View source for Newton's method

{{Short description|Algorithm for finding zeros of functions}}
{{About|Newton's method for finding roots|Newton's method for finding minima|Newton's method in optimization}}
{{Use dmy dates|date=January 2020}}
[[File:Methode newton.png|thumb|An illustration of Newton's method.]]
In [[numerical analysis]], the '''Newton–Raphson method''', also known simply as '''Newton's method''', named after [[Isaac Newton]] and [[Joseph Raphson]], is a [[root-finding algorithm]] which produces successively better [[Numerical analysis|approximations]] to the [[root of a function|roots]] (or zeroes) of a [[Real number|real]]-valued [[function (mathematics)|function]]. The most basic version starts with a [[real-valued function]]  {{mvar|f}}, its [[derivative]] {{mvar|{{prime|f}}}}, and an initial guess {{math|{{var|x}}{{sub|0}}}} for a [[Zero of a function|root]] of {{mvar|f}}. If {{mvar|f}} satisfies certain assumptions and the initial guess is close, then

<math display="block">x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)}</math>

is a better approximation of the root than {{math|{{var|x}}{{sub|0}}}}. Geometrically, {{math|({{var|x}}{{sub|1}}, 0)}} is the [[x-intercept]] of the [[tangent]] of the [[graph of a function|graph]] of {{mvar|f}} at {{math|({{var|x}}{{sub|0}}, {{var|f}}({{var|x}}{{sub|0}}))}}: that is, the improved guess, {{math|{{var|x}}{{sub|1}}}}, is the unique root of the [[linear approximation]] of {{mvar|f}}  at the initial guess, {{math|{{var|x}}{{sub|0}}}}. The process is repeated as

<math display="block">x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}</math>

until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of [[Householder's method]]s, and was succeeded by [[Halley's method]]. The method can also be extended to [[Complex-valued function|complex functions]] and to [[systems of equations]].

== Description ==

The idea is to start with an initial guess, then to approximate the function by its [[tangent line]], and finally to compute the {{mvar|x}}-intercept of this tangent line. This {{mvar|x}}-intercept will typically be a better approximation to the original function's root than the first guess, and the method can be [[iterative method|iterated]].

[[Image:newton iteration.svg|alt=Illustration of Newton's method|thumb|right|upright=1.4|{{math|{{var|x}}{{sub|{{var|n}}+1}}}} is a better approximation than {{math|{{var|x}}{{sub|{{var|n}}}}}} for the root {{mvar|x}} of the function {{mvar|f}} (blue curve)]]

If the [[tangent line]] to the curve {{math|{{var|f}}({{var|x}})}} at {{math|1={{var|x}} = {{var|x}}{{sub|{{var|n}}}}}} intercepts the {{math|''x''}}-axis at {{math|{{var|x}}{{sub|{{var|n}}+1}}}} then the slope is

<math display="block">f'(x_n) = \dfrac{f(x_n)-0} {x_n-x_{n+1}}. </math>

Solving for {{math|{{var|x}}{{sub|{{var|n}}+1}}}} gives

<math display="block">x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. </math>

[[Image:NewtonIteration Ani.gif|alt=Illustration of Newton's method|thumb|right|upright=1.4|Iteration typically improves the approximation]]

We start the process with some arbitrary initial value {{math|''x''<sub>0</sub>}}. (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 the [[intermediate value theorem]].) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that {{math|{{var|{{prime|f}}}}({{var|x}}{{sub|0}}) ≠ 0}}. Furthermore, for a zero of [[Multiplicity (mathematics)|multiplicity]]&nbsp;1, the convergence is at least quadratic (see ''[[Rate of convergence]]'') in a [[neighbourhood (mathematics)|neighbourhood]] of the zero, which intuitively means that the number of correct digits roughly doubles in every step. More details can be found in ''{{section link|#Analysis}}'' below.

[[Householder's method]]s 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 if {{mvar|f}} or its derivatives are computationally expensive to evaluate.

==History==
The name "Newton's method" is derived from [[Isaac Newton]]'s description of a special case of the method in ''[[De analysi per aequationes numero terminorum infinitas]]'' (written in 1669, published in 1711 by [[William Jones (mathematician)|William Jones]]) and in ''De metodis fluxionum et serierum infinitarum'' (written in 1671, translated and published as ''[[Method of Fluxions]]'' in 1736 by [[John Colson]]). However, his method differs substantially from the modern method given above. Newton 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, producing [[Taylor series]] in the latter case.

