Innumerical analysis, theNewton–Raphson method, also known simply asNewton's method, named afterIsaac Newton andJoseph Raphson, is aroot-finding algorithm which produces successively betterapproximations to theroots (or zeroes) of areal-valuedfunction. The most basic version starts with areal-valued functionf, itsderivativef′, and an initial guessx₀ for aroot off. Iff satisfies certain assumptions and the initial guess is close, then

$${\displaystyle x_1=x_0-\frac{f(x_0)}{f^{\prime }(x_0)}}$$

is a better approximation of the root thanx₀. Geometrically,(x₁, 0) is thex-intercept of thetangent of thegraph off at(x₀,f(x₀)): that is, the improved guess,x₁, is the unique root of thelinear approximation off at the initial guess,x₀. The process is repeated as

$${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}}$$

until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class ofHouseholder's methods, and was succeeded byHalley's method. The method can also be extended tocomplex functions and tosystems of equations.

The idea is to start with an initial guess, then to approximate the function by itstangent line, and finally to compute thex-intercept of this tangent line. Thisx-intercept will typically be a better approximation to the original function's root than the first guess, and the method can beiterated.

If thetangent line to the curvef(x) atx =x_n intercepts thex-axis atx_n+1 then the slope is

$${\displaystyle f^{\prime }(x_n)={\displaystyle \frac{f(x_n)}{x_n-x_{n+1}}}.}$$ 

Solving forx_n+1 gives

$${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}.}$$ 

We start the process with some arbitrary initial valuex₀. (The closer to the zero, the better. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to theintermediate value theorem.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and thatf′(x₀) ≠ 0. Furthermore, for a zero ofmultiplicity 1, the convergence is at least quadratic (seeRate of convergence) in aneighbourhood of the zero, which intuitively means that the number of correct digits roughly doubles in every step. More details can be found in§ Analysis below.

Householder's methods are similar but have higher order for even faster convergence. However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly iff or its derivatives are computationally expensive to evaluate.

In theOld Babylonian period (19th–16th century BCE), the side of a square of known area could be effectively approximated, and this is conjectured to have been done using a special case of Newton's method,described algebraically below, by iteratively improving an initial estimate; an equivalent method can be found inHeron of Alexandria'sMetrica (1st–2nd century CE), so is often calledHeron's method.^[1]Jamshīd al-Kāshī used a method to solvex^P −N = 0 to find roots ofN, a method that was algebraically equivalent to Newton's method, and in which a similar method was found inTrigonometria Britannica, published byHenry Briggs in 1633.^[2]

The method first appeared roughly inIsaac Newton's work inDe analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 byWilliam Jones) and inDe metodis fluxionum et serierum infinitarum (written in 1671, translated and published asMethod of Fluxions in 1736 byJohn Colson).^[3][4] However, while Newton gave the basic ideas, his method differs from the modern method given above. He applied the method only to polynomials, starting with an initial root estimate and extracting a sequence of error corrections. He used each correction to rewrite the polynomial in terms of the remaining error, and then solved for a new correction by neglecting higher-degree terms. He did not explicitly connect the method with derivatives or present a general formula. Newton applied this method to both numerical and algebraic problems, producingTaylor series in the latter case.

Newton may have derived his method from a similar, less precise method by mathematicianFrançois Viète, however, the two methods are not the same.^[3] The essence of Viète's own method can be found in the work of the mathematicianSharaf al-Din al-Tusi.^[5]

The Japanese mathematicianSeki Kōwa used a form of Newton's method in the 1680s to solve single-variable equations, though the connection with calculus was missing.^[6]

Newton's method was first published in 1685 inA Treatise of Algebra both Historical and Practical byJohn Wallis.^[7] In 1690,Joseph Raphson published a simplified description inAnalysis aequationum universalis.^[8] Raphson also applied the method only to polynomials, but he avoided Newton's tedious rewriting process by extracting each successive correction from the original polynomial. This allowed him to derive a reusable iterative expression for each problem. Finally, in 1740,Thomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.

Arthur Cayley in 1879 inThe Newton–Fourier imaginary problem was the first to notice the difficulties in generalizing Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values. This opened the way to the study of thetheory of iterations of rational functions.

