Ana Sayfa
Matematikçiler
Makaleler
Matematik Seçkileri
Fraktallar
=> Apollonian Gasket
=> Barnsley's Fern
=> Barnsley's Tree
=> Batrachion
=> Blancmange Function
=> Box Fractal
=> Brown Function
=> Cactus Fractal
=> Cantor Dust
=> Cantor Function
=> Cantor Set
=> Cantor Square Fractal
=> Capacity Dimension
=> Carotid-Kundalini Fractal
=> Cesàro Fractal
=> Chaos Game
=> Circles-and-Squares Fractal
=> Coastline Paradox
=> Correlation Exponent
=> Count
=> Cross-Stitch Curve
=> Curlicue Fractal
=> Delannoy Number
=> Dendrite Fractal
=> Devil's Staircase
=> Douady's Rabbit Fractal
=> Dragon Curve
=> Elephant Valley
=> Exterior Snowflake
=> Gosper Island
=> H-Fractal
=> Haferman Carpet
=> Hénon Map
=> Hilbert Curve
=> Householder's Method
=> Ice Fractal
=> Julia Set
=> Koch Antisnowflake
=> Koch Snowflake
=> Lévy Fractal
=> Lévy Tapestry
=> Lindenmayer System
=> Mandelbrot Set
=> Mandelbrot Set Lemniscate
=> Mandelbrot Tree
=> Menger Sponge
=> Minkowski Sausage
=> Mira Fractal
=> Newton's Method
=> Peano Curve
=> Peano-Gosper Curve
=> Pentaflake
=> Plane-Filling Function
=> Pythagoras Tree
=> Randelbrot Set
=> Rep-Tile
=> Reverend Back's Abbey Floor
=> San Marco Fractal
=> Sea Horse Valley
=> Siegel Disk Fractal
=> Sierpiński Arrowhead Curve
=> Sierpiński Carpet
=> Sierpiński Curve
=> Sierpiński Sieve
=> Star Fractal
=> Strange Attractor
=> Tetrix
Paradokslar
Sayılar Teorisi
Ziyaretçi defteri
 

Newton's Method

Newton's method, also called the Newton-Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function f(x) in the vicinity of a suspected root. Newton's method is sometimes also known as Newton's iteration, although in this work the latter term is reserved to the application of Newton's method for computing square roots.

For f(x) a polynomial, Newton's method is essentially the same as Horner's method.

The Taylor series of f(x) about the point x=x_0+epsilon is given by

 f(x_0+epsilon)=f(x_0)+f^'(x_0)epsilon+1/2f^('')(x_0)epsilon^2+....
(1)

Keeping terms only to first order,

 f(x_0+epsilon) approx f(x_0)+f^'(x_0)epsilon.
(2)

This expression can be used to estimate the amount of offset epsilon needed to land closer to the root starting from an initial guess x_0. Setting f(x_0+epsilon)=0 and solving (2) for epsilon=epsilon_0 gives

 epsilon_0=-(f(x_0))/(f^'(x_0)),
(3)

which is the first-order adjustment to the root's position. By letting x_1=x_0+epsilon_0, calculating a new epsilon_1, and so on, the process can be repeated until it converges to a fixed point (which is precisely a root) using

 epsilon_n=-(f(x_n))/(f^'(x_n)).
(4)

Unfortunately, this procedure can be unstable near a horizontal asymptote or a local extremum. However, with a good initial choice of the root's position, the algorithm can be applied iteratively to obtain

 x_(n+1)=x_n-(f(x_n))/(f^'(x_n))
(5)

for n=1, 2, 3, .... An initial point x_0 that provides safe convergence of Newton's method is called an approximate zero.

Newton's method can be implemented in Mathematica as

NewtonsMethodList[f_, {x_, x0_}, n_] :=
NestList[# - Function[x, f][#]/
Derivative[1][Function[x, f]][#]& , x0, n]

The error epsilon_(n+1) after the (n+1)st iteration is given by

epsilon_(n+1) = epsilon_n+(x_(n+1)-x_n)
(6)
= epsilon_n-(f(x_n))/(f^'(x_n)).
(7)

But

f(x_n) = f(x_(n-1))+f^'(x_(n-1))epsilon_n+1/2f^('')(x_(n-1))epsilon_n^2+...
(8)
 
(9)
= f^'(x_(n-1))epsilon_n+1/2f^('')(x_(n-1))epsilon_n^2+...
(10)
f^'(x_n) = f^'(x_(n-1))+f^('')(x)epsilon_n+...,
(11)

so

(f(x_n))/(f^'(x_n)) = (f^'(x_(n-1))epsilon_n+1/2f^('')(x_(n-1))epsilon_n^2+...)/(f^'(x_(n-1))+f^('')(x_(n-1))epsilon_n+...)
(12)
 approx (f^'(x_(n-1))epsilon_n+1/2f^('')(x_(n-1))epsilon_n^2)/(f^'(x_(n-1)))
(13)
= epsilon_n+(f^('')(x_(n-1)))/(2f^'(x_(n-1)))epsilon_n^2,
(14)

and (◇) becomes

epsilon_(n+1) = epsilon_n-[epsilon_n+(f^('')(x_(n-1)))/(2f^'(x_(n-1)))epsilon_n^2]
(15)
= -(f^('')(x_(n-1)))/(2f^'(x_(n-1)))epsilon_n^2.
(16)

Therefore, when the method converges, it does so quadratically.

NewtonsMethodBasins

Applying Newton's method to the roots of any polynomial of degree two or higher yields a rational map of C, and the Julia set of this map is a fractal whenever there are three or more distinct roots. Iterating the method for the roots of z^n-1=0 with starting point z_0 gives

 z_(i+1)=z_i-(z_i^n-1)/(nz_i^(n-1))
(17)

(Mandelbrot 1983, Gleick 1988, Peitgen and Saupe 1988, Press et al. 1992, Dickau 1997). Since this is an nth order polynomial, there are n roots to which the algorithm can converge. Coloring the basin of attraction (the set of initial points z_0 that converge to the same root) for each root a different color then gives the above plots.

NewtonsMethodCross

Fractals typically arise from non-polynomial maps as well. The plot above shows the number of iterations needed for Newton's method to converge for the function z^2-2^z (D. Cross, pers. comm., Jan. 10, 2005) and z^3-3^z.

 

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