Newton may have derived his method from a similar, less precise method by [[Franciscus Vieta|Vieta]]. The essence of Vieta's method can be found in the work of the [[Mathematics in medieval Islam|Persian mathematician]] [[Sharaf al-Din al-Tusi]], while his successor [[Jamshīd al-Kāshī]] used a form of Newton's method to solve {{math|{{var|x}}{{sup|{{var|P}}}} − {{var|N}} {{=}} 0}} to find roots of {{mvar|N}} (Ypma 1995). A special case of Newton's method for calculating square roots was known since ancient times and is often called the [[Babylonian method]].

Newton's method was used by 17th-century Japanese mathematician [[Seki Kōwa]] to solve single-variable equations, though the connection with calculus was missing.<ref>{{cite web |title=Chapter 2. Seki Takakazu |url=http://www.ndl.go.jp/math/e/s1/2.html |website=Japanese Mathematics in the Edo Period |publisher=National Diet Library |access-date=24 February 2019}}</ref>

Newton's method was first published in 1685 in ''A Treatise of Algebra both Historical and Practical'' by [[John Wallis]].<ref>{{cite book |first=John |last=Wallis |author-link=John Wallis |title=A Treatise of Algebra, both Historical and Practical |publisher=Richard Davis |location=Oxford |date=1685 |url=http://www.e-rara.ch/zut/content/titleinfo/2507537 |doi=10.3931/e-rara-8842}}</ref> In 1690, [[Joseph Raphson]] published a simplified description in ''Analysis aequationum universalis''.<ref>{{cite book |last=Raphson |first=Joseph |author-link=Joseph Raphson |title=Analysis Æequationum Universalis |edition=2nd |language=la |date=1697 |publisher=Thomas Bradyll |location=London |url=https://archive.org/details/bub_gb_4nlbAAAAQAAJ |doi=10.3931/e-rara-13516}}</ref> 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 in ''The 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 the [[Julia set|theory of iterations]] of rational functions.

==Practical considerations==
Newton's method is a powerful technique—in general the [[rate of convergence|convergence]] is 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.

===Difficulty in calculating the derivative of a function===
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 the [[secant method]] whose convergence is slower than that of Newton's method.

===Failure of the method to converge to the root===
It is important to review the [[#Proof of quadratic convergence for Newton's iterative method|proof of quadratic convergence]] of Newton's method before implementing it. Specifically, one should review the assumptions made in the proof. For [[#Failure analysis|situations where the method fails to converge]], it is because the assumptions made in this proof are not met.

For example, [[#Divergence even when initialization is close to the root|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 using [[successive over-relaxation#Other applications of the method|successive 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.

===Slow convergence for roots of multiplicity greater than 1===
If the root being sought has [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]] 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 multiplicity {{mvar|m}} of the root is known, the following modified algorithm preserves the quadratic convergence rate:<ref>{{cite web|title=Accelerated and Modified Newton Methods|url=http://mathfaculty.fullerton.edu/mathews/n2003/newtonacceleratemod.html|access-date=4 March 2016|archive-url=https://web.archive.org/web/20190524083302/http://mathfaculty.fullerton.edu/mathews/n2003/NewtonAccelerateMod.html|archive-date=24 May 2019|url-status=dead}}</ref>

<math display="block">x_{n+1} = x_n - m\frac{f(x_n)}{f'(x_n)}. </math>

This is equivalent to using [[successive over-relaxation#Other applications of the method|successive over-relaxation]]. On the other hand, if the multiplicity {{mvar|m}} of the root is not known, it is possible to estimate {{mvar|m}} after carrying out one or two iterations, and then use that value to increase the rate of convergence.