Newton's method is a powerful technique—if the derivative of the function at the root is nonzero, then theconvergence is at least quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method.

Newton's method requires that the derivative can be calculated directly. An analytical expression for the derivative may not be easily obtainable or could be expensive to evaluate. In these situations, it may be appropriate to approximate the derivative by using the slope of a line through two nearby points on the function. Using this approximation would result in something like thesecant method whose convergence is slower than that of Newton's method.

It is important to review theproof of quadratic convergence of Newton's method before implementing it. Specifically, one should review the assumptions made in the proof. Forsituations where the method fails to converge, it is because the assumptions made in this proof are not met.

For example,in some cases, if the first derivative is not well behaved in the neighborhood of a particular root, then it is possible that Newton's method will fail to converge no matter where the initialization is set. In some cases, Newton's method can be stabilized by usingsuccessive over-relaxation, or the speed of convergence can be increased by using the same method.

In a robust implementation of Newton's method, it is common to place limits on the number of iterations, bound the solution to an interval known to contain the root, and combine the method with a more robust root finding method.

If the root being sought hasmultiplicity greater than one, the convergence rate is merely linear (errors reduced by a constant factor at each step) unless special steps are taken. When there are two or more roots that are close together then it may take many iterations before the iterates get close enough to one of them for the quadratic convergence to be apparent. However, if the multiplicitym of the root is known, the following modified algorithm preserves the quadratic convergence rate:^[9]

$${\displaystyle x_{n+1}=x_n-m\frac{f(x_n)}{f^{\prime }(x_n)}.}$$ 

This is equivalent to usingsuccessive over-relaxation. On the other hand, if the multiplicitym of the root is not known, it is possible to estimatem after carrying out one or two iterations, and then use that value to increase the rate of convergence.

If the multiplicitym of the root is finite theng(x) =⁠f(x)/f′(x)⁠ will have a root at the same location with multiplicity 1. Applying Newton's method to find the root ofg(x) recovers quadratic convergence in many cases although it generally involves the second derivative off(x). In a particularly simple case, iff(x) =x^m theng(x) =⁠x/m⁠ and Newton's method finds the root in a single iteration with

$${\displaystyle x_{n+1}=x_n-\frac{g(x_n)}{g^{\prime }(x_n)}=x_n-\frac{\frac{x_n}{m}}{\frac{1}{m}}=0.}$$ 

Suppose that the functionf has a zero atα, i.e.,f(α) = 0, andf is differentiable in aneighborhood ofα.

Iff is continuously differentiable and its derivative is nonzero at α, then there exists aneighborhood ofα such that for all starting valuesx₀ in that neighborhood, thesequence(x_n) willconverge toα.^[10]

Iff is continuously differentiable, its derivative is nonzero at α,and it has asecond derivative at α, then the convergence is quadratic or faster. If the second derivative is not 0 atα then the convergence is merely quadratic. If the third derivative exists and is bounded in a neighborhood ofα, then:

$${\displaystyle \mathrm{\Delta }{x}_{i+1}=\frac{f^{″}(\alpha )}{2f^{\prime }(\alpha )}{\left(\mathrm{\Delta }{x}_i\right)}^2+O{\left(\mathrm{\Delta }{x}_i\right)}^3,}$$ 

where

$${\displaystyle \mathrm{\Delta }{x}_i\triangleq x_i-\alpha .}$$ 

If the derivative is 0 atα, then the convergence is usually only linear. Specifically, iff is twice continuously differentiable,f′(α) = 0 andf″(α) ≠ 0, then there exists a neighborhood ofα such that, for all starting valuesx₀ in that neighborhood, the sequence of iterates converges linearly, withrate⁠1/2⁠.^[11] Alternatively, iff′(α) = 0 andf′(x) ≠ 0 forx ≠α,x in aneighborhoodU ofα,α being a zero ofmultiplicityr, and iff ∈C^r(U), then there exists a neighborhood ofα such that, for all starting valuesx₀ in that neighborhood, the sequence of iterates converges linearly.

However, even linear convergence is not guaranteed in pathological situations.

