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)
Derivative free algorithm for solving nonlinear equations. (English) Zbl 1232.65073
The authors develop a simple yet practical approach for constructing derivative-free iterative methods for nonlinear equations. The approach starts from the classical Newton method and replaces the first-order derivative by finite difference approximation depending on a parameter.By appropriately selecting the parameter, methods with desired convergence orders are generated.
65H05Single nonlinear equations (numerical methods)
[1]Sharma JR (2005) A composite third order Newton–Steffensen method for solving nonlinear equations. Appl Math Comput 169(1): 242–246 · Zbl 1084.65054 · doi:10.1016/j.amc.2004.10.040
[2]Argyros IK (2007) Computational theory of iterative methods, series: studies in computational mathematics. In: Chui CK, Wuytack L (eds) Elsevier, New York, vol 15
[3]Li X, Mu C, Ma J, Wang C (2009) Sixteenth order method for nonlinear equations. Appl Math Comput 215(10): 3769–4054
[4]Neta B (1981) On a family of multipoint methods for non-linear equations. Int J Comput Math 9: 353–361 · Zbl 0466.65027 · doi:10.1080/00207168108803257
[5]King RF (1973) A family of fourth order methods for nonlinear equations. SIAM J Numer Anal 10(5): 876–879 · Zbl 0266.65040 · doi:10.1137/0710072
[6]Traub JF (1977) Iterative methods for the solution of equations. Chelsea Publishing Company, New York
[7]Argyros IK, Chen D, Qian Q (1994) The Jarrat method in Banach space setting. J Comput Appl Math 51: 103–106 · Zbl 0809.65054 · doi:10.1016/0377-0427(94)90093-0
[8]Chun C (2006) Construction of Newton-like iteration methods for solving nonlinear equations. Numer Math 104(3): 297–315 · Zbl 1126.65042 · doi:10.1007/s00211-006-0025-2
[9]Ren H, Wu Q, Bi W (2009) New variants of Jarratt’s method with sixth-order convergence. Numer Algorithms 52: 585–603 · Zbl 1187.65052 · doi:10.1007/s11075-009-9302-3
[10]Wang X, Kou J, Li Y (2008) A variant of Jarratt method with sixth-order convergence. Appl Math Comput 204: 14–19 · Zbl 1168.65346 · doi:10.1016/j.amc.2008.05.112
[11]Sharma JR, Guha RK (2007) A family of modified Ostrowski methods with accelerated sixth order convergence. Appl Math Comput 190: 111–115 · Zbl 1126.65046 · doi:10.1016/j.amc.2007.01.009
[12]Chun C, Ham Y (2003) Some sixth-order variants of Ostrowski root-finding methods. Appl Math Comput 193: 389–394 · Zbl 1193.65055 · doi:10.1016/j.amc.2007.03.074
[13]Bi W, Ren H, Wu Q (2009) Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J Comput Appl Math. 225: 105–112 · Zbl 1161.65039 · doi:10.1016/j.cam.2008.07.004
[14]ARPREC, C++/Fortran-90 arbitrary precision package. http://crd.lbl.gov/dhbailey/mpdist/
[15]Hernández MA (2000) Second-derivative-free variant of the Chebyshev method for nonlinear equations. J Optim Theory Appl 104(3): 501–515 · Zbl 1034.65039 · doi:10.1023/A:1004618223538
[16]Hernández MA (2001) Chebyshev’s approximation algorithms, applications. Comput Math Appl 41: 433–445 · Zbl 0985.65058 · doi:10.1016/S0898-1221(00)00286-8
[17]Kou J, Li Y, Wang X (2006) A uniparametric Chebyshev-type method free from second derivatives. Appl Math Comput 179: 296–300 · Zbl 1102.65055 · doi:10.1016/j.amc.2005.11.110
[18]Wu QB, Zhao YQ (2006) The convergence theorem for a family deformed Chebyshev method in Banach space. Appl Math Comput 182: 1369–1376 · Zbl 1151.65051 · doi:10.1016/j.amc.2006.05.022
[19]Frontini M, Sormani E (2004) Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl Math Comput 149: 771–782 · Zbl 1050.65055 · doi:10.1016/S0096-3003(03)00178-4
[20]Chun C (2007) Some second-derivative-free variants of Chebyshev–Halley methods. Appl Math Comput 191(2): 410–414 · Zbl 1193.65054 · doi:10.1016/j.amc.2007.02.105