If the multiplicity {{mvar|m}} of the root is finite then {{math|1={{var|g}}({{var|x}}) = {{sfrac|{{var|f}}({{var|x}})|{{var|{{prime|f}}}}({{var|x}})}}}} will have a root at the same location with multiplicity 1.  Applying Newton's method to find the root of {{math|{{var|g}}({{var|x}})}} recovers quadratic convergence in many cases although it generally involves the second derivative of {{math|{{var|f}}({{var|x}})}}.  In a particularly simple case, if {{math|1={{var|f}}({{var|x}}) = {{var|x}}{{sup|{{var|m}}}}}} then {{math|1={{var|g}}({{var|x}}) = {{sfrac|{{var|x}}|{{var|m}}}}}} and Newton's method finds the root in a single iteration with

<math display="block">x_{n+1} = x_n - \frac{g(x_n)}{g'(x_n)} = x_n - \frac{\;\frac{x_n}{m}\;}{\frac{1}{m}} = 0\,.</math>

==Analysis==
Suppose that the function {{mvar|f}} has a zero at {{mvar|α}}, i.e., {{math|{{var|f}}({{var|α}}) {{=}} 0}}, and {{mvar|f}} is differentiable in a [[topological neighborhood|neighborhood]] of {{mvar|α}}.

If {{mvar|f}} is continuously differentiable and its derivative is nonzero at&nbsp;{{mvar|α}}, then there exists a [[topological neighborhood|neighborhood]] of {{mvar|α}} such that for all starting values {{math|{{var|x}}{{sub|0}}}} in that neighborhood, the [[sequence]] {{math|({{var|x}}{{sub|{{var|n}}}})}} will [[limit of a sequence|converge]] to {{mvar|α}}.<ref>{{citation|title=A Theoretical Introduction to Numerical Analysis|first1=Victor S.|last1=Ryaben'kii|first2=Semyon V.|last2=Tsynkov|publisher=CRC Press|year=2006|isbn=9781584886075|page=243|url=https://books.google.com/books?id=V8gWP031-hEC&pg=PA243}}.</ref>

If {{mvar|f}} is continuously differentiable, its derivative is nonzero at&nbsp;{{mvar|α}}, ''and'' it has a [[second derivative]] at&nbsp;{{mvar|α}}, then the convergence is quadratic or faster. If the second derivative is not 0 at {{mvar|α}} then the convergence is merely quadratic. If the third derivative exists and is bounded in a neighborhood of {{mvar|α}}, then:

<math display="block">\Delta x_{i+1} = \frac{f'' (\alpha)}{2 f' (\alpha)} \left(\Delta x_{i}\right)^2 + O\left(\Delta x_{i}\right)^3 \,,</math>

where

<math display="block">\Delta x_i \triangleq x_i - \alpha \,.</math>

If the derivative is 0 at {{mvar|α}}, then the convergence is usually only linear. Specifically, if {{mvar|f}} is twice continuously differentiable, {{math|{{var|{{prime|f}}}}({{var|α}}) {{=}} 0}} and {{math|{{var|{{prime|f|c=&Prime;}}}}({{var|α}}) ≠ 0}}, then there exists a neighborhood of {{mvar|α}} such that, for all starting values {{math|{{var|x}}{{sub|0}}}} in that neighborhood, the sequence of iterates converges linearly, with [[rate of convergence|rate]] {{sfrac|1|2}}.<ref>{{harvnb|Süli|Mayers|2003|loc=Exercise 1.6}}</ref> Alternatively, if {{math|{{var|{{prime|f}}}}({{var|α}}) {{=}} 0}} and {{math|{{var|{{prime|f}}}}({{var|x}}) ≠ 0}} for {{math|{{var|x}} ≠ {{var|α}}}}, {{mvar|x}}&nbsp;in a [[topological neighborhood|neighborhood]] {{mvar|U}} of {{mvar|α}}, {{mvar|α}} being a zero of [[Multiplicity (mathematics)|multiplicity]] {{mvar|r}}, and if {{math|{{var|f}} ∈ {{var|C}}{{isup|{{var|r}}}}({{var|U}})}}, then there exists a neighborhood of {{mvar|α}} such that, for all starting values {{math|{{var|x}}{{sub|0}}}} 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 neighborhood {{math|{{var|U}}{{sub|+}}}} of {{mvar|α}}, if {{mvar|f}} is twice differentiable in {{math|{{var|U}}{{sub|+}}}} and if {{math|{{var|{{prime|f}}}} ≠ 0}}, {{math|{{var|f}} · {{var|{{prime|f|c=&Prime;}}}} > 0}} in {{math|{{var|U}}{{sub|+}}}}, then, for each {{math|{{var|x}}{{sub|0}}}} in {{math|{{var|U}}{{sub|+}}}} the sequence {{math|{{var|x}}{{sub|{{var|k}}}}}} is monotonically decreasing to {{mvar|α}}.

