Linear algebra algorithms as dynamical systems

Acta Numerica

Volumn 17, Issue , 2008, Pages 1-86

  Chu, Moody T ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 58149396665     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492906340019     Document Type: Article

References (166)

1. P.-A. Absil and K. Kurdyka (2006), 'On the stable equilibrium poin ts of gradient systems', Systems Control Lett. 55, 573-577.
2. P.-A. Absil, R. Mahony and B. Andrews (2005), 'Convergence of the iterates of descent methods for analytic cost functions', SIAM J. Optim. 16, 531-547 (electronic).
3. N. I. Akhiezer (1965), The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York. Translated by N. Kemmer.
4. E. Allgower and K. Georg (1980), 'Simplicial and continuation methods for approximating fixed points and solutions to systems of equations', SIAM Rev. 22, 28-85.
5. E. L. Allgower and K. Georg (2003), Introduction to Numerical Continuation Methods, Vol. 45 of Classics in Applied Mathematics, SIAM, Philadelphia, PA.
6. A. C. Antoulas (2005), Approximation of Large-Scale Dynamical Systems, Vol. 6 of Advances in Design and Control, SIAM, Philadelphia, PA.
7. A. I. Aptckarcv, A. Branquinho and F. Marcellán (1997), 'Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation', J. Comput. Appl. Math. 78, 139-160.
8. P. Arbenz and G. H. Golub (1995), 'Matrix shapes invariant under the symmetric QR algorithm', Numer. Linear Algebra Appl. 2, 87-93.
9. V. I. Arnold (1988), Geometrical Methods in the Theory of Ordinary Differential Equations, Vol. 250 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), second cdn, Springer, New York. Translated from the Russian by Joseph Szücs (József M. Szu″cs).
10. D. A. Ashlock, K. R. Driessel and I. R. Hentzel (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.
11. A. Baker (2002), Matrix Groups: An Introduction to Lie Group Theory, Springer Undergraduate Mathematics Series, Springer, London.
12. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijk-hout, R. Pozo, C. Romine and H. Van der Vorst (1994), Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA.
13. T. Beelen and P. Van Dooren (1990), Computational aspects of the Jordan canonical form, in Reliable Numerical Computation, Oxford University Press, New York, pp. 57-72.
14. P. Benner and D. Kressner (2006), 'Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II', ACM Trans. Math. Software 32, 352-373.
15. P. Benner, R. Byers, V. Mehrmann and H. Xu (2002), 'Numerical computation of deflating subspaccs of skew-Hamiltonian/Hamiltonian pencils', SIAM J. Matrix Anal Appl. 24, 165-190 (electronic).
16. P. Benner, D. Kressner and V. Mehrmann (2005), Skcw-Hamiltonian and Hamiltonian eigenvalue problems: theory, algorithms and applications, in Proc. Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, pp. 3-39.
17. P. Benner, V. Mehrmann and H. Xu (1998), 'A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or sym-plcctic pencils', Numer. Math. 78, 329-358.
18. A. Bhaya and E. Kaszkurewicz (2006), Control Perspectives on Numerical Algorithms and Matrix Problems, Vol. 10 of Advances in Design and Control, SIAM, Philadelphia, PA.
19. S. Blanes, F. Casas, J. A. Oteo and J. Ros (1998), 'Magnus and Fer expansions for matrix differential equations: The convergence problem', J. Phys. A 31, 259-268.
20. A. M. Bloch and A. Iserles (2006), 'On an isospectral Lie-Poisson system and its Lie algebra', Found. Comput. Math. 6, 121-144.
21. A. M. Bloch, R. W. Brockett and T. S. Ratiu (1992), 'Completely integrable gradient flows', Comm. Math. Phys. 147, 57-74.
22. J. H. Bramble (1993), Multigrid Methods, Vol. 294 of Pitman Research Notes in Mathematics Series, Longman, Harlow.
23. W. L. Briggs (1987), A Multigrid Tutorial, SIAM, Philadelphia, PA.
24. R. W. Brockett (1991), 'Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems', Linear Algebra Appl. 146, 79-91.
25. R. W. Brockett (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.
26. A. Bunse-Gerstner, R. Byers, V. Mehrmann and N. K. Nichols (1991), 'Numerical computation of an analytic singular value decomposition of a matrix valued function', Numer. Math. 60, 1-39.
