
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
Monica Bianchini, Stefano Fanelli, Marco Gori, "Optimal Algorithms for WellConditioned Nonlinear Systems of Equations," IEEE Transactions on Computers, vol. 50, no. 7, pp. 689698, July, 2001.  
BibTex  x  
@article{ 10.1109/12.936235, author = {Monica Bianchini and Stefano Fanelli and Marco Gori}, title = {Optimal Algorithms for WellConditioned Nonlinear Systems of Equations}, journal ={IEEE Transactions on Computers}, volume = {50}, number = {7}, issn = {00189340}, year = {2001}, pages = {689698}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.936235}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  Optimal Algorithms for WellConditioned Nonlinear Systems of Equations IS  7 SN  00189340 SP689 EP698 EPD  689698 A1  Monica Bianchini, A1  Stefano Fanelli, A1  Marco Gori, PY  2001 KW  Computational complexity KW  nonlinear systems of equations KW  parallel processing KW  terminal attraction dynamics. VL  50 JA  IEEE Transactions on Computers ER   
Abstract—We propose solving nonlinear systems of equations by function optimization and we give an optimal algorithm which relies on a special canonical form of gradient descent. The algorithm can be applied under certain assumptions on the function to be optimized, that is, an upper bound must exist for the norm of the Hessian, whereas the norm of the gradient must be lower bounded. Due to its intrinsic structure, the algorithm looks particularly appealing for a parallel implementation. As a particular case, more specific results are given for linear systems. We prove that reaching a solution with a degree of precision
[1] J.J. Moré and M.Y. Cosnard, “Numerical Solution of NonlinearEquations,” ACM Trans. Math. Software, vol. 5, no. 1, pp. 6485, 1979.
[2] J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, 1996.
[3] A. Eiger, K. Sikorski, and F. Stenger, “A Bisection Method for Systems of Nonlinear Equations,” ACM Trans. Math. Software, vol. 10, no. 4, pp. 367377, 1984.
[4] A.A.M. Cuyt, “Computational Implementation of the Multivariate Halley Method for Solving Nonlinear Systems of Equations,” ACM Trans. Math. Software, vol. 11, no. 1, pp. 2036, 1985.
[5] P. Deuflhard, Newton Techniques for Highly Nonlinear Problems—Theory and Algorithms. Academic Press, 1996.
[6] J.J. Moré, B.S. Garbow, and K.E. Hillstrom, “User's Guide for MINPACK1,” Technical Report ANL8074, Argonne Nat'l Laboratory, Argonne, Ill., 1980.
[7] L.T. Watson, S.C. Billups, and A.P. Morgan, “Algorithm 652: HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms,” ACM Trans. Math. Software, vol. 13, pp. 281310, 1987.
[8] F. Zirilli, “The Solutions of Nonlinear Systems of Equations by Second Order Systems of Ordinary Differential Equations and Linearly Implicit A–Stable Techniques,” SIAM J. Numerical Analysis, vol. 19, no. 4, pp. 800815, 1982.
[9] H. Jarausch and W. Mackens, “Solving Large Nonlinear Systems of Equations by an Adaptive Condensation Process,” Numerical Math., vol. 50, no. 6, pp. 633653, 1987.
[10] J. Abaffy, C.G. Broyden, and E. Spedicato, “A Class of Direct Methods for Linear Systems,” Numerical Math., vol. 45, no. 3, pp. 361376, 1984.
[11] J. Abaffy, A. Galántai, and E. Spedicato, “The Local Convergence of ABS Methods for Nonlinear Algebraic Equations,” Numerical Math., vol. 51, no. 4, pp. 429439, 1987.
[12] R.P. Brent, “Some Efficient Algorithms for Solving Systems of Nonlinear Equations,” SIAM J. Numerical Analysis, vol. 10, no. 2, pp. 327344, 1973.
[13] K.M. Brown, “A Quadratically Convergent NewtonLike Method Based upon GaussianElimination,” SIAM J. Numerical Analysis, vol. 6, no. 4, pp. 560569, 1969.
[14] J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, Inc., San Diego, CA, 1970.
[15] O. Axelsson and A.T. Chronopoulos, “On Nonlinear Generalized Conjugate Gradient Methods,” Numerical Math., vol. 69, no. 1, pp. 115, 1994.