In practice, these results are local, and the neighborhood of convergence is not known in advance. But there are also some results on global convergence: for instance, given a right neighborhoodU₊ ofα, iff is twice differentiable inU₊ and iff′ ≠ 0,f ·f″ > 0 inU₊, then, for eachx₀ inU₊ the sequencex_k is monotonically decreasing toα.

According toTaylor's theorem, any functionf(x) which has a continuous second derivative can be represented by an expansion about a point that is close to a root off(x). Suppose this root isα. Then the expansion off(α) aboutx_n is:

where theLagrange form of the Taylor series expansion remainder is

$${\displaystyle R_1=\frac{1}{2!}{f}^{″}({\xi }_n){\left(\alpha -x_n\right)}^2,}$$ 

whereξ_n is in betweenx_n andα.

Sinceα is the root, (1) becomes:

Dividing equation (2) byf′(x_n) and rearranging gives

Remembering thatx_{n + 1} is defined by

one finds that

$${\displaystyle \underset{{\epsilon }_{n+1}}{\underbrace{\alpha -x_{n+1}}}=\frac{-f^{″}({\xi }_n)}{2f^{\prime }(x_n)}{(\underset{{\epsilon }_n}{\underbrace{\alpha -x_n}})}^2.}$$ 

That is,

Taking the absolute value of both sides gives

Equation (6) shows that theorder of convergence is at least quadratic if the following conditions are satisfied:

1. f′(x) ≠ 0; for allx ∈I, whereI is the interval[α − |ε₀|,α + |ε₀|];
2. f″(x) is continuous, for allx ∈I;
3. M |ε₀| < 1

whereM is given by

$${\displaystyle M=\frac{1}{2}\left(\underset{x\in I}{sup}|f^{″}(x)|\right)\left(\underset{x\in I}{sup}\frac{1}{|f^{\prime }(x)|}\right).}$$ 

If these conditions hold,

$${\displaystyle |{\epsilon }_{n+1}|\le M\cdot {\epsilon }_n^2.}$$ 

Suppose thatf(x) is aconcave function on an interval, which isstrictly increasing. If it is negative at the left endpoint and positive at the right endpoint, theintermediate value theorem guarantees that there is a zeroζ off somewhere in the interval. From geometrical principles, it can be seen that the Newton iterationx_i starting at the left endpoint ismonotonically increasing and convergent, necessarily toζ.^[12]

Joseph Fourier introduced a modification of Newton's method starting at the right endpoint:

${\displaystyle y_{i+1}=y_i-\frac{f(y_i)}{f^{\prime }(x_i)}.}$ 

This sequence is monotonically decreasing and convergent. By passing to the limit in this definition, it can be seen that the limit ofy_i must also be the zeroζ.^[12]

So, in the case of a concave increasing function with a zero, initialization is largely irrelevant. Newton iteration starting anywhere left of the zero will converge, as will Fourier's modified Newton iteration starting anywhere right of the zero. The accuracy at any step of the iteration can be determined directly from the difference between the location of the iteration from the left and the location of the iteration from the right. Iff is twice continuously differentiable, it can be proved usingTaylor's theorem that

${\displaystyle \underset{i\to \mathrm{\infty }}{lim}\frac{y_{i+1}-x_{i+1}}{(y_i-x_i{)}^2}=-\frac{1}{2}\frac{f^{″}(\zeta )}{f^{\prime }(\zeta )},}$ 

showing that this difference in locations converges quadratically to zero.^[12]

All of the above can be extended to systems of equations in multiple variables, although in that context the relevant concepts ofmonotonicity and concavity are more subtle to formulate.^[13] In the case of single equations in a single variable, the above monotonic convergence of Newton's method can also be generalized to replace concavity by positivity or negativity conditions on an arbitrary higher-order derivative off. However, in this generalization, Newton's iteration is modified so as to be based onTaylor polynomials rather than thetangent line. In the case of concavity, this modification coincides with the standard Newton method.^[14]