===Proof of quadratic convergence for Newton's iterative method===
According to [[Taylor's theorem]], any function {{math|{{var|f}}({{var|x}})}} which has a continuous second derivative can be represented by an expansion about a point that is close to a root of {{math|{{var|f}}({{var|x}})}}. Suppose this root is {{mvar|α}}. Then the expansion of {{math|{{var|f}}({{var|α}})}} about {{math|{{var|x}}{{sub|{{var|n}}}}}} is:
{{NumBlk|:|<math>f(\alpha) = f(x_n) + f'(x_n)(\alpha - x_n) + R_1 \,</math>|{{EquationRef|1}}}}

where the [[Lagrange remainder|Lagrange form of the Taylor series expansion remainder]] is

<math display="block">R_1 = \frac{1}{2!}f''(\xi_n)\left(\alpha - x_n\right)^{2} \,,</math>

where {{math|{{var|ξ}}{{sub|{{var|n}}}}}} is in between {{math|{{var|x}}{{sub|{{var|n}}}}}} and {{mvar|α}}.

Since {{mvar|α}} is the root, ({{EquationNote|1}}) becomes:

{{NumBlk|:|<math>0 = f(\alpha) = f(x_n) + f'(x_n)(\alpha - x_n) + \tfrac12f''(\xi_n)\left(\alpha - x_n\right)^2 \,</math>|{{EquationRef|2}}}}

Dividing equation ({{EquationNote|2}}) by {{math|{{var|{{prime|f}}}}({{var|x}}{{sub|{{var|n}}}})}} and rearranging gives

{{NumBlk|:|<math> \frac {f(x_n)}{f'(x_n)}+\left(\alpha-x_n\right) = \frac {- f'' (\xi_n)}{2 f'(x_n)}\left(\alpha-x_n\right)^2 </math>|{{EquationRef|3}}}}

Remembering that {{math|{{var|x}}{{sub|{{var|n}} + 1}}}} is defined by

{{NumBlk|:|<math> x_{n+1} = x_{n} - \frac {f(x_n)}{f'(x_n)} \,,</math>|{{EquationRef|4}}}}

one finds that

<math display="block"> \underbrace{\alpha - x_{n+1}}_{\varepsilon_{n+1}} = \frac {- f'' (\xi_n)}{2 f'(x_n)} {(\,\underbrace{\alpha - x_n}_{\varepsilon_{n}}\,)}^2 \,.</math>

That is,

{{NumBlk|:|<math> \varepsilon_{n+1} = \frac {- f'' (\xi_n)}{2 f'(x_n)} \cdot \varepsilon_n^2 \,.</math>|{{EquationRef|5}}}}

Taking the absolute value of both sides gives

{{NumBlk|:|<math> \left| {\varepsilon_{n+1}}\right| = \frac {\left| f'' (\xi_n) \right| }{2 \left| f'(x_n) \right|} \cdot \varepsilon_n^2 \,. </math> |{{EquationRef|6}}}}

Equation ({{EquationNote|6}}) shows that the [[order of convergence]] is at least quadratic if the following conditions are satisfied:

# {{math|{{var|{{prime|f}}}}({{var|x}}) ≠ 0}}; for all {{math|{{var|x}} ∈ {{var|I}}}}, where {{mvar|I}} is the interval {{math|[{{var|α}} − {{abs|{{var|ε}}{{sub|0}}}}, {{var|α}} + {{abs|{{var|ε}}{{sub|0}}}}]}};
# {{math|{{var|{{prime|f|c=&Prime;}}}}({{var|x}})}} is continuous, for all {{math|{{var|x}} ∈ {{var|I}}}};
# {{math|{{var|M}} {{abs|{{var|ε}}{{sub|0}}}} < 1}}

where {{mvar|M}} is given by

<math display="block"> M = \frac12 \left( \sup_{x \in I} \vert f'' (x) \vert \right) \left( \sup_{x \in I} \frac {1}{ \vert f'(x) \vert } \right) . \,</math>

If these conditions hold,

<math display="block"> \vert \varepsilon_{n+1} \vert \leq M \cdot \varepsilon_n^2 \,. </math>

