discretization of double-bracket flows’, Found. Comput. Math. 2, 305–329.CrossRefGoogle Scholar
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. and Zanna, A. (2000), Lie-group methods, in Acta Numerica, Vol. 9, Cambridge University Press, pp. 215–365.Google Scholar
Iwasaki, M. and Nakamura, Y. (2002), ‘On the convergence of a solution of the discrete Lotka–Volterra system’, Inverse Problems 18, 1569–1578.CrossRefGoogle Scholar
Iwasaki, M. and Nakamura, Y. (2004), ‘An application of the discrete Lotka–Volterra system with variable step-size to singular value computation’, Inverse Problems 20, 553–563.CrossRefGoogle Scholar
Iwasaki, M. and Nakamura, Y. (2006), ‘Accurate computation of singular values in terms of shifted integrable schemes’, Japan J. Indust. Appl. Math. 23, 239–259.CrossRefGoogle Scholar
Kahan, W. (1972), Conserving confluence curbs ill-condition. Technical report 6, Computer Science Department, University of California, Berkeley.Google Scholar
Karmarkar, N. (1984), ‘A new polynomial-time algorithm for linear programming’, Combinatorica 4, 373–395.CrossRefGoogle Scholar
Kelley, C. T. and Keyes, D. E. (1998), ‘Convergence analysis of pseudo-transient continuation’, SIAM J. Numer. Anal. 35, 508–523 (electronic).CrossRefGoogle Scholar
Kelley, C. T., Liao, L.-Z., Qi, L., Chu, M. T., Reese, J. P. and Winton, C. (2007), Projected pseudo-transient continuation. Preprint, North Carolina State University.Google Scholar
Kulenović, M. R. S. and Merino, O. (2002), Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman & Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Lax, P. D. (1968), ‘Integrals of nonlinear equations of evolution and solitary waves’, Comm. Pure Appl. Math. 21, 467–490.CrossRefGoogle Scholar
Lin, W.-W. and Xu, S.-F. (2006), ‘Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations’, SIAM J. Matrix Anal. Appl. 28, 26–39 (electronic).CrossRefGoogle Scholar
Lin, W.-W., Mehrmann, V. and Xu, H. (1999), ‘Canonical forms for Hamiltonian and symplectic matrices and pencils’, Linear Algebra Appl. 302/303, 469–533.CrossRefGoogle Scholar
Lojasiewicz, S. (1963), Une propriété topologique des sous-ensembles analytiques réels, in Les Equations aux Dérivées Partielles, Paris 1962, Éditions du Centre National de la Recherche Scientifique, Paris, pp. 87–89.Google Scholar
Mackey, D. S., Mackey, N. and Tisseur, F. (2003), ‘Structured tools for structured matrices’, Electron. J. Linear Algebra 10, 106–145 (electronic).CrossRefGoogle Scholar
Mehl, C. (1999), ‘Condensed forms for skew-Hamiltonian/Hamiltonian pencils’, SIAM J. Matrix Anal. Appl. 21, 454–476 (electronic).CrossRefGoogle Scholar
Mehrmann, V. and Watkins, D. (2000), ‘Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils’, SIAM J. Sci. Comput. 22, 1905–1925 (electronic).CrossRefGoogle Scholar
Meurant, G. (2006), The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations, Vol. 19 of Software, Environments, and Tools, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Morgan, A. (1987), Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Moser, J. (1975), ‘Three integrable Hamiltonian systems connected with isospectral deformations’, Adv. Math. 16, 197–220.CrossRefGoogle Scholar
Mulder, W. A. and van Leer, B. (1985), ‘Experiments with implicit upwind methods for the Euler equations’, J. Comput. Phys. 59, 232–246.CrossRefGoogle Scholar
Munthe-Kaas, H. (1998), ‘Runge–Kutta methods on Lie groups’, BIT 38, 92–111.CrossRefGoogle Scholar
Nakamura, Y. (2004), A new approach to numerical algorithms in terms of integrable systems, in Proc. International Conference on Informatics Research for Development of Knowledge Society Infrastructure: ICKS 2004 (Ibaraki, T., Inui, T. and Tanaka, K. K., eds), IEEE Computer Society Press, pp. 194–205.Google Scholar