If we seek the root of a single function${\displaystyle f:{\mathbf{R}}^n\to \mathbf{R}}$ then the error${\displaystyle {ϵ}_n=x_n-\alpha }$  is a vector such that its components obey${\displaystyle {ϵ}_k^{(n+1)}=\frac{1}{2}({ϵ}^{(n)}{)}^T{Q}_k{ϵ}^{(n)}+O(\Vert {ϵ}^{(n)}{\Vert }^3)}$  where${\displaystyle Q_k}$  is a quadratic form:${\displaystyle (Q_k{)}_{i,j}=\sum _{\ell }((D^{2}f{)}^{-1}{)}_{i,\ell }\frac{{\mathrm{\partial }}^{3}f}{\mathrm{\partial }{x}_j\mathrm{\partial }{x}_k\mathrm{\partial }{x}_{\ell }}}$ evaluated at the root${\displaystyle \alpha }$ (where${\displaystyle D^{2}f}$  is the 2nd derivative Hessian matrix).

Newton's method is one of many knownmethods of computing square roots. Given a positive numbera, the problem of finding a numberx such thatx² =a is equivalent to finding a root of the functionf(x) =x² −a. The Newton iteration defined by this function is given by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=x_n-\frac{x_n^2-a}{2x_n}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right).}$ 

This happens to coincide with the"Babylonian" method of finding square roots, which consists of replacing an approximate rootx_n by thearithmetic mean ofx_n anda⁄x_n. By performing this iteration, it is possible to evaluate a square root to any desired accuracy by only using the basicarithmetic operations.

The following three tables show examples of the result of this computation for finding the square root of 612, with the iteration initialized at the values of 1, 10, and −20. Each row in a "x_n" column is obtained by applying the preceding formula to the entry above it, for instance

${\displaystyle 306.5=\frac{1}{2}\left(1+\frac{612}{1}\right).}$ 

The correct digits are underlined. It is seen that with only a few iterations one can obtain a solution accurate to many decimal places. The first table shows that this is true even if the Newton iteration were initialized by the very inaccurate guess of1.

When computing any nonzero square root, the first derivative off must be nonzero at the root, and thatf is a smooth function. So, even before any computation, it is known that any convergent Newton iteration has a quadratic rate of convergence. This is reflected in the above tables by the fact that once a Newton iterate gets close to the root, the number of correct digits approximately doubles with each iteration.

Consider the problem of finding the positive numberx withcosx =x³. We can rephrase that as finding the zero off(x) = cos(x) −x³. We havef′(x) = −sin(x) − 3x². Sincecos(x) ≤ 1 for allx andx³ > 1 forx > 1, we know that our solution lies between 0 and 1.

A starting value of 0 will lead to an undefined result which illustrates the importance of using a starting point close to the solution. For example, with an initial guessx₀ = 0.5, the sequence given by Newton's method is:

 

The correct digits are underlined in the above example. In particular,x₆ is correct to 12 decimal places. We see that the number of correct digits after the decimal point increases from 2 (forx₃) to 5 and 10, illustrating the quadratic convergence.

The functionf(x) =x² has a root at 0.^[15] Sincef is continuously differentiable at its root, the theory guarantees that Newton's method as initialized sufficiently close to the root will converge. However, since the derivativef ′ is zero at the root, quadratic convergence is not ensured by the theory. In this particular example, the Newton iteration is given by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=\frac{1}{2}{x}_n.}$ 

It is visible from this that Newton's method could be initialized anywhere and converge to zero, but at only a linear rate. If initialized at 1, dozens of iterations would be required before ten digits of accuracy are achieved.

The functionf(x) =x +x^4/3 also has a root at 0, where it is continuously differentiable. Although the first derivativef ′ is nonzero at the root, the second derivativef ′′ is nonexistent there, so that quadratic convergence cannot be guaranteed. In fact the Newton iteration is given by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=\frac{x^{4/3}}{3+4x^{1/3}}.}$ 

From this, it can be seen that the rate of convergence is superlinear but subquadratic. This can be seen in the following tables, the left of which shows Newton's method applied to the abovef(x) =x +x^4/3 and the right of which shows Newton's method applied tof(x) =x +x². The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1,2,3,5,10,20,39,...) being approximately doubled from row to row. While the convergence on the left is superlinear, the order of magnitude is only multiplied by about 4/3 from row to row (0,1,2,4,5,7,10,13,...).

