# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A modified Newton method with cubic convergence: the multivariate case. (English) Zbl 1059.65044
Let $m,n\in\bbfN$, $F: \bbfR^m\to\bbfR^n$ be a sufficiently differentiable mapping, $x_0, x_n\in\bbfR^m$, $F_n:= F(x_n)$, and $F_n':= F'(x_n)$ the Jacobi matrix at $x_n$. The author considers the following multipoint method for the approximate computation of a zero of $F$: $$\text{For }n=1,2,\dots: F_n+ F_n'c_n= 0\Rightarrow c_n,\ F_n+ F'(x_n+\tfrac12 c_n) d_n= 0\Rightarrow d_n,\ x_{n+1}:= x_n+ d_n.\tag +$$ He shows that (+) under appropiate conditions converges locally with order three to a simple zero of $F$. Moreover, he uses two nontrivial examples to compare the computational results of (+) with those of Newton’s method but not, e.g. with the respective results of a similar multipoint method of -- under appropiate assumptions -- order three which differs from (+) in the second equation which is replaced by $F(x_n+ c_n)+ F_n'c_n= 0$ (and then $x_{n+1}:= x_n+ c_n+ d_n)$. In this case, the two systems of linear equations in question to be solved in each iteration step have the same coefficient matrix, respectively [Comp., e.g., {\it W. E. Bosarge, jun.} and {\it P. L. Falb}, J. Optimization Theory Appl. 4, 155--166 (1969; Zbl 0172.18703)]. Finally, the author shortly discusses how certain numerical difficulties -- e.g., if the Jadobi matrix at the zero of $F$ is singular -- could be dealt with, and under what conditions (+) may converge globally.

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods)
Maple
Full Text:
##### References:
 [1] Char, B. W.; Geddes, K. O.; Gonnet, G. H.; Leong, B. L.; Monagan, M. B.; Watt, S. M.: Maple V language reference manual. (1991) [2] Char, B. W.; Geddes, K. O.; Gonnet, G. H.; Leong, B. L.; Monagan, M. B.; Watt, S. M.: Maple V library reference manual. (1991) [3] D. Dent, M. Paprzycki, A. Kucaba-Pietal, Performance of solvers for systems of nonlinear equations, in: Proceedings of the 15th Annual Conference on Applied Mathematics, University of Central Oklahoma, Edmond, OK, 1999, pp. 67--77. · Zbl 0943.65058 [4] Dent, D.; Paprzycki, M.; Kucaba-Pietal, A.: Studying the numerical properties of solvers for systems of nonlinear equations. Proceedings of the ninth colloquium on difference equations, 113-118 (1999) · Zbl 0943.65058 [5] D. Dent, M. Paprzycki, A. Kucaba-Pietal, Testing convergence of nonlinear system solvers, in: Proceedings of the First Southern Symposium on Computing, The University of Southern Mississippi, December 4--5, 1998, 1999. http://pax.st.usm.edu/cmi/fscc98_html/processed/. [6] Dent, D.; Paprzycki, M.; Kucaba-Pietal, A.: Recent advances in solvers for nonlinear algebraic equations. Comput. assist. Mech. eng. Sci. (CAMES) 7, 493-505 (2000) · Zbl 0974.65051 [7] Dent, D.; Paprzycki, M.; Kucaba-Pietal, A.: Studying the performance nonlinear systems solvers applied to the random vibration test. Lecture notes in computer science 2179, 471-478 (2001) · Zbl 1031.65068 [8] D. Dent, M. Paprzycki, A. Kucaba-Pietal, Comparing solvers for large systems of nonlinear algebraic equations, in: Proceedings of the Southern Conference on Computing, The University of Southern Mississippi, October 26--28, 2000, 2002. http://www.sc.usm.edu/conferences/scc2/papers/Dent-etal.ps. · Zbl 0974.65051 [9] Deuflhard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. math. 22, 289-315 (1974) · Zbl 0313.65070 [10] Deuflhard, P.: A relaxation strategy for the modified Newton method. Lecture notes in mathematics 477, 59-73 (1975) [11] Homeier, H. H. H: A modified Newton method for root finding with cubic convergence. J. comput. Appl. math. 157, 227-230 (2003) · Zbl 1070.65541 [12] Meißner, H.; Paldus, J.: Direct iterative solution of the generalized Bloch equation. II. A general formalism for many-electron systems. J. chem. Phys. 113, No. 7, 2594-2611 (2000) [13] Meißner, H.; Paldus, J.: Direct iterative solution of the generalized Bloch equation. III. application to H2-cluster models. J. chem. Phys. 113, No. 7, 2612-2621 (2000) [14] Meißner, H.; Paldus, J.: Direct iterative solution of the generalized Bloch equation. IV. application to H2 lih, beh, and CH2. J. chem. Phys. 113, No. 7, 2622-2637 (2000) [15] U. Nowak, L. Weimann, GIANT--a software package for the numerical solution of very large systems of highly nonlinear equations, Technical Report TR 90-11, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, 1990. available from the ZIB ftp-server. [16] U. Nowak, L. Weimann, A family of Newton codes for systems of highly nonlinear equations, Technical Report TR 91-10, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, 1991. available from the ZIB ftp-server. [17] Paldus, J.: Coupled cluster theory. Methods in computational molecular physics, 99-194 (1992) [18] Powell, M. J. D: A hybrid method for nonlinear equations. Numerical methods for nonlinear algebraic equations, 87-114 (1970) [19] Rheinboldt, W. C.: Methods for solving system of nonlinear equations. (1998) · Zbl 0906.65051