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Difference between revisions of "Newton method"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,  "Functional analysis and applied mathematics"  ''Nat. Bur. Sci. Rep.'' , '''1509'''  (1952)  ''Uspekhi Mat. Nauk'' , '''3''' :  6  (1948)  pp. 89–185</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Collatz,  "Funktionalanalysis und numerische Mathematik" , Springer  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Krasnosel'skii,  G.M. Vainikko,  P.P. Zabreiko,  et al.,  "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,  "Functional analysis and applied mathematics"  ''Nat. Bur. Sci. Rep.'' , '''1509'''  (1952)  ''Uspekhi Mat. Nauk'' , '''3''' :  6  (1948)  pp. 89–185 {{MR|0053389}} {{ZBL|0034.21203}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Collatz,  "Funktionalanalysis und numerische Mathematik" , Springer  (1964) {{MR|0165651}} {{ZBL|0139.09802}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Krasnosel'skii,  G.M. Vainikko,  P.P. Zabreiko,  et al.,  "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian) {{MR|385655}} {{ZBL|0231.41024}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian) {{MR|0362811}} {{ZBL|0524.65001}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Lax,  S. Burstein,  A. Lax,  "Calculus with applications and computing" , New York Univ. Inst. Math. Mech.  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E., jr. Dennis,  R. Schnable,  "Least change secant updates for quasi-Newton methods"  ''SIAM Review'' , '''21'''  (1979)  pp. 443–459</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M. Ortega,  W.C. Rheinboldt,  "Iterative solution of nonlinear equations" , Acad. Press  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , Dover, reprint  (1987)  pp. Chapt. 8</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Lax,  S. Burstein,  A. Lax,  "Calculus with applications and computing" , New York Univ. Inst. Math. Mech.  (1972) {{MR|0354951}} {{ZBL|0264.26002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E., jr. Dennis,  R. Schnable,  "Least change secant updates for quasi-Newton methods"  ''SIAM Review'' , '''21'''  (1979)  pp. 443–459 {{MR|0545880}} {{ZBL|0424.65020}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M. Ortega,  W.C. Rheinboldt,  "Iterative solution of nonlinear equations" , Acad. Press  (1970) {{MR|0273810}} {{ZBL|0241.65046}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , Dover, reprint  (1987)  pp. Chapt. 8 {{MR|0895822}} {{ZBL|0641.65001}} </TD></TR></table>

Revision as of 12:12, 27 September 2012

method of tangents

A method for the approximation of the location of the roots of a real equation

(1)

where is a differentiable function. The successive approximations of Newton's method are computed by the formulas

(2)

If is twice continuously differentiable, is a simple root of (1) and the initial approximation lies sufficiently close to , then Newton's method has quadratic convergence, that is,

where is a constant depending only on and the initial approximation .

Frequently, for the solution of (1) one applies instead of (2) the so-called modified Newton method:

(3)

Under the same assumptions under which Newton's method has quadratic convergence, the method (3) has linear convergence, that is, it converges with the rate of a geometric progression with denominator less than 1.

In connection with solving a non-linear operator equation with an operator , where and are Banach spaces, a generalization of (2) is the Newton–Kantorovich method. Its formulas are of the form

where is the Fréchet derivative of at , which is an invertible operator acting from to . Under special assumptions the Newton–Kantorovich method has quadratic convergence, and the corresponding modified method has linear convergence (cf. also Kantorovich process).

I. Newton worked out his method in 1669.

References

[1] L.V. Kantorovich, "Functional analysis and applied mathematics" Nat. Bur. Sci. Rep. , 1509 (1952) Uspekhi Mat. Nauk , 3 : 6 (1948) pp. 89–185 MR0053389 Zbl 0034.21203
[2] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian)
[3] L. Collatz, "Funktionalanalysis und numerische Mathematik" , Springer (1964) MR0165651 Zbl 0139.09802
[4] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian) MR385655 Zbl 0231.41024
[5] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) MR0362811 Zbl 0524.65001


Comments

The Newton method is also known as the Newton–Raphson method, cf., e.g., [a4], Sect. (10.11) for single equations and Sect. (10.13) for systems of equations.

References

[a1] P. Lax, S. Burstein, A. Lax, "Calculus with applications and computing" , New York Univ. Inst. Math. Mech. (1972) MR0354951 Zbl 0264.26002
[a2] J.E., jr. Dennis, R. Schnable, "Least change secant updates for quasi-Newton methods" SIAM Review , 21 (1979) pp. 443–459 MR0545880 Zbl 0424.65020
[a3] J.M. Ortega, W.C. Rheinboldt, "Iterative solution of nonlinear equations" , Acad. Press (1970) MR0273810 Zbl 0241.65046
[a4] F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. Chapt. 8 MR0895822 Zbl 0641.65001
How to Cite This Entry:
Newton method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton_method&oldid=28253
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article