The rate of convergence is distinguished from the number of iterations required to reach a given accuracy. For example, the functionf(x) =x²⁰ − 1 has a root at 1. Sincef ′(1) ≠ 0 andf is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically. However, if initialized at 0.5, the first few iterates of Newton's method are approximately 26214, 24904, 23658, 22476, decreasing slowly, with only the 200th iterate being 1.0371. The following iterates are 1.0103, 1.00093, 1.0000082, and 1.00000000065, illustrating quadratic convergence. This highlights that quadratic convergence of a Newton iteration does not mean that only few iterates are required; this only applies once the sequence of iterates is sufficiently close to the root.^[16]

The functionf(x) =x(1 +x²)^−1/2 has a root at 0. The Newton iteration is given by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=-x_n^3.}$ 

From this, it can be seen that there are three possible phenomena for a Newton iteration. If initialized strictly between±1, the Newton iteration will converge (super-)quadratically to 0; if initialized exactly at1 or−1, the Newton iteration will oscillate endlessly between±1; if initialized anywhere else, the Newton iteration will diverge.^[17] This same trichotomy occurs forf(x) = arctanx.^[15]

In cases where the function in question has multiple roots, it can be difficult to control, via choice of initialization, which root (if any) is identified by Newton's method. For example, the functionf(x) =x(x² − 1)(x − 3)e^{−(x − 1)2/2} has roots at −1, 0, 1, and 3.^[18] If initialized at −1.488, the Newton iteration converges to 0; if initialized at −1.487, it diverges to∞; if initialized at −1.486, it converges to −1; if initialized at −1.485, it diverges to−∞; if initialized at −1.4843, it converges to 3; if initialized at −1.484, it converges to1. This kind of subtle dependence on initialization is not uncommon; it is frequently studied in thecomplex plane in the form of theNewton fractal.

Consider the problem of finding a root off(x) =x^1/3. The Newton iteration is

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=x_n-\frac{x_n^{1/3}}{\frac{1}{3}{x}_n^{-2/3}}=-2x_n.}$ 

Unless Newton's method is initialized at the exact root 0, it is seen that the sequence of iterates will fail to converge. For example, even if initialized at the reasonably accurate guess of 0.001, the first several iterates are −0.002, 0.004, −0.008, 0.016, reaching 1048.58, −2097.15, ... by the 20th iterate. This failure of convergence is not contradicted by the analytic theory, since in this casef is not differentiable at its root.

In the above example, failure of convergence is reflected by the failure off(x_n) to get closer to zero asn increases, as well as by the fact that successive iterates are growing further and further apart. However, the functionf(x) =x^1/3e^−x2 also has a root at 0. The Newton iteration is given by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=x_n\left(1-\frac{3}{1-6x_n^2}\right).}$ 

In this example, where againf is not differentiable at the root, any Newton iteration not starting exactly at the root will diverge, but with bothx_{n + 1} −x_n andf(x_n) converging to zero.^[19] This is seen in the following table showing the iterates with initialization 1:

Although the convergence ofx_{n + 1} −x_n in this case is not very rapid, it can be proved from the iteration formula. This example highlights the possibility that a stopping criterion for Newton's method based only on the smallness ofx_{n + 1} −x_n andf(x_n) might falsely identify a root.

It is easy to find situations for which Newton's method oscillates endlessly between two distinct values. For example, for Newton's method as applied to a functionf to oscillate between 0 and 1, it is only necessary that the tangent line tof at 0 intersects thex-axis at 1 and that the tangent line tof at 1 intersects thex-axis at 0.^[19] This is the case, for example, iff(x) =x³ − 2x + 2. For this function, it is even the case that Newton's iteration as initialized sufficiently close to 0 or 1 willasymptotically oscillate between these values. For example, Newton's method as initialized at 0.99 yields iterates 0.99, −0.06317, 1.00628, 0.03651, 1.00196, 0.01162, 1.00020, 0.00120, 1.000002, and so on. This behavior is present despite the presence of a root off approximately equal to −1.76929.