Nakamura, Y. (2006), Functionality of Integrable Systems, Kyoritsu Shuppan Co., Tokyo, Japan. (In Japanese.)Google Scholar
Orsi, R. (2006), ‘Numerical methods for solving inverse eigenvalue problems for nonnegative matrices’, SIAM J. Matrix Anal. Appl. 28, 190–212 (electronic).CrossRefGoogle Scholar
Ortega, J. M. and Rheinboldt, W. C. (2000), Iterative Solution of Nonlinear Equations in Several Variables, Vol. 30 of Classics in Applied Mathematics, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Owren, B. and Welfert, B. (2000), ‘The Newton iteration on Lie groups’, BIT 40, 121–145.CrossRefGoogle Scholar
Paige, C. and Van Loan, C. (1981), ‘A Schur decomposition for Hamiltonian matrices’, Linear Algebra Appl. 41, 11–32.CrossRefGoogle Scholar
Parlett, B. N. (1974), ‘The Rayleigh quotient iteration and some generalizations for nonnormal matrices’, Math. Comp. 28, 679–693.CrossRefGoogle Scholar
Parlett, B. N. and Marques, O. A. (2000), An implementation of the dqds algorithm (positive case), in Proc. International Workshop on Accurate Solution of Eigenvalue Problems, University Park 1998, Vol. 309, pp. 217–259.Google Scholar
Pöppe, C. (1989), ‘General determinants and the τ function for the Kadomtsev–Petviashvili hierarchy’, Inverse Problems 5, 613–630. See also corrigenda: 5, 1173.CrossRefGoogle Scholar
Potra, F. A. and Wright, S. J. (2000), ‘Interior-point methods’, J. Comput. Appl. Math. 124, 281–302.CrossRefGoogle Scholar
Ruhe, A. (1987), ‘Closest normal matrix finally found!’, BIT 27, 585–598.CrossRefGoogle Scholar
Rutishauser, H. (1954), ‘Der Quotienten-Differenzen-Algorithmus’, Z. Angew. Math. Physik 5, 233–251.CrossRefGoogle Scholar
Rutishauser, H. (1960), ‘Über eine kubisch konvergente Variante der LR-Transformation’, Z. Angew. Math. Mech. 40, 49–54.CrossRefGoogle Scholar
Saad, Y. and Schultz, M. H. (1986), ‘GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems’, SIAM J. Sci. Statist. Comput. 7, 856–869.CrossRefGoogle Scholar
Savinov, G. V. (1983), ‘The conjugate gradient method for systems of nonlinear equations’, J. Math. Sci. 23, 2012–2017. Translated from Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 70 (1977), 178–183.CrossRefGoogle Scholar
Sedaghat, H. (2003), Nonlinear Difference Equations: Theory with Applications to Social Science Models, Vol. 15 of Mathematical Modelling: Theory and Applications, Kluwer, Dordrecht.CrossRefGoogle Scholar
Shaw, R. (1982), Linear Algebra and Group Representations I: Linear Algebra and Introduction to Group Representations, Academic Press, London.Google Scholar
Simon, L. (1983), ‘Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems’, Ann. of Math. (2) 118, 525–571.CrossRefGoogle Scholar
Smale, S. (1977), Convergent process of price adjustment and global Newton methods, in Frontiers of Quantitative Economics: Invited papers, Econometric Soc., Third World Congress, Toronto 1975, Vol. IIIA, North-Holland, Amsterdam, pp. 191–205.Google Scholar
Smirnov, V. I. (1970), Linear Algebra and Group Theory, Dover, New York.Google Scholar
Smith, S. T. (1991), ‘Dynamical systems that perform the singular value decomposition’, Systems Control Lett. 16, 319–327.CrossRefGoogle Scholar
Stewart, G. W. (1993), ‘On the early history of the singular value decomposition’, SIAM Rev. 35, 551–566.CrossRefGoogle Scholar
Stuart, A. M. and Humphries, A. R. (1996), Dynamical Systems and Numerical Analysis, Vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge.Google Scholar
Symes, W. W. (1981/1982), ‘The QR algorithm and scattering for the finite nonperiodic Toda lattice’, Phys. D 4, 275–280.CrossRefGoogle Scholar
Szegő, G. (1975), Orthogonal Polynomials, fourth edn. Vol. XXIII of AMS Colloquium Series, AMS, Providence, RI.Google Scholar