27. M. P. Calvo, A. Iserles and A. Zanna (1997), 'Numerical solution of isospectral flows', Math. Comp. 66, 1461-1486.
28. E. Celledoni and A. Iserles (2000), 'Approximating the exponential from a Lie algebra to a Lie group', Math. Comp. 69, 1457-1480.
29. R. Chill (2003), 'On the Łojasicwicz-Simon gradient inequality', J. Funct. Anal. 201, 572-601.
30. D. Chu, X. Liu and V. Mehrmann (2007), 'A numerical method for computing the Hamiltonian Schur form', Numer. Math. 105, 375-412.
31. M. Chu (1986a), 'A continuous approximation to the generalized Schur decomposition', Linear Algebra Appl. 78, 119-132.
32. M. T. Chu (1986b), 'A differential equation approach to the singular value decomposition of bidiagonal matrices', Linear Algebra Appl. 80, 71-79.
33. M. T. Chu (1988), 'On the continuous realization of iterative processes', SIAM Rev. 30, 375-387.
34. M. T. Chu (1991), 'A continuous Jacobi-like approach to the simultaneous reduction of real matrices', Linear Algebra Appl. 147, 75-96.
35. M. T. Chu (1995), 'Constructing a Hermitian matrix from its diagonal entries and eigenvalues', SIAM J. Matrix Anal. Appl. 16, 207-217.
36. M. T. Chu and N. Del Buono (2008a), 'Total decoupling of general quadratic pencils I: Theory', J. Sound Vibration 309, 96-111.
37. M. T. Chu and N. Del Buono (2008b), 'Total decoupling of general quadratic pencils II: Structure preserving isospcctral flows', J. Sound Vibration 309, 112-128.
38. M. T. Chu and K. R. Driessel (1990), 'The projected gradient method for least squares matrix approximations with spectral constraints', SIAM J. Numer. Anal. 27, 1050-1060.
39. M. T. Chu and R. E. Fundcrlic (2002), 'The centroid decomposition: Relationships between discrete variational decompositions and SVDs', SIAM J. Matrix Anal. Appl. 23, 1025-1044 (electronic).
40. M. T. Chu and G. H. Golub (2002), Structured inverse eigenvalue problems, in Acta Numerica, Vol. 11, Cambridge University Press, pp. 1-71.
41. M. T. Chu and G. H. Golub (2005), Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York.
42. M. T. Chu and Q. Guo (1998), 'A numerical method for the inverse stochastic spectrum problem', SIAM J. Matrix Anal. Appl. 19, 1027-1039 (electronic).
43. M. T. Chu and L. K. Norris (1988), 'Isospcctral flows and abstract matrix factor izations', SIAM J. Numer. Anal. 25, 1383-1391.
44. M. T. Chu and N. T. Trendafilov (2001), 'The orthogonally constrained regression revisited', J. Comput. Graph. Statist. 10, 746-771.
45. M. T. Chu and S.-F. Xu (2005), 'On computing minimal realizable spectral radii of non-negative matrices', Numer. Linear Algebra Appl. 12, 77-86.
46. M. Chu, N. Del Buono, L. Lopez and T. Politi (2005), 'On the low-rank approximation of data on the unit sphere', SIAM J. Matrix Anal. Appl. 27, 46-60 (electronic).
47. T. F. Cox and M. A. A. Cox (1994), Multidimensional Scaling, Vol. 59 of Monographs on Statistics and Applied Probability, Chapman & Hall, London.
48. M. L. Curtis (1984), Matrix Groups, second edn, Universitcxt, Springer, New York.
49. J. W. Daniel (1967), 'The conjugate gradient method for linear and nonlinear operator equations', SIAM J. Numer. Anal. 4, 10-26.
50. E. Date, M. Kashiwara, M. Jimbo and T. Miwa (1983), Transformation groups for soliton equations, in Nonlinear Integrable Systems: Classical Theory and Quantum Theory, Kyoto 1981, World Scientific, Singapore, pp. 39-119.
51. P. Deift, J. Demmel, L. C. Li and C. Tomei (1991), 'The bidiagonal singular value decomposition and Hamiltonian mechanics', SIAM J. Numer. Anal. 28, 1463-1516.