In some cases, it is not even possible to perform the Newton iteration. For example, iff(x) =x² − 1, then the Newton iteration is defined by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=x_n-\frac{x_n^2-1}{2x_n}=\frac{x_n^2+1}{2x_n}.}$ 

So Newton's method cannot be initialized at 0, since this would makex₁ undefined. Geometrically, this is because the tangent line tof at 0 is horizontal (i.e.f ′(0) = 0), never intersecting thex-axis.

Even if the initialization is selected so that the Newton iteration can begin, the same phenomenon can block the iteration from being indefinitely continued.

Iff has an incomplete domain, it is possible for Newton's method to send the iterates outside of the domain, so that it is impossible to continue the iteration.^[19] For example, thenatural logarithm functionf(x) = lnx has a root at 1, and is defined only for positivex. Newton's iteration in this case is given by

${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=x_n(1-\ln x_n).}$ 

So if the iteration is initialized ate, the next iterate is 0; if the iteration is initialized at a value larger thane, then the next iterate is negative. In either case, the method cannot be continued.

One may also use Newton's method to solve systems ofk equations, which amounts to finding the (simultaneous) zeroes ofk continuously differentiable functions${\displaystyle f:{\mathbb{R}}^k\to \mathbb{R}.}$  This is equivalent to finding the zeroes of a single vector-valued function${\displaystyle F:{\mathbb{R}}^k\to {\mathbb{R}}^k.}$  In the formulation given above, the scalarsx_n are replaced by vectorsx_n and instead of dividing the functionf(x_n) by its derivativef′(x_n) one instead has to left multiply the functionF(x_n) by the inverse of itsk ×kJacobian matrixJ_F(x_n).^[20][21][22] This results in the expression

$${\displaystyle {\mathbf{x}}_{n+1}={\mathbf{x}}_n-J_F({\mathbf{x}}_n{)}^{-1}F({\mathbf{x}}_n).}$$ 

or, by solving thesystem of linear equations

$${\displaystyle J_F({\mathbf{x}}_n)({\mathbf{x}}_{n+1}-{\mathbf{x}}_n)=-F({\mathbf{x}}_n)}$$ 

for the unknownx_{n + 1} −x_n.^[23]

Thek-dimensional variant of Newton's method can be used to solve systems of greater thank (nonlinear) equations as well if the algorithm uses thegeneralized inverse of the non-squareJacobian matrixJ⁺ = (J^TJ)⁻¹J^T instead of the inverse ofJ. If thenonlinear system has no solution, the method attempts to find a solution in thenon-linear least squares sense. SeeGauss–Newton algorithm for more information.

For example, the following set of equations needs to be solved for vector of points${\displaystyle [x_1,x_2],}$  given the vector of known values${\displaystyle [2,3].}$ ^[24]

 

the function vector,${\displaystyle F(X_k),}$  and Jacobian Matrix,${\displaystyle J(X_k)}$  for iteration k, and the vector of known values,${\displaystyle Y,}$  are defined below.

 

Note that${\displaystyle F(X_k)}$  could have been rewritten to absorb${\displaystyle Y,}$  and thus eliminate${\displaystyle Y}$  from the equations. The equation to solve for each iteration are

 

and

${\displaystyle X_{k+1}=X_k-C_{k+1}}$ 

The iterations should be repeated until  where${\displaystyle E}$  is a value acceptably small enough to meet application requirements.

If vector${\displaystyle X_0}$  is initially chosen to be  that is,${\displaystyle x_1=1,}$  and${\displaystyle x_2=1,}$  and${\displaystyle E,}$  is chosen to be 1.10⁻³, then the example converges after four iterations to a value of${\displaystyle X_4=\left[0.567297,-0.309442\right].}$ 

The following iterations were made during the course of the solution.

Converging iteration sequence

Step	Variable	Value
0	x =
	f(x) =
1	J =
	c =
	x =
	f(x) =
2	J =
	c =
	x =
	f(x) =
3	J =
	c =
	x =
	f(x) =
4	J =
	c =
	x =
	f(x) =

When dealing withcomplex functions, Newton's method can be directly applied to find their zeroes.^[25] Each zero has abasin of attraction in the complex plane, the set of all starting values that cause the method to converge to that particular zero. These sets can be mapped as in the image shown. For many complex functions, the boundaries of the basins of attraction arefractals.