Takata, M., Iwasaki, M., Kimura, K. and Nakamura, Y. (2005), An evaluation of singular value computation by the discrete Lotka–Volterra system, in Proc. 2005 International Conference on Parallel and Distributed Processing Techniques and Applications: PDPTA2005, Vol. II, pp. 410–416.Google Scholar
Takata, M., Iwasaki, M., Kimura, K. and Nakamura, Y. (2006), Performance of a new scheme for bidiagonal singular value decomposition of large scale, in Proc. IASTED International Conference on Parallel and Distributed Computing and Networks: PDCN2006, pp. 304–309.Google Scholar
Tam, T.-Y. (2004), ‘Gradient flows and double bracket equations’, Differential Geom. Appl. 20, 209–224.CrossRefGoogle Scholar
Tisseur, F. and Meerbergen, K. (2001), ‘The quadratic eigenvalue problem’, SIAM Rev. 43, 235–286 (electronic).CrossRefGoogle Scholar
Toselli, A. and Widlund, O. (2005), Domain Decomposition Methods: Algorithms and Theory, Vol. 34 of Springer Series in Computational Mathematics, Springer, Berlin.CrossRefGoogle Scholar
Tsujimoto, S. (1995), Molecule solution of hungry Volterra equations, in Soliton Theory and Its Applications (Satsuma, J., ed.), University of Tokyo, Japan, pp. 53–56. (In Japanese.)Google Scholar
Tsujimoto, S., Nakamura, Y. and Iwasaki, M. (2001), ‘The discrete Lotka–Volterra system computes singular values’, Inverse Problems 17, 53–58.CrossRefGoogle Scholar
Tsypkin, Y. Z. (1971), Adaptation and Learning in Automatic Systems, Vol. 73 of Mathematics in Science and Engineering, Academic Press, New York. Translated from the Russian by Nikolic, Z. J..Google Scholar
Tsypkin, Y. Z. (1973), Foundations of the Theory of Learning Systems, Vol. 101 of Mathematics in Science and Engineering, Academic Press, New York/London. Translated from the Russian by Nikolic, Z. J..Google Scholar
van der Vorst, H. A. (2003), Iterative Krylov Methods for Large Linear Systems, Vol. 13 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Van Loan, C. F. (1984), ‘A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix’, Linear Alg. Appl. 61, 233–253.CrossRefGoogle Scholar
Varga, R. S. (2000), Matrix Iterative Analysis, Vol. 27 of Springer Series in Computational Mathematics, expanded edn, Springer, Berlin.CrossRefGoogle Scholar
Watkins, D. S. (1982), ‘Understanding the QR algorithm’, SIAM Rev. 24, 427–440.CrossRefGoogle Scholar
Watkins, D. S. and Elsner, L. (1988), ‘Self-similar flows’, Linear Algebra Appl. 110, 213–242.CrossRefGoogle Scholar
Wimp, J. (1984), Computation with Recurrence Relations, Applicable Mathematics Series, Pitman, Boston, MA.Google Scholar
Wright, K. (1992), ‘Differential equations for the analytic singular value decomposition of a matrix’, Numer. Math. 63, 283–295.CrossRefGoogle Scholar
Wright, M. H. (2005), ‘The interior-point revolution in optimization: history, recent developments, and lasting consequences’, Bull. Amer. Math. Soc. (N.S.) 42, 39–56 (electronic).CrossRefGoogle Scholar
Wright, S. J. (1997), Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Yabe, H. and Takano, M. (2004), ‘Global convergence properties of nonlinear conjugate gradient methods with modified secant condition’, Comput. Optim. Appl. 28, 203–225.CrossRefGoogle Scholar
Zeng, Z. and Li, T. Y. (2007), A numerical method for computing the Jordan canonical form. Preprint, Northeastern Illinois University.Google Scholar
Zha, H. and Zhang, Z. (1995), ‘A note on constructing a symmetric matrix with specified diagonal entries and eigenvalues’, BIT 35, 448–451.CrossRefGoogle Scholar
Zhang, S. and Deng, Z. (2005), ‘Geometric integration methods for general nonlinear dynamic equation based on Magnus and Fer expansions’, Progr. Natur. Sci. (English edn) 15, 304–314.CrossRefGoogle Scholar