# 臺灣博碩士論文加值系統

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 我們討論多重網格法應用在延續法中，來解半線性橢圓特徵值問題。在多重網格法的V循環，W循環以及滿近似法中，我們分別使用Lanczos法和MINRES法作為其中的鬆弛法。因為GMRES是MINRES法的一般化，我們可以利用Brown和Walker的結果來解在延續問題中產生的接近奇異地對稱線性系統。我們提供一些MINRES法的誤差界限值，比張和簡提供的更準確一點。對於非線性橢圓特徵值問題的曲線追蹤，我們提供一些多重網格延續演算法。由數值的結果證明我們提供的演算法是有效的，並且可以被輕易的實行，最後我們對所得結果做結論。
 We study multigrid methods in the context of continuation methods for semilinear elliptic eigenvalue problems, where the Lanczos method and its variant MINRES are used as the relaxation scheme for the V-cycle, W-cycle and full multigrid V-cycle schemes, respectively. Since the GMRES method is a generalization of the MINRES algorithm, the results of Brown and Walker \cite{BW} can be exploited to solve nearly singular symmetric linear systems that occur in continuation problems. We give some error bounds for the MINRES algorithm which are more accurate than those given by Chang and Chien \cite{CC2}. Some multigrid-continuation algorithms are proposed for curve-tracking in nonlinear elliptic eigenvalue problems. Our numerical results show the algorithms we propose are efficient and can be easily implemented. Finally, some concluding remarks are given.
 1. Introduction.....2 2. A brief review of the Lanczos type algorithms.....4 3. Solving symmetric (nearly) singular systems by MINRES....9 4. V-cycle, W-cycle and Full multigrid V-cycle methods....12 5. Numerical results....16 6. Conclusions....19
 \bibitem{AG1} E. L. Allgower, and K. Georg, {\it Numerical path following}, Acta Numerica, 2(1997), pp. 1-64.\bibitem{BK} J. H. Bolstad \& H. B. Keller, A multigrid continuation method for elliptic problems with folds, {\it SIAM J. Sci. Stat. Comput.}, 7(1986), pp.1081--1104.\bibitem{Br1} A. Brandt, Multi-level adaptive solutions to boundary value problems, {\it Math. Comput.}, 31(1977), pp.333--390.\bibitem{BHM} W. L. Briggs, Van Emden, Henson \& S. F. McCormic, {\it A Multigrid Tutorial}, Second Ed., SIAM Publications, Philadelphia, 2000.\bibitem{BW} P. N. Brown and H. F. Walker, GMRES on (nearly) singular systems, {\it SIAM J. Matrix Anal. Appl.}, 18(1997), pp.37--51.\bibitem{CK} T. F. Chan \& H. B. Keller, Arc-length continuation and multi-grid techniques for nonlinear elliptic eigenvalue problems, {\it SIAM J. Sci. Stat. Comput.}, 3(1982), pp.173--194.\bibitem{CC2} S.-L. Chang and C.-S. Chien, A multigrid-Lanczos algorithm for the numerical solutions of nonlinear eigenvalue problems, {\it Inter. J. Bifurcation and Chaos}, to appear.\bibitem{CC1} C.-S. Chien and S.-L. Chang, Application of the Lanczos algorithm for solving the linear systems that occur in continuation problems, {\it Numer. Linear Algebra Applic.}, to appear.\bibitem{CWS} C.-S. Chien, Z.-L. Weng and C.-L. Shen, Lanczos type methods for continuation problems, {\it Numer. Linear Algebra with Applic.}, 4(1997), pp.23--41.\bibitem{DIRWW} C. Desa, K. M. Irani, C. J. Ribbens, L. T. Watson and H. F. Walker, Preconditioned iterative methods for homotopy curve tracking, {\it SIAM J. Sci. Stat. Comput.}, 13(1992), pp.30--46.\bibitem{D97} E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede,\& X. J. Wang, {\it AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont)}, User's Guide, (Computer Science Department, Concordia University, Montreal, Canada) 1997.\bibitem{D20} E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, \& X. J. Wang, {\it AUTO2000: Continuation and bifurcation software for ordinary differential equations}, Report, (Applied Mathematics, Caltech, Pasadena) 2000.\bibitem{GV} G. H. Golub and C. F. Van Loan, {\it Matrix Computations}, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.\bibitem{G2} W. J. F. Govaerts, {\it Numerical Methods for Bifurcations of Dynamical Equilibria}, SIAM Publications, Philadelphia, 2000.\bibitem{H} W. Hackbusch, {\it Multigrid Methods and Applications,} Springer-Verlag, Berlin, 1985.\bibitem{IK} E. Isaacson and H. B. Keller, {\it Analysis of Numerical Methods}, John Wiley and Sons, New York, 1966.\bibitem{K} H. B. Keller, {\it Lectures on Numerical Methods in Bifurcation Problems}, Springer-Verlag, Berlin, 1987.\bibitem{L} C. Lanczos, An iteration method for the solution of the eigenvalue problems of linear differentiaal and integral operators, {\it J. Res. Nat. Bur. Stand.}, 45(1950), pp.255--282.\bibitem{MW} H. D. Mittelmann and H. Weber, Multi-grid solution of bifurcation problems, {\it SIAM J. Sci. Stat. Comput.}, 6(1985), pp.49--85.\bibitem{PS} C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, {\it SIAM J. Numer. Anal.}, 12(1975), pp.617--629.\bibitem{P1} B. N. Parlett, {\it The Symmetric Eigenvalue Problems}, Prentice Hall, Englewood Cliffs, NJ, U.S.A., 1980.\bibitem{Saad} Y. Saad, {\it Iterative Methods for Sparse Linear Systems}, PWS Publishing Co., Boston, 1996.\bibitem{Weber} H. Weber, Multigrid bifurcation iteration, {\it SIAM J. Numer. Anal.}, 22(1985), pp.262--279.
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