In some cases there are regions in the complex plane which are not in any of these basins of attraction, meaning the iterates do not converge. For example,^[26] if one uses a real initial condition to seek a root ofx² + 1, all subsequent iterates will be real numbers and so the iterations cannot converge to either root, since both roots are non-real. In this casealmost all real initial conditions lead tochaotic behavior, while some initial conditions iterate either to infinity or to repeating cycles of any finite length.

Curt McMullen has shown that for any possible purely iterative algorithm similar to Newton's method, the algorithm will diverge on some open regions of the complex plane when applied to some polynomial of degree 4 or higher. However, McMullen gave a generally convergent algorithm for polynomials of degree 3.^[27] Also, for any polynomial, Hubbard, Schleicher, and Sutherland gave a method for selecting a set of initial points such that Newton's method will certainly converge at one of them at least.^[28]

Another generalization is Newton's method to find a root of afunctionalF defined in aBanach space. In this case the formulation is

$${\displaystyle X_{n+1}=X_n-(F^{\prime }(X_n){)}^{-1}F(X_n),}$$ 

whereF′(X_n) is theFréchet derivative computed atX_n. One needs the Fréchet derivative to be boundedly invertible at eachX_n in order for the method to be applicable. A condition for existence of and convergence to a root is given by theNewton–Kantorovich theorem.^[29]

In the 1950s,John Nash developed a version of the Newton's method to apply to the problem of constructingisometric embeddings of generalRiemannian manifolds inEuclidean space. Theloss of derivatives problem, present in this context, made the standard Newton iteration inapplicable, since it could not be continued indefinitely (much less converge). Nash's solution involved the introduction ofsmoothing operators into the iteration. He was able to prove the convergence of his smoothed Newton method, for the purpose of proving animplicit function theorem for isometric embeddings. In the 1960s,Jürgen Moser showed that Nash's methods were flexible enough to apply to problems beyond isometric embedding, particularly incelestial mechanics. Since then, a number of mathematicians, includingMikhael Gromov andRichard Hamilton, have found generalized abstract versions of the Nash–Moser theory.^[30][31] In Hamilton's formulation, the Nash–Moser theorem forms a generalization of the Banach space Newton method which takes place in certainFréchet spaces.

When the Jacobian is unavailable or too expensive to compute at every iteration, aquasi-Newton method can be used.

Since higher-order Taylor expansions offer more accurate local approximations of a functionf, it is reasonable to ask why Newton’s method relies only on a second-order Taylor approximation. In the 19th century, Russian mathematician Pafnuty Chebyshev explored this idea by developing a variant of Newton’s method that used cubic approximations.^[32][33][34]

Inp-adic analysis, the standard method to show a polynomial equation in one variable has ap-adic root isHensel's lemma, which uses the recursion from Newton's method on thep-adic numbers. Because of the more stable behavior of addition and multiplication in thep-adic numbers compared to the real numbers (specifically, the unit ball in thep-adics is a ring), convergence in Hensel's lemma can be guaranteed under much simpler hypotheses than in the classical Newton's method on the real line.

Newton's method can be generalized with theq-analog of the usual derivative.^[35]

A nonlinear equation has multiple solutions in general. But if the initial value is not appropriate, Newton's method may not converge to the desired solution or may converge to the same solution found earlier. When we have already foundN solutions of${\displaystyle f(x)=0}$ , then the next root can be found by applying Newton's method to the next equation:^[36][37]

$${\displaystyle F(x)=\frac{f(x)}{\prod _{i=1}^N(x-x_i)}=0.}$$ 

This method is applied to obtain zeros of theBessel function of the second kind.^[38]

Hirano's modified Newton method is a modification conserving the convergence of Newton method and avoiding unstableness.^[39] It is developed to solve complex polynomials.