52. P. Deift, T. Nanda and C. Tomei (1983), 'Ordinary differential equations and the symmetric eigenvalue problem', SIAM J. Numer. Anal. 20, 1-22.
53. N. Del Buono and L. Lopez (2002), 'Geometric integration on manifold of square oblique rotation matrices', SIAM J. Matrix Anal. Appl. 23, 974-989 (electronic).
54. J. Della-Dora (1975), 'Numerical linear algorithms and group theory', Linear Algebra Appl. 10, 267-283.
55. J. Dcmmcl and W. Kahan (1990), 'Accurate singular values of bidiagonal matrices', SIAM J. Sci. Statist. Comput. 11, 873-912.
56. J. Dcmmcl, M. Gu, S. Eisenstat, I. Slapničar, K. Vcsclić and Z. Drmač (1999), 'Computing the singular value decomposition with high relative accuracy', Linear Algebra Appl. 299, 21-80.
57. R. L. Devaney (1992), A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley Studies in Nonlincarity, Addison-Wesley, Reading, MA.
58. L. Dieci, R. D. Russell and E. S. Van Vleck (1994), 'Unitary integrators and applications to continuous orthonormalization techniques', SIAM J. Numer. Anal. 31, 261-281.
59. A. Edelman, T. A. Arias and S. T. Smith (1999), 'The geometry of algorithms with orthogonality constraints', SIAM J. Matrix Anal. Appl. 20, 303-353 (electronic).
60. S. Elaydi (2005), An Introduction to Difference Equations, Undergraduate Texts in Mathematics, third edn, Springer, New York.
61. K. Engø (2003), 'Partitioned Runge-Kutta methods in Lie-group setting', BIT 43, 21-39.
62. H. Faßbcnder, D. S. Mackey, N. Mackcy and H. Xu (1999), 'Hamiltonian square roots of skew-Hamiltonian matrices', Linear Algebra Appl. 287, 125-159.
63. L. Faybusovich (1991), 'Hamiltonian structure of dynamical systems which solve linear programming problems', Phys. D 53, 217-232.
64. K. V. Fernando and B. N. Parlett (1994), 'Accurate singular values and differential qd algorithms', Numer. Math. 67, 191-229.
65. J. G. F. Francis (1961/1962), 'The QR transformation: A unitary analogue to the LR transformation I, Comput. J. 4, 265-271.
66. R. W. Freund and N. M. Nachtigal (1991), 'QMR: A quasi-minimal residual method for non-Hermitian linear systems', Numer. Math. 60, 315-339.
67. O. Galor (2005), 'Discrete dynamical systems', GE, Growth. Math Methods. Econ-WPA. Available at http://ideas.repec.Org/p/wpa/wuwpge/0504001. html.
68. C. B. García and F. J. Gould (1980), 'Relations between several path following algorithms and local and global Newton methods', SIAM Rev. 22, 263-274.
69. S. D. Garvcy, M. I. Friswcll and U. Prclls (2002a), 'Co-ordinate transformations for second order systems I: General transformations', J. Sound Vibration 258, 885-909.
70. S. D. Garvey, M. I. Friswell and U. Prells (2002b), 'Co-ordinate transformations for second order systems II: Elementary structure-preserving transformations', J. Sound Vibration 258, 911-930.
71. C. W. Gear (1981), 'Numerical solution of ordinary differential equations: Is there anything left to do?', SIAM Rev. 23, 10-24.
72. I. Gohberg, P. Lancaster and L. Rodman (1982), Matrix Polynomials, Academic Press, New York.
73. D. Goldberg (1991), 'What every computer scientist should know about floatingpoint arithmetic', ACM Computing Surveys 23, 5-48.
74. G. Golub and W. Kahan (1965), 'Calculating the singular values and pseudo-inverse of a matrix', J. Soc. Indust. Appl. Math. Ser. B, Numer. Anal. 2, 205-224.
75. G. H. Golub and C. F. Van Loan (1996), Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, third edn, Johns Hopkins University Press, Baltimore, MD.
76. G. H. Golub and J. H. Wilkinson (1976), 'Ill-conditioned cigensystems and the computation of the Jordan canonical form', SIAM Rev. 18, 578-619.
77. A. Greenbaum (1997), Iterative Methods for Solving Linear Systems, Vol. 17 of Frontiers in Applied Mathematics, SIAM, Philadelphia, PA.
