The algorithm is second in the class of Householder's methods, after Newton's method. Like the latter, it iteratively produces a sequence of approximations to the root; their rate of convergence to the root is cubic. Multidimensional versions of this method exist.
Halley's method exactly finds the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method or the Secant method which approximate the function linearly, or Muller's method which approximates the function quadratically.[1]
Edmond Halley was an English mathematician who introduced the method now called by his name. Halley's method is a numerical algorithm for solving the nonlinear equation f(x) = 0. In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations:
If f is a three times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood of a, the iterates xn satisfy:
This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence is cubic.[3]
The following alternative formulation shows the similarity between Halley's method and Newton's method. The expression is computed only once, and it is particularly useful when can be simplified:
When the second derivative is very close to zero, the Halley's method iteration is almost the same as the Newton's method iteration.
DerivationEdit
Consider the function
Any root of f which is not a root of its derivative is a root of g; and any root r of g must be a root of f provided the derivative of f at r is not infinite. Applying Newton's method to g gives
with
and the result follows. Notice that if f′ (c) = 0, then one cannot apply this at c because g(c) would be undefined.
Cubic convergenceEdit
Suppose a is a root of f but not of its derivative. And suppose that the third derivative of f exists and is continuous in a neighborhood of a and xn is in that neighborhood. Then Taylor's theorem implies:
and also
where ξ and η are numbers lying between a and xn. Multiply the first equation by and subtract from it the second equation times to give:
Canceling and re-organizing terms yields:
Put the second term on the left side and divide through by
to get:
Thus:
The limit of the coefficient on the right side as xn → a is:
If we take K to be a little larger than the absolute value of this, we can take absolute values of both sides of the formula and replace the absolute value of coefficient by its upper bound near a to get:
^Boyd, J. P. (2013). "Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix". SIAM Review. 55 (2): 375–396. doi:10.1137/110838297.
^Scavo, T. R.; Thoo, J. B. (1995). "On the geometry of Halley's method". American Mathematical Monthly. 102 (5): 417–426. doi:10.2307/2975033. JSTOR 2975033.
^Alefeld, G. (1981). "On the convergence of Halley's method". American Mathematical Monthly. 88 (7): 530–536. doi:10.2307/2321760. JSTOR 2321760.
^Proinov, Petko D.; Ivanov, Stoil I. (2015). "On the convergence of Halley's method for simultaneous computation of polynomial zeros". J. Numer. Math. 23 (4): 379–394. doi:10.1515/jnma-2015-0026. S2CID 10356202.
Newton's method and high order iterations, Pascal Sebah and Xavier Gourdon, 2001 (the site has a link to a Postscript version for better formula display)
October 14, 2023
halley, method, numerical, analysis, root, finding, algorithm, used, functions, real, variable, with, continuous, second, derivative, named, after, inventor, edmond, halley, algorithm, second, class, householder, methods, after, newton, method, like, latter, i. In numerical analysis Halley s method is a root finding algorithm used for functions of one real variable with a continuous second derivative It is named after its inventor Edmond Halley The algorithm is second in the class of Householder s methods after Newton s method Like the latter it iteratively produces a sequence of approximations to the root their rate of convergence to the root is cubic Multidimensional versions of this method exist Halley s method exactly finds the roots of a linear over linear Pade approximation to the function in contrast to Newton s method or the Secant method which approximate the function linearly or Muller s method which approximates the function quadratically 1 Contents 1 Method 2 Derivation 3 Cubic convergence 4 References 5 External linksMethod EditEdmond Halley was an English mathematician who introduced the method now called by his name Halley s method is a numerical algorithm for solving the nonlinear equation f x 0 In this case the function f has to be a function of one real variable The method consists of a sequence of iterations x n 1 x n 2 f x n f x n 2 f x n 2 f x n f x n displaystyle x n 1 x n frac 2f x n f x n 2 f x n 2 f x n f x n nbsp beginning with an initial guess x0 2 If f is a three times continuously differentiable function and a is a zero of f but not of its derivative then in a neighborhood of a the iterates xn satisfy x n 1 a K x