Combining Newton's method withinterval arithmetic is very useful in some contexts. This provides a stopping criterion that is more reliable than the usual ones (which are a small value of the function or a small variation of the variable between consecutive iterations). Also, this may detect cases where Newton's method converges theoretically but diverges numerically because of an insufficientfloating-point precision (this is typically the case for polynomials of large degree, where a very small change of the variable may change dramatically the value of the function; seeWilkinson's polynomial).^[40][41]

Considerf →C¹(X), whereX is a real interval, and suppose that we have an interval extensionF′ off′, meaning thatF′ takes as input an intervalY ⊆X and outputs an intervalF′(Y) such that:

 

We also assume that0 ∉F′(X), so in particularf has at most one root inX.We then define the interval Newton operator by:

$${\displaystyle N(Y)=m-\frac{f(m)}{F^{\prime }(Y)}=\left\{m-\frac{f(m)}{z}|z\in F^{\prime }(Y)\right\}}$$ 

wherem ∈Y. Note that the hypothesis onF′ implies thatN(Y) is well defined and is an interval (seeinterval arithmetic for further details on interval operations). This naturally leads to the following sequence:

 

Themean value theorem ensures that if there is a root off inX_k, then it is also inX_{k + 1}. Moreover, the hypothesis onF′ ensures thatX_{k + 1} is at most half the size ofX_k whenm is the midpoint ofY, so this sequence converges towards[x*,x*], wherex* is the root off inX.

IfF′(X) strictly contains 0, the use of extended interval division produces a union of two intervals forN(X); multiple roots are therefore automatically separated and bounded.

Newton's method can be used to find a minimum or maximum of a functionf(x). The derivative is zero at a minimum or maximum, so local minima and maxima can be found by applying Newton's method to the derivative.^[42] The iteration becomes:

$${\displaystyle x_{n+1}=x_n-\frac{f^{\prime }(x_n)}{f^{″}(x_n)}.}$$ 

An important application isNewton–Raphson division, which can be used to quickly find thereciprocal of a numbera, using only multiplication and subtraction, that is to say the numberx such that⁠1/x⁠ =a. We can rephrase that as finding the zero off(x) =⁠1/x⁠ −a. We havef′(x) = −⁠1/x²⁠.

Newton's iteration is

$${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}=x_n+\frac{\frac{1}{x_n}-a}{\frac{1}{x_n^2}}=x_n(2-a{x}_n).}$$ 

Therefore, Newton's iteration needs only two multiplications and one subtraction.

This method is also very efficient to compute the multiplicative inverse of apower series.

Manytranscendental equations can be solved up to an arbitrary precision by using Newton's method. For example, finding the cumulativeprobability density function, such as aNormal distribution to fit a known probability generally involves integral functions with no known means to solve in closed form. However, computing the derivatives needed to solve them numerically with Newton's method is generally known, making numerical solutions possible. For an example, see the numerical solution to theinverse Normal cumulative distribution.

A numerical verification for solutions of nonlinear equations has been established by using Newton's method multiple times and forming a set of solution candidates.^{[citation needed]}

The following is an example of a possible implementation of Newton's method in thePython (version 3.x) programming language for finding a root of a functionf which has derivativef_prime.

The initial guess will bex₀ = 1 and the function will bef(x) =x² − 2 so thatf′(x) = 2x.

Each new iteration of Newton's method will be denoted byx1. We will check during the computation whether the denominator (yprime) becomes too small (smaller thanepsilon), which would be the case iff′(x_n) ≈ 0, since otherwise a large amount of error could be introduced.

deff(x):returnx**2-2# f(x) = x^2 - 2deff_prime(x):return2*x# f'(x) = 2xdefnewtons_method(x0,f,f_prime,tolerance,epsilon,max_iterations):"""Newton's method    Args:      x0:              The initial guess      f:               The function whose root we are trying to find      f_prime:         The derivative of the function      tolerance:       Stop when iterations change by less than this      epsilon:         Do not divide by a number smaller than this      max_iterations:  The maximum number of iterations to compute    """for_inrange(max_iterations):y=f(x0)yprime=f_prime(x0)ifabs(yprime)<epsilon:# Give up if the denominator is too smallbreakx1=x0-y/yprime# Do Newton's computationifabs(x1-x0)<=tolerance:# Stop when the result is within the desired tolerancereturnx1# x1 is a solution within tolerance and maximum number of iterationsx0=x1# Update x0 to start the process againreturnNone# Newton's method did not converge

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