In numerical Analysis, the method of Newton , or method of Newton-Raphson , is a effective algorithm to find Approximation S of one zero (or root) of a function of a real variable with actual values. The algorithm consists in linearizing a function F in a point and to take the point of cancellation of this linearization like approximation of the zero required one. One reiterates this procedure in the approximation obtained. In the successful outcomes, the successive approximations obtained convergent with a quadratic speed. In an abstract way, the number of correct decimals doubles with each stage.

Applied to derived from a function, this algorithm makes it possible to obtain an evaluation of the critical points. The method of Newton spreads in higher dimension. The reason resides in a use of the Théorème of the point fixes, which however is not necessary to include/understand the direction of the result.

This method bears the name of the English mathematicians Isaac Newton (1643 - 1727) and Joseph Raphson, which was the first to describe it to apply it to the research of the zeros of a polynomial equation.

History

The method of Newton was described satisfactorily by the mathematician English Isaac Newton (1643 - 1727) in Of analysi per aequationes number terminorum infinitas , written in 1669 and published in 1711 by William Jones (1675 - 1749) . It was again described in Of metodis fluxionum and serierum infinitarum (Of the method of the Fluxion S and the infinite consequences), written in 1671, translated and published under the title Methods off Fluxions in 1736 per John Colson. However, Newton applied the method only to the only polynomials. As the concept of derived and thus of linearization was not defined at that time, the adopted approach differs: Newton sought to refine a rough approximation of one zero of a polynomial by a polynomial calculation.

For example, 2 is one zero exact of $X^3-2X^2-3X+6$ and thus can be taken as one zero approximate of $P\; (X)\; =X^3-2X^2-3X+7$. A small disturbance must be written 2+Y. But, $P\; (2+Y)\; =1-7Y+2Y^2+Y^3=1-7Y$ by neglecting the terms of order at least 2. The cancellation of P by 3+Y imposes $Y=1/7$.

This method was the object of former publications. In 1685, John Wallis (1616 - 1703) published the first description in of it has off Treatise Algebra both Historical and Practical . In 1690, Joseph Raphson published of it a description simplifiee in Analysis aequationum universalis . Raphson always regarded the method of Newton as a purely algebraic method and restricted also its use with the only polynomials. However, it highlighted the recursive calculation of the successive approximations of one zero of a polynomial instead of regarding as Newton a succession of polynomials.

It is only in 1740 that Thomas Simpson (1710 - 1761 described this method of calculating iterative to approach the solutions of a nonlinear equation, using calculation fluxionnel. Simpson applied the method of Newton has problems of optimization. Arthur Cayley (1821 - 1895) was the first to note the difficulty in generalizing the method of Newton to the complex variables in 1879, for example with the polynomials of degree higher than 3.

Method

On the basis of a reasonable approximate value of one zero of a function of a real variable, one approximate to the first order the function by its tangent in this point. This tangent is a Fonction closely connected which one can find the single zero (analyzes elementary). This zero of the tangent will be generally closer to the zero of the function. By this operation, one can thus hope to improve the approximation by Itération S successive. The advantage of this method is to be able to be explained graphically. However, the establishment of a program requires to clarify calculations to be carried out. That is to say a function F : '' B '' → R definite and derivable and on the interval '' B '', and with real values . The regularity of F is minimal here. Let us take X ₀ an arbitrary reality. By recurrence, one defines the continuation X _N by:

$x\_\; \{n+1\}\; =\; x\_n\; -\; \backslash \; frac\; \{F\; (x\_n)\}\{f\text{'}\; (x\_n)\}$ where F   'indicates the Dérivée from the function F . It may be that the recurrence finishes, so at the stage N , X _N does not belong to the field of definition.

If the zero unknown α is isolated, then there exists a Voisinage of α such as for all the starting values X ₀ in this vicinity, the continuation ( X _N ) will converge towards α . Moreover, if F   '( α ) ≠ 0, then convergence is quadratic, which means intuitively that the number of correct figures is roughly doubled with each stage.

Although the method is very effective, certain practical aspects must be taken into account. Above all, the method of Newton requires that the derivative is actually calculated. Whenever the derivative is only estimated by taking the slope between two points of the function, the method takes the name of Méthode of the secant, less effective (of order 1,618 which is the Golden section) and lower than other algorithms. In addition, if the starting value is too far away from truth zero, the method of Newton can enter in infinite loop without producing improved approximation. Because of that, very implemented of the method of Newton must include a check code of the iteration count.

Example

To illustrate the method, let us seek the positive number X checking cos ( X ) = X ³. Let us reformulate the question to introduce a function having to cancel itself: one seeks the zero positive one (the root) of F ( X )   =  cos ( X )   −  X ³. Derivation gives F   '( X )   =  −  sin ( X )   −  3 X ².

Like cos ( X )   ≤  1 for all X and X ³  >  1 for X   >  1, we know that our zero range between 0 and 1. We test a starting value of X ₀  =  0,5.

