zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 third-order modification of Newton’s method for multiple roots. (English) Zbl 1162.65342
Summary: We present a new third-order modification of Newton’s method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots.
65H05Single nonlinear equations (numerical methods)
[1]Schröder, E.: Über unendlich viele algorithmen zur auflösung der gleichungen, Math. ann. 2, 317-365 (1870) · Zbl 02.0042.02
[2]Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204
[3]Hansen, E.; Patrick, M.: A family of root finding methods, Numer. math. 27, 257-269 (1977) · Zbl 0361.65041 · doi:10.1007/BF01396176
[4]Victory, H. D.; Neta, B.: A higher order method for multiple zeros of nonlinear functions, Int. J. Comput. math. 12, 329-335 (1983) · Zbl 0499.65026 · doi:10.1080/00207168208803346
[5]Dong, C.: A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation, Math. numer. Sinica 11, 445-450 (1982) · Zbl 0511.65030
[6]Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations, Int. J. Comput. math. 21, 363-367 (1987) · Zbl 0656.65050 · doi:10.1080/00207168708803576
[7]Neta, B.; Johnson, A. N.: High order nonlinear solver for multiple roots, Comput. math. Appl. 55, 2012-2017 (2008) · Zbl 1142.65044 · doi:10.1016/j.camwa.2007.09.001
[8]B. Neta, Extension of Murakami’s High order nonlinear solver to multiple roots, Int. J. Comput. Math., in press, doi:10.1080/00207160802272263.
[9]Werner, W.: Iterationsverfahren höherer ordnung zur lösung nicht linearer gleichungen, Z. angew. Math. mech. 61, T322-T324 (1981) · Zbl 0494.65024
[10]B. Neta, Numerical Methods for the Solution of Equations, Net-A-Sof, California, 1983.
[11]Halley, E.: A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Phil. trans. Roy. soc. London 18, 136-148 (1694)
[12]King, R. F.: A family of fourth order methods for nonlinear equations, SIAM J. Numer. anal. 10, 876-879 (1973) · Zbl 0266.65040 · doi:10.1137/0710072
[13]Jarratt, P.: Some fourth order multipoint methods for solving equations, Math. comput. 20, 434-437 (1966) · Zbl 0229.65049 · doi:10.2307/2003602
[14]Osada, N.: An optimal multiple root-finding method of order three, J. comput. Appl. math. 51, 131-133 (1994) · Zbl 0814.65045 · doi:10.1016/0377-0427(94)00044-1
[15]Jarratt, P.: Multipoint iterative methods for solving certain equations, Comput. J. 8, 398-400 (1966) · Zbl 0141.13404
[16]Murakami, T.: Some fifth order multipoint iterative formulae for solving equations, J. inform. Process. 1, 138-139 (1978) · Zbl 0394.65015
[17]Chun, C.: A simply constructed third-order modifications of Newton’s method, J. comput. Appl. math. (2007)
[18]Redfern, D.: The Maple handbook, (1994)
[19]Neta, B.: New third order nonlinear solvers for multiple roots, Appl. math. Comput. 202, 162-170 (2008) · Zbl 1151.65041 · doi:10.1016/j.amc.2008.01.031