78. J. Guckenheimer (2002), Numerical analysis of dynamical systems, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, pp. 345-390.
79. L. A. Hageman and D. M. Young (1981), Applied Iterative Methods, Academic Press, New York. Also, unabridged republication of the 1981 original: Dover, Mineola, NY (2004).
80. E. Hairer and G. Wanner (1996), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Vol. 14 of Springer Series in Computational Mathematics, second edn, Springer, Berlin.
81. E. Hairer, C. Lubich and G. Wanner (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, second edn, Springer, Berlin.
82. E. Hairer, S. P. Nørsett and G. Wanner (1993), Solving ordinary differential equations I: Nonstiff Problems, Vol. 8 of Springer Series in Computational Mathematics, second edn, Springer, Berlin.
83. R. Hauser and J. Nedić (2007), 'On the relationship between the convergence rates of iterative and continuous processes', SIAM J. Optim. 18, 52-64 (electronic).
84. U. Helmke and J. B. Moore (1994), Optimization and Dynamical Systems, Communications and Control Engineering Series, Springer, London.
85. U. Helmke and F. Wirth (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.
86. U. Helmke and F. Wirth (2001), 'On controllability of the real shifted inverse power iteration', Systems Control Lett. 43, 9-23.
87. U. Helmke, J. B. Moore and J. E. Perkins (1994), 'Dynamical systems that compute balanced realizations and the singular value decomposition', SIAM J. Matrix Anal. Appl. 15, 733-754.
88. M. R. Hestenes and E. Stiefel (1952), 'Methods of conjugate gradients for solving linear systems', J. Research Nat. Bur. Standards 49, 409-436.
89. N. J. Higham (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.
90. R. Hirota, S. Tsujimoto and T. Imai (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.
91. M. W. Hirsch and S. Smale (1979), 'On algorithms for solving f(x) = 0', Comm. Pure Appl. Math. 32, 281-313.
92. R. A. Horn and C. R. Johnson (1990), Matrix Analysis, Cambridge University Press, Cambridge.
93. P. Horst (1965), Factor Analysis of Data Matrices, Holt, Rinehart and Winston, New York.
94. R. Howe (1983), 'Very basic Lie theory', Amer. Math. Monthly 90, 600-623.
95. Correction to 'Very basic Lie theory', 91 (1984), 247.
96. A. Iscrlcs (2002), 'On the discretization of double-bracket flows', Found. Comput. Math. 2, 305-329.
97. A. Iserles, H. Z. Munthe-Kaas, S. P. N0rsett and A. Zanna (2000), Lie-group methods, in Acta Numerica, Vol. 9, Cambridge University Press, pp. 215-365.
98. M. Iwasaki and Y. Nakamura (2002), 'On the convergence of a solution of the discrete Lotka-Volterra system', Inverse Problems 18, 1569-1578.
99. M. Iwasaki and Y. Nakamura (2004), 'An application of the discrete Lotka-Volterra system with variable step-size to singular value computation', Inverse Problems 20, 553-563.
100. M. Iwasaki and Y. Nakamura (2006), 'Accurate computation of singular values in terms of shifted integrable schemes', Japan J. Indust. Appl. Math. 23, 239-259.
101. W. Kahan (1972), Conserving confluence curbs ill-condition. Technical report 6, Computer Science Department, University of California, Berkeley.
102. N. Karmarkar (1984), 'A new polynomial-time algorithm for linear programming', Combinatorica 4, 373-395.
103. C. T. Kelley and D. E. Keyes (1998), 'Convergence analysis of pseudo-transient continuation', SIAM J. Numer. Anal. 35, 508-523 (electronic).
104. C. T. Kelley, L.-Z. Liao, L. Qi, M. T. Chu, J. P. Reese and C. Winton (2007), Projected pscudo-transient continuation. Preprint, North Carolina State University.
105. M. R. S. Kulenović and O. Merino (2002), Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman & Hall/CRC, Boca Raton, FL.
106. P. D. Lax (1968), 'Integrals of nonlinear equations of evolution and solitary waves', Comm. Pure Appl. Math. 21, 467-490.
107. W.-W. Lin and S.-F. Xu (2006), 'Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations', SIAM J. Matrix Anal. Appl. 28, 26-39 (electronic).