[16] Y. Saad and M.H. Schultz, "GMRES: A Generalized Minimal Residual Algorithm for Solving Nnonsymmetric Linear Systems," SIAM J. Scientific and Statistical Computing, Vol. 7, No. 3, 1986, pp. 856869.
[17] D.R. Fokkema, G.L.G. Sleijpen, and H.A. Van der Vorst, “Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations,” SIAM J. Scientific Computing, vol. 19, no. 2, pp. 657674, 1998.
[18] S.Y. Berkovich, “An Overlaying Technique for Solving Linear Equations in RealTime Computing,” IEEE Trans. Computers, vol. 42, no. 5, pp. 513517, 1993.
[19] R.W. Freund and N.M. Nachtigal, “QMR: A QuasiMinimal Residual Method for NonHermitian Linear Systems,” Numerical Math., vol. 60, pp. 315339, 1991.
[20] R.W. Freund, “A TransposeFree QuasiMinimal Residual Algorithm for NonHermitian Linear Systems,” SIAM J. Scientific Computing, vol. 14, no. 2, pp. 470482, 1993.
[21] M. Zak, “Terminal Attractors for Addressable Memory in Neural Networks,” Physics Letters A, vol. 133, no. 1.2, pp. 1822, 1988.
[22] M. Zak, “Terminal Attractors in Neural Networks,” Neural Networks, vol. 2, no. 4, pp. 259274, 1989.
[23] S. Wang and C.H. Hsu, “Terminal Attractor Learning Algorithms for Backpropagation Neural Networks,” Proc. Int'l Joint Conf. Neural Networks, pp. 183189, Nov. 1991.
[24] J.J. Moré, B.S. Garbow, and K.E. Hillstrom, “Testing Unconstrained Optimization Software,” ACM Trans. Math. Software, vol. 7, no. 1, pp. 1741, 1981.
[25] V. Kumar, A. Grama, A. Gupta, and G. Karypis, Introduction to Parallel Computing: Design and Analysis of Algorithms. Benjamin Cummings, 1994.
[26] C. Aykanat, F. Ozguner, F. Ercal, and P. Sadayappan, “Iterative Algorithms for Solution of Large Sparse Systems of Linear Equations on Hypercubes,” IEEE Trans. Computers, vol. 37, no. 12, pp. 1,554–1,567, Dec. 1988.
[27] D.H. Lawrie and A.H. Sameh, “The Computation and Communication Complexity of a Parallel Banded System Solver,” ACM Trans. Math. Software, vol. 10, pp. 185195, 1984.
[28] E. Dekker and L. Dekker, “Parallel Minimal Norm Method for Tridiagonal Linear Systems,” IEEE Trans. Computers, vol. 44, no. 7, pp. 942946, July 1995.
[29] A. RoyChowdhury, N. Bellas, and P. Banerjee, AlgorithmBased Error Detection Schemes for Iterative Solution of Partial Differential Equations IEEE Trans. Computers, vol. 45, no. 4, pp. 394407, Apr. 1996.
[30] A.K. Cline, C.B. Moler, G.W. Stewart, and J.H. Wilkinson, “An Estimate for the Condition Number of a Matrix,” SIAM J. Numerical Analysis, vol. 16, no. 2, pp. 368375, 1979.
[31] D.P. O'Leary, “Estimating Matrix Condition Numbers,” SIAM J. Scientific and Statistical Computing, vol. 1, pp. 205209, 1980.
[32] A.K. Cline and R.K. Rew, “A Set of CounterExamples to Three Condition Number Estimators,” SIAM J. Scientific and Statistical Computing, vol. 4, no. 4, pp. 602611, 1983.
[33] W.W. Hager, “Condition Estimates,” SIAM J. Scientific and Statistical Computing, vol. 5, no. 2, pp. 311316, 1984.
[34] “MINPACK Documentation,” Applied Math. Division, Argonne Nat'l Laboratory, Argonne III.,http://www.nctlib.org/minpackreadme, 2001.
[35] F.P. Preparata and M.I. Shamos, Computational Geometry. SpringerVerlag, 1985.
[36] G.H. Golub and J.M. Ortega, Scientific Computing and Differential Equations: An Introduction to Numerical Methods. Academic Press, 1992.