On solving a special class of weakly nonlinear finite-difference systems

References

• Apostol , T. M. 1974 . Mathematical Analysis , 2 , Reading, MA : Addison-Wesley Publ .
   Google Scholar
• Argyros , I. K. 2005 . Concerning the ‘Terra Incognita’ between convergence regions of two Newton methods . Nonlinear Anal. , 62 : 179 – 194 .
   Web of Science ®Google Scholar
• Aris , R. 1975 . The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts , Vol. I, II , Oxford : Clarendon Press .
   Google Scholar
• Armijo , L. 1966 . Minimsation of functions having Lipschitz-continuous first partial derivatives . Pac. J. Math. , 16 : 1 – 3 .
   Web of Science ®Google Scholar
• Chen , J. and Li , W. 2006 . Convergence behaviour of inexact Newton methods under weak Lipschitz condition . J. Comput. Appl. Math. , 191 : 143 – 164 .
   Google Scholar
• Dembo , R. S. , Eisenstat , S. C. and Steihaug , T. 1982 . Inexact Newton methods . SIAM J. Numer. Anal. , 19 : 400 – 408 .
   Web of Science ®Google Scholar
• Den Heijer , C. and Rheinboldt , W. C. 1981 . On steplength algorithms for a class of continuation methods . SIAM J. Numer. Anal. , 18 : 925 – 948 .
   Web of Science ®Google Scholar
• Dennis , J. E. Jr. and Schnabel , R. B. 1983 . Numerical Methods for Unconstrained Optimisation and Nonlinear Equations , Englewwod Cliffs, NJ : Prentice-Hall .
   Google Scholar
• Deuflhard , P. 2004 . Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms , Berlin : Springer-Verlag .
   Google Scholar
• Deuflhard , P. and Heindl , G. 1979 . Affine invariant convergence theorems for Newton's method and extensions to related methods . SIAM J. Numer. Anal. , 16 : 1 – 10 .
   Web of Science ®Google Scholar
• Deuflhard , P. and Hohmann , A. 2000 . Numerical Analysis in Modern Scientific Computing: An Introduction , 2 , Berlin : Springer-Verlag .
   Google Scholar
• Deuflhard , P. and Potra , F. A. 1992 . Asymptotic mesh independence of Newton–Galerkin methods via a refined Mysovskii theorem . SIAM J. Numer. Anal. , 29 : 1395 – 1412 .
   Web of Science ®Google Scholar
• Durazzi , C. 2000 . On the Newton interior-point method for nonlinear programming problems . J. Optim. Theory Appl. , 104 : 73 – 90 .
   Google Scholar
• Eisenstat , S. C. and Walker , H. F. 1994 . Globally convergent inexact Newton methods . SIAM J. Optim. , 4 : 393 – 422 .
   Web of Science ®Google Scholar
• Eisenstat , S. C. and Walker , H. F. 1996 . Choosing the forcing terms in an inexact Newton method . SIAM J. Sci. Comput. , 17 : 16 – 32 .
   Web of Science ®Google Scholar
• El–Bakry , A. S. 1996 . On the formulation and theory of the Newton interior-point method for nonlinear programming . J. Optim. Theory Appl. , 89 : 507 – 541 .
   Web of Science ®Google Scholar
• Evans , D. J. 1985 . Group explicit iterative methods for solving large linear systems . Int. J. Comput. Math. , 17 : 81 – 108 .
   Web of Science ®Google Scholar
• Evans , D. J. and Yousif , W. S. 1988 . The solution of two-point boundary value problems by the Alternating Group Explicit (AGE) method . SIAM J. Sci. Stat. Comput. , 9 : 474 – 484 .
   Web of Science ®Google Scholar
• Fife , P. C. 1979 . Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes Biomathematics , Vol. 28 , New York : Springer .
   Google Scholar
• Galligani , E. 2003 . The Newton-arithmetic mean method for the solution of systems of nonlinear equations . Appl. Math. Comput. , 134 : 9 – 34 .
   Web of Science ®Google Scholar
• Galligani , E. and Ruggiero , V. 1997 . The two-stage arithmetic mean method . Appl. Math. Comput. , 85 : 245 – 264 .
   Web of Science ®Google Scholar
• Henrici , P. 1962 . Discrete Variable Methods in Ordinary Differential Equations , New York : J. Wiley & Sons .
   Google Scholar
• Isaacson , E. and Keller , H. 1966 . Analysis of Numerical Methods , New York : J. Wiley & Sons .
   Google Scholar
• Kelley , C. T. 1995 . Iterative Methods for Linear and Nonlinear Equations , Philadelphia : SIAM .
   Google Scholar
• Kernevez , J. P. 1980 . Enzyme Mathematics , Amsterdam : North Holland .
   Google Scholar
• Meyer , G. H. 1973 . “ The numerical solution of quasilinear equations ” . In Numerical Solution of Systems of Nonlinear Algebraic Equations , Edited by: Byrne , G. and Hall , C. A. 27 – 62 . New York : Academic Press .
   Google Scholar
• Murray , J. D. 2003 . Mathematical Biology , Vol. I, II , Berlin : Springer-Verlag .
   Google Scholar
• Nocedal , J. and Wright , S. J. 1999 . Numerical Optimisation , New York : Springer-Verlag .
   Google Scholar
• O'Leary , D. P. and White , R. E. 1985 . Multisplitting of matrices and parallel solution of linear systems . SIAM J. Algebr Discret Methods , 6 : 630 – 640 .
   Web of Science ®Google Scholar
• Ortega , J. M. 1972 . Numerical Analysis: A Second Course , New York : Academic Press .
   Google Scholar
• Ortega , J. M. and Rheinboldt , W. C. 1970 . Iterative Solution of Nonlinear Equations in Several Variables , New York : Academic Press .
   Google Scholar
• Peressini , A. L. , Sullivan , F. E. and Uhl , J. J. Jr. 1988 . The Mathematics of Nonlinear Programming , New York : Springer-Verlag .
   Google Scholar
• Ruggiero , V. and Galligani , E. 1992 . A parallel algorithm for solving block tridiagonal linear systems . Comput. Math. Appl. , 24 : 15 – 21 .
   Web of Science ®Google Scholar
• Stoer , J. and Bulirsch , R. 1980 . Introduction to Numerical Analysis , New York : Springer-Verlag .
   Google Scholar
• Svirezhev , Y. M. and Logofet , D. O. 1983 . Stability of Biological Communities , Moscow : MIR .
   Google Scholar
• van Dorsselaer , J. L.M. and Spijker , M. N. 1994 . The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems . IMA J. Numer. Anal. , 14 : 183 – 209 .
   Google Scholar
• Wolfe , P. 1969 . Convergence conditions for ascent methods . SIAM Rev. , 11 : 226 – 235 .
   Web of Science ®Google Scholar
• Wolfe , P. 1971 . Convergence conditions for ascent methods. II: some corrections . SIAM Rev. , 13 : 185 – 188 .
   Web of Science ®Google Scholar
• Ypma , T. J. 1982 . Affine invariant convergence results for Newton's method . BIT , 22 : 108 – 118 .
   Web of Science ®Google Scholar
• Ypma , T. J. 1983 . Local convergence of difference Newton-like methods . Math. Comput. , 41 : 527 – 536 .
   Google Scholar
• Ypma , T. J. 1984 . Local convergence of inexact Newton methods . SIAM J. Numer. Anal. , 21 : 583 – 590 .
   Web of Science ®Google Scholar