108. W.-W. Lin, V. Mehrmann and H. Xu (1999), 'Canonical forms for Hamiltonian and symplcctic matrices and pencils', Linear Algebra Appl. 302/303, 469-533.
109. S. Łojasiewicz (1963), Unc propriété topologique des sous-enscmbles analytiques réels, in Les Equations aux Dérivées Partielles, Paris 1962, Éditions du Centre National de la Recherche Scientifique, Paris, pp. 87-89.
110. D. S. Mackcy, N. Mackcy and F. Tisseur (2003), 'Structured tools for structured matrices', Electron. J. Linear Algebra 10, 106-145 (electronic).
111. C. Mehl (1999), 'Condensed forms for skew-Hamiltonian/Hamiltonian pencils', SIAM J. Matrix Anal. Appl. 21, 454-476 (electronic).
112. V. Mehrmann and D. Watkins (2000), 'Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils', SIAM J. Sci. Comput. 22, 1905-1925 (electronic).
113. G. Meurant (2006), The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations, Vol. 19 of Software. Environments, and Tools, SIAM, Philadelphia, PA.
114. A. Morgan (1987), Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice-Hall, Englewood Cliffs, NJ.
115. J. Moser (1975), 'Three integrable Hamiltonian systems connected with isospectral deformations', Adv. Math. 16, 197-220.
116. W. A. Mulder and B. van Leer (1985), 'Experiments with implicit upwind methods for the Euler equations', J. Comput. Phys. 59, 232-246.
117. H. Munthc-Kaas (1998), 'Runge-Kutta methods on Lie groups', BIT 38, 92-111.
118. Y. Nakamura (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 (T. Ibaraki, T. Inui and K. K. Tanaka, eds), IEEE Computer Society Press, pp. 194-205.
119. Y. Nakamura (2006), Functionality of Integrable Systems, Kyoritsu Shuppan Co., Tokyo, Japan. (In Japanese.)
120. R. Orsi (2006), 'Numerical methods for solving inverse eigenvalue problems for nonnegative matrices', SIAM J. Matrix Anal. Appl. 28, 190-212 (electronic).
121. J. M. Ortega and W. C. Rheinboldt (2000), Iterative Solution of Nonlinear Equations in Several Variables, Vol. 30 of Classics in Applied Mathematics, SIAM, Philadelphia, PA.
122. B. Owren and B. Welfert (2000), 'The Newton iteration on Lie groups', BIT 40, 121-145.
123. C. Paige and C. Van Loan (1981), 'A Schur decomposition for Hamiltonian matrices', Linear Algebra Appl. 41, 11-32.
124. B. N. Parlett (1974), 'The Rayleigh quotient iteration and some generalizations for nonnormal matrices', Math. Comp. 28, 679-693.
125. B. N. Parlett and O. A. Marques (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.
126. C. Pöppe (1989), 'General determinants and the τfunction for the Kadomtsev-Petviashvili hierarchy', Inverse Problems 5, 613-630.
127. See also corrigenda: 5, 1173
128. F. A. Potra and S. J. Wright (2000), 'Interior-point methods', J. Comput. Appl. Math. 124, 281-302.
129. A. Ruhe (1987), 'Closest normal matrix finally found!', BIT 27, 585-598.
130. H. Rutishauser (1954), 'Der Quotienten-Differenzen-Algorithmus', Z. Angew. Math. Physik 5, 233-251.
131. H. Rutishauser (1960), 'Über eine kubisch konvergente Variante der LR-Trans-formation', Z. Angew. Math. Mech. 40, 49-54.
132. Y. Saad and M. H. Schultz (1986), 'GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems', SIAM J. Sci. Statist. Comput. 7, 856-869.
133. G. V. Savinov (1983), 'The conjugate gradient method for systems of nonlinear equations', J. Math. Sci. 23, 2012-2017.
134. Translated from Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 70 (1977), 178-183.
135. H. Sedaghat (2003), Nonlinear Difference Equations: Theory with Applications to Social Science Models, Vol. 15 of Mathematical Modelling: Theory and Applications, Kluwer, Dordrecht.
136. R. Shaw (1982), Linear Algebra and Group Representations I: Linear Algebra and Introduction to Group Representations, Academic Press, London.