$\backslash \; begin\; \{matrix\}$

x_1 & = & x_0 - \ frac {F (x_0)}{f' (x_0)} & = &0,5- \ frac {\ cos (0,5) - 0,5^3} {- \ sin (0,5) - 3 \ times 0,5^2} & \ simeq & 1,112 \, 141 \, 637 \, 1 \ \ x_2 & = & x_1 - \ frac {F (x_1)}{f' (x_1)} & & \ vdots & \ simeq & 0,909 \, 672 \, 693 \, 736 \ \ x_3 & & \ vdots & & \ vdots & \ simeq & 0,866 \, 263 \, 818 \, 209 \ \ x_4 & & \ vdots & & \ vdots & \ simeq & 0,865 \, 477 \, 135 \, 298 \ \ x_5 & & \ vdots & & \ vdots & \ simeq & 0,865 \, 474 \, 033 \, 111 \ \ x_6 & & \ vdots & & \ vdots & \ simeq & 0,865 \, 474 \, 033 \, 101 \ \ x_7 & & \ vdots & & \ vdots & \ simeq & 0,865 \, 474 \, 033 \, 102 \end{matrix} The first 7 figures of this value coincide with the first 7 figures of truth zero.

Speed of convergence

The speed of convergence of a continuation $x\_n$ obtained by the method of Newton can be obtained like application of the formula of Taylor. It is a question of evaluating an increase of $\backslash \; log|x\_n-a|$.

$f$ is a continuously differentiable function twice defined in the vicinity of has . It is supposed that has is being one zero of F which one tries to approximate by the method of Newton. One makes the assumption that has is one zero of order 1, in other words that F '(A) is nonnull. The formula of Taylor is written:

$0=f\; (A)\; =f\; (X)\; +f\text{'}\; (X)\; (center)\; +\; \backslash \; frac\; \{F$ (\ xi)}{2} {(center) ^2} , with $\backslash \; xi$ between X and has . On the basis of the approximation X , the method of Newton provides at the end of an iteration: $N\_f\; (X)\; -\; a=x-\; \backslash \; frac\; \{F\; (X)\}\{f\text{'}\; (X)\}\; -\; a=\; \backslash \; frac\; \{F$ (\ xi)}{2 \, f' (X)}(x-a) ^2. For a compact interval I container X and has and included in the field of definition of F , one poses: $\backslash \; textstyle\; m\_1=\; \backslash \; min\_\; \{X\; \backslash \; in\; I\}\; |f\text{'}\; (X)\; |$ like $\backslash \; textstyle\; M\_2=\; \backslash \; max\_\; \{X\; \backslash \; in\; I\}\; |F$ (X) |. Then, for all $x\; \backslash \; in\; I$: $\backslash \; Bigl|N\_f\; (X)\; -\; has\; \backslash \; Bigr|\; \backslash \; the\; \backslash \; frac\; \{M\_2\}\; \{2m\_1\}\; |x-a|^2$. By immediate recurrence, it comes: or $K=\; \backslash \; tfrac\; \{M\_2\}\; \{2m\_1\}$. While passing to the Logarithm: The convergence of X N towards has is thus quadratic.

Complex variable

The method can apply without difficulty to the functions of a complex variable. The problems of convergence become more serious. It must be mentioned other limiting behaviors:

• the continuation obtained by recurrence diverges towards the infinite one;
• the continuation present of the limiting cycles in other words, it converges alternatively worms of the points which are not of the zeros of the function considered;
• the continuation can approach the zeros of the function asymptotically, but with each stage of the iteration, one finds oneself towards new a zero.

The whole of the points from which can be obtained a continuation which converges towards a fixed zêro calls the basin of attraction from this zero.

Generalization

One can also want to use the method of Newton to solve systems of N equations (not necessarily linear), which amounts finding zero of functions the continuously derivable ones F : R ^K → R ^K . In the formulation given above, it is necessary to multiply by the reverse of the matrix Jacobienne K   x  K F   “( X _N ) instead of dividing by F   ” ( X _N ). Rather than to calculate the reverse of the matrix now, one can save time by solving the Système of linear equations

$F\; \backslash ,\; \text{'}(x\_n)\; (x\_\; \{n+1\}\; -\; x\_n)\; =\; -\; F\; (x\_n)$

for the unknown factor X _{N +1}  −  X _N . Once again, this method functions only for one initial value X ₀ sufficiently near to truth zero. Thus, one can start by locating a favorable area by a coarse method, then to apply the method of Newton to polish the precision.

The method can also be applied to find of the zeros of complex functions. For much of complex functions, the whole of all the starting values which make it possible the method to converge towards truth zero (the “basin of attraction”) is a Fractale. In the particular case where the complex function is a polynomial function, the method converges starting from any starting value except if the polynomial admits double roots.

See too

• Method of Newton on math-linux.com.
• Method of the secant
• Method of the false position (or Method Controlled Falsi)

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