Discretization of implicit ODEs for singular root-finding problems
Tài liệu tham khảo
Allgower, 1990
Boggs, 1971, The solution of nonlinear systems of equations by A-stable integration techniques, SIAM J. Numer. Anal., 8, 767, 10.1137/0708071
Branin, 1972, Widely convergent methods for finding multiple solutions of simultaneous nonlinear equations, IBM J. Res. Develop., 16, 502, 10.1147/rd.165.0504
Brenan, 1996
Chua, 1989, Impasse points, I. Numerical aspects, Internat. J. Circuit Theory Appl., 17, 213, 10.1002/cta.4490170207
Chua, 1989, Impasse points, II. Analytical aspects, Internat. J. Circuit Theory Appl., 17, 271, 10.1002/cta.4490170303
D.F. Davidenko, On a new method of numerical solution of systems of non-linear equations, Dokl. Akad. Nauk. SSSR 88 1953 601–602 (in Russian). Math. Rev. 14 (1953) 906.
Decker, 1983, Convergence rates for Newton's method at singular points, SIAM J. Numer. Anal., 20, 296, 10.1137/0720020
Decker, 1983, Sublinear convergence of the chord method at singular points, Numer. Math., 42, 147, 10.1007/BF01395307
Decker, 1985, Expanded convergence domains for Newton's method at nearly singular roots, SIAM J. Sci. Statist. Comput., 6, 951, 10.1137/0906064
Decker, 1985, Broyden's method for a class of problems having singular Jacobian at the root, SIAM J. Numer. Anal., 22, 566, 10.1137/0722034
Diener, 1995, Trajectory methods in global optimization, 649
Gantmacher, 1959
Garcı́a, 1980, Relations between several path following algorithms and local and global Newton methods, SIAM Rev., 22, 263, 10.1137/1022055
M.K. Gavurin, Nonlinear functional equations and continuous analogs of iterative methods, Izv. Vyss. Ucebnh. Zaved. Mat. 6 (1958) 18–31 in Russian. Technical Report 68–70, University of Maryland, College Park, 1968.
Griewank, 1980, Starlike domains of convergence for Newton's method at singularities, Numer. Math., 35, 95, 10.1007/BF01396373
Griewank, 1985, On solving nonlinear equations with simple singularities or nearly singular solutions, SIAM Rev., 27, 537, 10.1137/1027141
Hairer, 1996
Incerti, 1979, A new method for solving nonlinear simultaneous equations, SIAM J. Numer. Anal, 16, 779, 10.1137/0716057
Keller, 1978, Global homotopies and Newton methods, 73
Keller, 1981, Geometrically isolated nonisolated solutions and their approximation, SIAM J. Numer. Anal., 18, 822, 10.1137/0718056
Kelley, 1986, A Shamanskii-like acceleration scheme for nonlinear equations at singular roots, Math. Comput., 47, 609
Kelley, 1983, A new acceleration method for Newton's method at singular points, SIAM J. Numer. Anal., 20, 1001, 10.1137/0720070
Kelley, 1993, Inexact Newton methods for singular problems, Optim. Methods Software, 2, 249, 10.1080/10556789308805545
Ortega, 1970
Peitgen, 1988, Global aspects of the continuous and discrete Newton method: a case study, Acta Appl. Math., 13, 123, 10.1007/BF00047504
Rabier, 1989, Implicit differential equations near a singular point, J. Math. Anal. Appl., 144, 425, 10.1016/0022-247X(89)90344-2
Rabier, 1994, On impasse points of quasi-linear differential-algebraic equations, J. Math. Anal. Appl., 181, 429, 10.1006/jmaa.1994.1033
Rabier, 1994, On the computation of impasse points of quasi-linear differential-algebraic equations, Math. Comput., 62, 133
Reddien, 1978, On Newton's method for singular problems, SIAM J. Numer. Anal., 15, 993, 10.1137/0715064
R. Riaza, Linearly implicit differential equations: qualitative theory, computational aspects and applications, Ph.D. Thesis, Universidad Politécnica de Madrid, 2000 (in Spanish).
Riaza, 2000, On singular equilibria of index-1 DAEs, Circuits Systems Signal Process., 19, 131, 10.1007/BF01212467
Riaza, 1999, Adaptive cellular integration of linearly implicit differential equations, J. Comput. Appl. Math., 111, 305, 10.1016/S0377-0427(99)00151-X
Riaza, 1999, Weak singularities and the continuous Newton method, J. Math. Anal. Appl., 236, 438, 10.1006/jmaa.1999.6453
R. Riaza, P.J. Zufiria, Strong singularities and the continuous Newton method, Proceedings of the EQUADIFF’99, Berlin, Germany, Vol. 2, 1999, pp. 925–927.
Riaza, 2001, Stability of singular equilibria in quasilinear implicit differential equations, J. Differential Equations, 171, 24, 10.1006/jdeq.2000.3832
Schropp, 1995, Using dynamical systems methods to solve minimization problems, Appl. Numer. Math., 18, 321, 10.1016/0168-9274(95)00065-3
Smale, 1976, A convergent process of price adjustment and global Newton methods, J. Math. Econom., 3, 107, 10.1016/0304-4068(76)90019-7
H. von Sosen, Folds and bifurcations in the solutions of semi-explicit differential-algebraic equations, Ph.D. Thesis, Part I, California Institute of Technology, 1994.
Sotomayor, 1997, Structurally stable differential systems of the form A(x)x′=F(x), Differential Equations Dynamic Systems, 5, 415
Stuart, 1996
Tanabe, 1979, Continuous Newton-Raphson method for solving an underdetermined system of nonlinear equations, Nonlinear Anal. TMA., 3, 495, 10.1016/0362-546X(79)90064-6
Tanabe, 1985, Global analysis of continuous analogues of the Levenberg-Marquardt and Newton-Raphson methods for solving nonlinear equations, Ann. Inst. Statist. Math. Part B, 37, 189, 10.1007/BF02481091
Venkatasubramanian, 1995, Local bifurcations and feasibility regions in differential-algebraic systems, IEEE Trans. Automat. Control, 40, 1992, 10.1109/9.478226
Watson, 1997, Algorithm 777: HOMPACK90: A suite of Fortran 90 codes for globally convergent homotopy algorithms, ACM Trans. Math. Software, 23, 514, 10.1145/279232.279235
P.J. Zufiria, Global behavior of a class of nonlinear dynamical systems: analytical, computational and control aspects, Ph.D. Thesis, University of Southern California, 1989.
Zufiria, 1990, On an application of dynamical system theory to determine all the zeros of a vector function, J. Math. Anal. Appl., 152, 269, 10.1016/0022-247X(90)90103-M