The convergence of an algorithm for solving sparse nonlinear systems
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- by C. G. Broyden PDF
- Math. Comp. 25 (1971), 285-294 Request permission
Abstract:
A new algorithm for solving systems of nonlinear equations where the Jacobian is known to be sparse is shown to converge locally if a sufficiently good initial estimate of the solution is available and if the Jacobian satisfies a Lipschitz condition. The results of numerical experiments are quoted in which systems of up to 600 equations have been solved by the method.References
- C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 (1965), 577–593. MR 198670, DOI 10.1090/S0025-5718-1965-0198670-6
- C. G. Broyden, A new method of solving nonlinear simultaneous equations, Comput. J. 12 (1969/70), 94–99. MR 245197, DOI 10.1093/comjnl/12.1.94
- C. G. Broyden, The convergence of single-rank quasi-Newton methods, Math. Comp. 24 (1970), 365–382. MR 279993, DOI 10.1090/S0025-5718-1970-0279993-0 A. Chang, Applications of Sparse Matrix Methods in Electric Power Systems Analysis, Proc. Sympos. on Sparse Matrices and Their Applications (IBM Watson Research Center, 1968), RA 1 #11707, Watson Research Center, Yorktown Heights, New York, 1969.
- D. F. Davidenko, The application of the method of the variation of a parameter to the construction of iteration formulae of heightened precision for the determination of numerical solutions of non-linear integral equations, Dokl. Akad. Nauk SSSR 162 (1965), 499–502 (Russian). MR 0178581
- Allen A. Goldstein, Constructive real analysis, Harper & Row, Publishers, New York-London, 1967. MR 0217616 F. G. Gustavson, W. Liniger & R. Willoughby, "Symbolic generation of an optimal Crout algorithm for sparse systems of linear equations," J. Assoc. Comput. Mach., v. 17, 1970, pp. 87-109. R. P. Tewarson, The Gaussian Elimination and Sparse Systems, Proc. Sympos. on Sparse Matrices and Their Applications (IBM Watson Research Center, 1968), Watson Research Center, Yorktown Heights, New York, 1968. W. F. Tinney & C. E. Hart, "Power flow solutions by Newton’s method," IEEE Trans., v. PAS-86, 1967, pp. 1449-1460.
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
- John M. Bennett, Triangular factors of modified matrices, Numer. Math. 7 (1965), 217–221. MR 177503, DOI 10.1007/BF01436076
- J. E. Dennis Jr., On the convergence of Newton-like methods, Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ. Essex, Colchester, 1969) Gordon and Breach, London, 1970, pp. 163–181. MR 0353662
- L. V. Kantorovič and G. P. Akilov, Funktsional′nyĭ analiz v normirovannykh prostranstvakh, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959 (Russian). MR 0119071
- L. K. Schubert, Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian, Math. Comp. 24 (1970), 27–30. MR 258276, DOI 10.1090/S0025-5718-1970-0258276-9
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Math. Comp. 25 (1971), 285-294
- MSC: Primary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-1971-0297122-5
- MathSciNet review: 0297122