===Fourier conditions===

Suppose that {{math|''f''(''x'')}} is a [[concave function]] on an interval, which is [[strictly increasing]]. If it is negative at the left endpoint and positive at the right endpoint, the [[intermediate value theorem]] guarantees that there is a zero {{math|&zeta;}} of {{mvar|f}} somewhere in the interval. From geometrical principles, it can be seen that the Newton iteration {{math|''x''<sub>''i''</sub>}} starting at the left endpoint is [[monotonic function|monotonically increasing]] and convergent, necessarily to {{math|&zeta;}}.<ref name="ostrowski">{{cite book|last1=Ostrowski|first1=A. M.|title=Solution of equations in Euclidean and Banach spaces|year=1973|mr=0359306|series=Pure and Applied Mathematics|volume=9|publisher=[[Academic Press]]|location=New York–London|author-link1=Alexander Ostrowski|zbl=0304.65002|edition=Third edition of 1960 original}}</ref>

[[Joseph Fourier]] introduced a modification of Newton's method starting at the right endpoint:
:<math>y_{i+1}=y_i-\frac{f(y_i)}{f'(x_i)}.</math>
This sequence is monotonically decreasing and convergent. By passing to the limit in this definition, it can be seen that the limit of {{math|''y''<sub>''i''</sub>}} must also be the zero {{math|&zeta;}}.<ref name="ostrowski" /> 

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. If {{mvar|f}} is twice continuously differentiable, it can be proved using [[Taylor's theorem]] that
:<math>\lim_{i\to\infty}\frac{y_{i+1}-x_{i+1}}{(y_i-x_i)^2}=-\frac{1}{2}\frac{f''(\zeta)}{f'(\zeta)},</math>
showing that this difference in locations converges quadratically to zero.<ref name="ostrowski" />

All of the above can be extended to systems of equations in multiple variables, although in that context the relevant concepts of [[monotonicity]] and concavity are more subtle to formulate.<ref>Ortega and Rheinboldt, Section 13.3</ref> 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 of {{mvar|f}}. However, in this generalization, Newton's iteration is modified so as to be based on [[Taylor polynomial]]s rather than the [[tangent line]]. In the case of concavity, this modification coincides with the standard Newton method.<ref>{{cite book|last1=Traub|first1=J. F.|title=Iterative methods for the solution of equations|year=1964|series=Prentice-Hall Series in Automatic Computation|publisher=[[Prentice-Hall, Inc.]]|location=Englewood Cliffs, NJ|mr=0169356|author-link1=Joseph F. Traub|zbl=0121.11204}}</ref>

==Examples==
===Use of Newton's method to compute square roots===
Newton's method is one of many known [[methods of computing square roots]]. Given a positive number {{mvar|a}}, the problem of finding a number {{mvar|x}} such that {{math|''x''<sup>2</sup> {{=}} ''a''}} is equivalent to finding a root of the function {{math|''f''(''x'') {{=}} ''x''<sup>2</sup> − ''a''}}. The Newton iteration defined by this function is given by
:<math>x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)} = x_n-\frac{x_n^2-a}{2x_n} = \frac{1}{2}\left(x_n+\frac{a}{x_n}\right).</math>
This happens to coincide with the [[Methods of computing square roots#Heron's method|"Babylonian" method of finding square roots]], which consists of replacing an approximate root {{math|''x''<sub>''n''</sub>}} by the [[arithmetic mean]] of {{math|{{var|x}}{{sub|{{var|n}}}}}} and {{math|{{frac|{{var|a}}|{{var|x}}{{sub|{{var|n}}}}}}}}. By performing this iteration, it is possible to evaluate a square root to any desired accuracy by only using the basic [[arithmetic 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 "{{math|''x''<sub>''n''</sub>}}" column is obtained by applying the preceding formula to the entry above it, for instance
:<math>306.5 = \frac{1}{2} \left( 1+\frac{612}{1} \right).</math>