[21]Kou J, Li Y, Wang X (2006) Modified Halleys method free from second derivative. Appl Math Comput 183(1): 704–708 · Zbl 1109.65047 · doi:10.1016/j.amc.2006.05.097
[22]Yueqing Z, Wu Q (2008) Newton-Kantorovich theorem for a family of modified Halleys methodnext term under Hlder continuity conditions in Banach space. Appl Math Comput 202(1): 243–251 · Zbl 1148.65036 · doi:10.1016/j.amc.2008.02.004
[23]Noor MA, Khana WA, Hussaina A (2007) A new modified Halley method without second derivatives for nonlinear equation. Appl Math Comput 189(2): 1268–1273 · Zbl 1243.65053 · doi:10.1016/j.amc.2006.12.011
[24]Osada N (2008) Chebyshev-Halley methods for analytic functions. J Comput Appl Math 216(2): 585–599 · Zbl 1146.65043 · doi:10.1016/j.cam.2007.06.020
[25]Ezquerro JA, Hernándeza MA (2007) Halley’s method for operators with unbounded second derivative. Appl Numer Math 57(3): 354–360 · Zbl 1252.65098 · doi:10.1016/j.apnum.2006.05.001
[26]Xiaojian Z (2008) Modified Chebyshev–Halley methods term free from second derivative. Appl Math Comput 203(2): 824–827 · Zbl 1157.65373 · doi:10.1016/j.amc.2008.05.092
[27]Kanwar V, Tomar SK (2007) Modified families of Newton, Halley and Chebyshev methods. Appl Math Comput 192(1): 20–26 · Zbl 1193.65065 · doi:10.1016/j.amc.2007.02.119
[28]Xu X, Ling Y (2009) Semilocal convergence for Halley’s method under weak Lipschitz condition. Appl Math Comput 215(8): 3057–3067 · Zbl 1187.65059 · doi:10.1016/j.amc.2009.09.055
[29]Noor KI, Noor MA (2007) Predictor-corrector Halley method for nonlinear equations. Appl Math Comput 188(2): 1587–1591 · Zbl 1119.65038 · doi:10.1016/j.amc.2006.11.023
[30]Li S, Li H, Cheng L (2009) Some second-derivative-free variants of Halleys method for multiple roots. Appl Math Comput 215(6): 2192–2198 · Zbl 1181.65070 · doi:10.1016/j.amc.2009.08.010
[31]Ezquerro JA, Hernándeza MA (2004) On Halley-type iterations with free second derivative. J Comput Appl Math 170(2): 455–459 · Zbl 1053.65042 · doi:10.1016/j.cam.2004.02.020
[32]Argyros IK (1997) The super-Halley method using divided differences. Appl Math Lett 10(4): 91–95 · Zbl 0883.65048 · doi:10.1016/S0893-9659(97)00065-7
[33]Ye X, Li C (2006) Convergence of the family of the deformed Euler–Halley iterations under the Hlder condition of the second derivative. J Comput Appl Math 194(2): 294–308 · Zbl 1100.47057 · doi:10.1016/j.cam.2005.07.019
[34]Rafiq A, Awais M, Zafar F (2007) Modified efficient variant of super-Halley method. Appl Math Comput 189(2): 2004–2010 · Zbl 1122.65345 · doi:10.1016/j.amc.2006.12.086
[35]Reeves R (1991) A note on Halley’s method. Comput Graph 15(1): 89–90 · doi:10.1016/0097-8493(91)90034-F
[36]Gutiérrez JM, Hernándeza MA (2001) An acceleration of Newton’s method: super-Halley method. Appl Math Comput 117(2–3): 223–239 · Zbl 1023.65051 · doi:10.1016/S0096-3003(99)00175-7
[37]Ezquerro JA, Hernándeza MA (2000) A modification of the super-Halley method under mild differentiability conditions. J Comput Appl Math 114(2): 405–409 · Zbl 0959.65074 · doi:10.1016/S0377-0427(99)00348-9
[38]Amat S, Bermúdez C, Busquier S, Mestiri D (2009) A family of Halley’s Chebyshev iterative schemes for non-Frchet differentiable operators. J Comput Appl Math 228(1): 486–493 · Zbl 1173.65036 · doi:10.1016/j.cam.2008.09.005
[39]Amat S, Busquier S, Gutiérrez JM (2003) Geometric constructions of iterative functions to solve nonlinear equations. J Comput Appl Math 157(1): 197–205 · Zbl 1024.65040 · doi:10.1016/S0377-0427(03)00420-5
[40]Grau M, Noguera M (2004) A variant of Cauchy’s method with accelerated fifth-order convergence. Appl Math Lett 17: 509–517 · Zbl 1070.65034 · doi:10.1016/S0893-9659(04)90119-X
[41]Kou J (2008) Some variants of Cauchy’s method with accelerated fourth-order convergence. J Comput Appl Math 213(1): 71–78 · Zbl 1135.65024 · doi:10.1016/j.cam.2007.01.031