Applications of Smooth Orthogonal Factorizations of Matrices

The purpose of this paper is to give instances where computation of smooth orthogonal factorization of matrices is encountered. Applications arising from dynamical systems are emphasized.

[1]  Ulrich Parlitz,et al.  Comparison of Different Methods for Computing Lyapunov Exponents , 1990 .

[2]  K. Wright.  Differential equations for the analytic singular value decomposition of a matrix , 1992 .

[3]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[4]  Gunter H. Meyer,et al.  Continuous orthonormalization for boundary value problems , 1986 .

[5]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[6]  W. Rheinboldt.  On the computation of multi-dimensional solution manifolds of parametrized equations , 1988 .

[7]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[8]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[9]  M. Chu.  On the Continuous Realization of Iterative Processes , 1988 .

[10]  R. Russell,et al.  Unitary integrators and applications to continuous orthonormalization techniques , 1994 .

[11]  P. Deift,et al.  Ordinary differential equations and the symmetric eigenvalue problem , 1983 .

[12]  A. Davey,et al.  An automatic orthonormalization method for solving stiff boundary-value problems , 1983 .

[13]  Antonella Zanna,et al.  Numerical solution of isospectral flows , 1997, Math. Comput..

[14]  L. Dieci,et al.  Positive definiteness in the numerical solution of Riccati differential equations , 1994 .

[15]  Hinke M. Osinga,et al.  Computing invariant manifolds , 1996 .

[16]  Lixin Liu,et al.  Computation and Continuation of Homoclinic and Heteroclinic Orbits with Arclength Parameterization , 1997, SIAM J. Sci. Comput..

[17]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[18]  Mark J. Friedman,et al.  Numerical computation of heteroclinic orbits , 1989 .

[19]  L. Dieci,et al.  Computation of a few Lyapunov exponents for continuous and discrete dynamical systems , 1995 .

[20]  B. Anderson,et al.  Linear Optimal Control , 1971 .

[21]  W. Beyn.  Numerical methods for dynamical systems , 1991 .

[22]  Luca Dieci,et al.  Computation of invariant tori by the method of characteristics , 1995 .

[23]  Gert Vegter,et al.  On the computation of normally hyperbolic invariant manifolds , 1996 .

[24]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[25]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[26]  D. Higham.  Time-stepping and preserving orthonormality , 1997 .

[27]  J. Greene,et al.  The calculation of Lyapunov spectra , 1987 .

[28]  Luca Dieci,et al.  Computation of orthonormal factors for fundamental solution matrices , 1999, Numerische Mathematik.

[29]  Luca Dieci,et al.  Continuous Orthonormalization for Linear Two-Point Boundary Value Problems Revisited , 2000 .

[30]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Connecting Orbits in Dynamical Systems , 1990 .

[31]  A. Bunse-Gerstner,et al.  Numerical computation of an analytic singular value decomposition of a matrix valued function , 1991 .

[32]  Wolf-Jürgen Beyn,et al.  On well-posed problems for connecting orbits in dynamical systems , 1994 .

[33]  Peter Kunkel,et al.  Continuation and collocation for parameter-dependent boundary value problems , 1989 .

[34]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[35]  G. Vegter,et al.  Algorithms for computing normally hyperbolic invariant manifolds , 1997 .

[36]  G. Moore,et al.  Computation and Parametrisation of Invariant Curves and Tori , 1996 .

[37]  Uri M. Ascher,et al.  Sequential Regularization Methods for Nonlinear Higher-Index DAEs , 1997, SIAM J. Sci. Comput..

[38]  Li︠u︡dmila I︠A︡kovlevna Adrianova.  Introduction to linear systems of differential equations , 1995 .

[39]  Danny C. Sorensen,et al.  A note on the computation of an orthonormal basis for the null space of a matrix , 1982, Math. Program..

[40]  R. Russell,et al.  On the Compuation of Lyapunov Exponents for Continuous Dynamical Systems , 1997 .