n a 3 for some K gt 0 displaystyle x n 1 a leq K cdot x n a 3 text for some K gt 0 nbsp This means that the iterates converge to the zero if the initial guess is sufficiently close and that the convergence is cubic 3 The following alternative formulation shows the similarity between Halley s method and Newton s method The expression f x n f x n displaystyle f x n f x n nbsp is computed only once and it is particularly useful when f x n f x n displaystyle f x n f x n nbsp can be simplified x n 1 x n f x n f x n f x n f x n f x n 2 x n f x n f x n 1 f x n f x n f x n 2 f x n 1 displaystyle x n 1 x n frac f x n f x n frac f x n f x n frac f x n 2 x n frac f x n f x n left 1 frac f x n f x n cdot frac f x n 2f x n right 1 nbsp When the second derivative is very close to zero the Halley s method iteration is almost the same as the Newton s method iteration Derivation EditConsider the function g x f x f x displaystyle g x frac f x sqrt f x nbsp Any root of f which is not a root of its derivative is a root of g and any root r of g must be a root of f provided the derivative of f at r is not infinite Applying Newton s method to g gives x n 1 x n g x n g x n displaystyle x n 1 x n frac g x n g x n nbsp with g x 2 f x 2 f x f x 2 f x f x displaystyle g x frac 2 f x 2 f x f x 2f x sqrt f x nbsp and the result follows Notice that if f c 0 then one cannot apply this at c because g c would be undefined Cubic convergence EditSuppose a is a root of f but not of its derivative And suppose that the third derivative of f exists and is continuous in a neighborhood of a and xn is in that neighborhood Then Taylor s theorem implies 0 f a f x n f x n a x n f x n 2 a x n 2 f 3 6 a x n 3 displaystyle 0 f a f x n f x n a x n frac f x n 2 a x n 2 frac f xi 6 a x n 3 nbsp and also 0 f a f x n f x n a x n f h 2 a x n 2 displaystyle 0 f a f x n f x n a x n frac f eta 2 a x n 2 nbsp where 3 and h are numbers lying between a and xn Multiply the first equation by 2 f x n displaystyle 2f x n nbsp and subtract from it the second equation times f x n a x n displaystyle f x n a x n nbsp to give 0 2 f x n f x n 2 f x n 2 a x n f x n f x n a x n 2 f x n f 3 3 a x n 3 f x n f x n a x n f x n f x n a x n 2 f x n f h 2 a x n 3 displaystyle begin aligned 0 amp 2f x n f x n 2 f x n 2 a x n f x n f x n a x n 2 frac f x n f xi 3 a x n 3 amp qquad f x n f x n a x n f x n f x n a x n 2 frac f x n f eta 2 a x n 3 end aligned nbsp Canceling f x n f x n a x n 2 displaystyle f x n f x n a x n 2 nbsp and re organizing terms yields 0 2 f x n f x n 2 f x n 2 f x n f x n a x n f x n f 3 3 f x n f h 2 a x n 3 displaystyle 0 2f x n f x n left 2 f x n 2 f x n f x n right a x n left frac f x n f xi 3 frac f x n f eta 2 right a x n 3 nbsp Put the second term on the left side and divide through by 2 f x n 2 f x n f x n displaystyle 2 f x n 2 f x n f x n nbsp to get a x n 2 f x n f x n 2 f x n 2 f x n f x n 2 f x n f 3 3 f x n f h 6 2 f x n 2 f x n f x n a x n 3 displaystyle a x n frac 2f x n f x n 2 f x n 2 f x n f x n frac 2f x n f xi 3f x n f eta 6 2 f x n 2 f x n f x n a x n 3 nbsp Thus a x n 1 2 f x n f 3 3 f x n f h 12 f x n 2 6 f x n f x n a x n 3 displaystyle a x n 1 frac 2f x n f xi 3f x n f eta 12 f x n 2 6f x n f x n a x n 3 nbsp The limit of the coefficient on the right side as xn a is 2 f a f a 3 f a f a 12 f a 2 displaystyle frac 2f a f a 3f a f a 12 f a 2 nbsp If we take K to be a little larger than the absolute value of this we can take absolute values of both sides of the formula and replace the absolute value of coefficient by its upper bound near a to get a x n 1 K a x n 3 displaystyle a x n 1 leq K a x n 3 nbsp which is what was to be proved To summarize D x i 1 3 f 2 2 f f 12 f 2 D x i 3 O D x i 4 D x i x i a displaystyle Delta x i 1 frac 3 f 2 2f f 12 f 2 Delta x i 3 O Delta x i 4 qquad Delta x i triangleq x i a nbsp 4 References Edit Boyd J P 2013 Finding the Zeros of a Univariate Equation Proxy Rootfinders Chebyshev Interpolation and the Companion Matrix SIAM Review 55 2 375 396 doi 10 1137 110838297 Scavo T R Thoo J B 1995 On the geometry of Halley s method American Mathematical Monthly 102 5 417 426 doi 10 2307 2975033 JSTOR 2975033 Alefeld G 1981 On the convergence of Halley s method American Mathematical Monthly 88 7 530 536 doi 10 2307 2321760 JSTOR 2321760 Proinov Petko D Ivanov Stoil I 2015 On the convergence of Halley s method for simultaneous computation of polynomial zeros J Numer Math 23 4 379 394 doi 10 1515 jnma 2015 0026 S2CID 10356202 External links EditWeisstein Eric W Halley s method MathWorld Newton s method and high order iterations Pascal Sebah and Xavier Gourdon 2001 the site has a link to a Postscript version for better formula display Retrieved from https en wikipedia org w index php title Halley 27s method amp oldid 1115714627, wikipedia, wiki, book, books, library,