137. L. Simon (1983), 'Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems', Ann. of Math. (2) 118, 525-571.
138. S. Smale (1977), Convergent process of price adjustment and global Newton methods, in Frontiers of Quantitative Economics: Invited papers, Econometric Soc., Third World Congress, Toronto 1915, Vol. IIIA, North-Holland, Amsterdam, pp. 191-205.
139. I. Smirnov (1970), Linear Algebra and Group Theory, Dover, New York.
140. S. T. Smith (1991), 'Dynamical systems that perform the singular value decomposition', Systems Control Lett. 16, 319-327.
141. G. W. Stewart (1993), 'On the early history of the singular value decomposition', SIAM Rev. 35, 551-566.
142. A. M. Stuart and A. R. Humphries (1996), Dynamical Systems and Numerical Analysis, Vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge.
143. W. W. Symes (1981/82), 'The QR algorithm and scattering for the finite nonperi-odic Toda lattice', Phys. D 4, 275-280.
144. G. Szego′ (1975), Orthogonal Polynomials, fourth edn. Vol. XXIII of AMS Colloquium Series, AMS, Providence, RI.
145. M. Takata, M. Iwasaki, K. Kimura and Y. Nakamura (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.
146. M. Takata, M. Iwasaki, K. Kimura and Y. Nakamura (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.
147. T.-Y. Tam (2004), 'Gradient flows and double bracket equations', Differential Geom. Appl. 20, 209-224.
148. F. Tisseur and K. Mccrbcrgcn (2001), 'The quadratic eigenvalue problem', SIAM Rev. 43, 235-286 (electronic).
149. A. Toselli and O. Widlund (2005), Domain Decomposition Methods: Algorithms and Theory, Vol. 34 of Springer Series in Computational Mathematics, Springer, Berlin.
150. S. Tsujimoto (1995), Molecule solution of hungry Volterra equations, in Soliton Theory and Its Applications (J. Satsuma, ed.), University of Tokyo, Japan, pp. 53-56. (In Japanese.)
151. S. Tsujimoto, Y. Nakamura and M. Iwasaki (2001), 'The discrete Lotka-Volterra system computes singular values', Inverse Problems 17, 53-58.
152. Y. Z. Tsypkin (1971), Adaptation and Learning in Automatic Systems, Vol. 73 of Mathematics in Science and Engineering, Academic Press, New York. Translated from the Russian by Z. J. Nikolic.
153. Y. Z. Tsypkin (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 Z. J. Nikolic.
154. H. A. van der Vorst (2003), Iterative Krylov Methods for Large Linear Systems, Vol. 13 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge.
155. C. F. Van Loan (1984), 'A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix', Linear Alg. Appl. 61, 233-253.
156. R. S. Varga (2000), Matrix Iterative Analysis, Vol. 27 of Springer Series in Computational Mathematics, expanded edn, Springer, Berlin.
157. D. S. Wat kins (1982), 'Understanding the QR algorithm', SIAM Rev. 24, 427-440.
158. D. S. Watkins and L. Elsner (1988), 'Self-similar flows', Linear Algebra Appl. 110, 213-242.
159. J. Wimp (1984), Computation with Recurrence Relations, Applicable Mathematics Series, Pitman, Boston, MA.
160. K. Wright (1992), 'Differential equations for the analytic singular value decomposition of a matrix', Numer. Math. 63, 283-295.
161. M. H. Wright (2005), 'The interior-point revolution in optimization: history, recent developments, and lasting consequences', Bull. Amer. Math. Soc. (N.S.) 42, 39-56 (electronic).
162. S. J. Wright (1997), Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA.
163. H. Yabe and M. Takano (2004), 'Global convergence properties of nonlinear conjugate gradient methods with modified secant condition', Comput. Optim. Appl. 28, 203-225.
164. Z. Zeng and T. Y. Li (2007), A numerical method for computing the Jordan canonical form. Preprint, Northeastern Illinois University.
165. H. Zha and Z. Zhang (1995), 'A note on constructing a symmetric matrix with specified diagonal entries and eigenvalues', BIT 35, 448-451.
166. S. Zhang and Z. Deng (2005), 'Geometric integration methods for general nonlinear dynamic equation based on Magnus and Fer expansions', Progr. Natur. Sci. (English edn) 15, 304-314.