{| class=wikitable style="border: none;"
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''f''(''x''<sub>''n''</sub>)}}
| rowspan=7 style="border: none;"|
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''f''(''x''<sub>''n''</sub>)}}
| rowspan=6 style="border: none;"|
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''f''(''x''<sub>''n''</sub>)}}
|-
| 1 || −611 || 10 || −512 || <u>−2</u>0 || −212
|-
| 306.5 || 9.3330 × 10<sup>4</sup> || 35.6 || 655.36 || <u>−2</u>5.3 || 28.09 
|-
| 154.2483686786 || 2.3180 × 10<sup>4</sup> || <u>2</u>6.3955056180 || 84.722 || <u>−24.7</u>448616601 || 0.30818 
|-
| 79.1079978644 || 5.6461 × 10<sup>3</sup> || <u>24.7</u>906354925 || 2.5756 || <u>−24.73863</u>45374 || 3.8777 × 10<sup>−5</sup>
|- 
| 43.4221286822 || 1.2735 × 10<sup>3</sup> || <u>24.7386</u>882941 || 2.6985 × 10<sup>−3</sup> || <u>−24.7386337537</u> || 6.1424 × 10<sup>−13</sup>
|-
| <u>2</u>8.7581624288 || 215.03 || <u>24.738633753</u>8 || 2.9746 × 10<sup>−9</sup> || style="border-style:none;" | 
|-
| <u>2</u>5.0195385369 || 13.977
|-
| <u>24.7</u>402106712 || 7.8024 × 10<sup>−2</sup>
|-
| <u>24.738633</u>8040 || 2.4865 × 10<sup>−6</sup>
|-
| <u>24.7386337537</u> || 2.5256 × 10<sup>−15</sup>
|}

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 of {{math|1}}.

When computing any nonzero square root, the first derivative of {{mvar|f}} must be nonzero at the root, and that {{mvar|f}} 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.

=== Solution of {{math|1=cos({{var|x}}) = {{var|x}}{{sup|3}}}} using Newton's method ===
Consider the problem of finding the positive number {{mvar|x}} with {{math|1=cos {{var|x}} = {{var|x}}{{sup|3}}}}. We can rephrase that as finding the zero of {{math|1={{var|f}}({{var|x}}) = cos({{var|x}}) &minus; {{var|x}}{{sup|3}}}}. We have {{math|1={{var|{{prime|f}}}}({{var|x}}) = &minus;sin({{var|x}}) &minus; 3{{var|x}}{{sup|2}}}}. Since {{math|cos({{var|x}}) ≤ 1}} for all {{mvar|x}} and {{math|{{var|x}}{{sup|3}} > 1}} for {{math|{{var|x}} > 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 guess {{math|1={{var|x}}{{sub|0}} = 0.5}}, the sequence given by Newton's method is:

<math display="block">\begin{matrix}
 x_1 & = & x_0 - \dfrac{f(x_0)}{f'(x_0)} & = & 0.5 - \dfrac{\cos 0.5 - 0.5^3}{-\sin 0.5 - 3 \times 0.5^2} & = & 1.112\,141\,637\,097\dots \\
 x_2 & = & x_1 - \dfrac{f(x_1)}{f'(x_1)} & = & \vdots & = & \underline{0.}909\,672\,693\,736\dots \\
 x_3 & = & \vdots & = & \vdots & = & \underline{0.86}7\,263\,818\,209\dots \\
 x_4 & = & \vdots & = & \vdots & = & \underline{0.865\,47}7\,135\,298\dots \\
 x_5 & = & \vdots & = & \vdots & = & \underline{0.865\,474\,033\,1}11\dots \\
 x_6 & = & \vdots & = & \vdots & = & \underline{0.865\,474\,033\,102}\dots
\end{matrix}
</math>

The correct digits are underlined in the above example. In particular, {{math|{{var|x}}{{sub|6}}}} is correct to 12 decimal places. We see that the number of correct digits after the decimal point increases from 2 (for {{math|{{var|x}}{{sub|3}}}}) to 5 and 10, illustrating the quadratic convergence.

===Slow convergence===
The function {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}} has a root at 0.<ref name="dennis">J. E. Dennis, Jr. and Robert B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations. SIAM</ref> Since {{mvar|f}} 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 derivative {{math|''f'' ′}} is zero at the root, quadratic convergence is not ensured by the theory. In this particular example, the Newton iteration is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=\frac{1}{2}x_n.</math>
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 function {{math|''f''(''x'') {{=}} ''x'' + ''x''<sup>4/3</sup>}} also has a root at 0, where it is continuously differentiable. Although the first derivative {{mvar|''f'' ′}} is nonzero at the root, the second derivative {{mvar|''f'' ′′}} is nonexistent there, so that quadratic convergence cannot be guaranteed. In fact the Newton iteration is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=\frac{x^{4/3}}{3+4x^{1/3}}.</math>
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 above {{math|''f''(''x'') {{=}} ''x'' + ''x''<sup>4/3</sup>}} and the right of which shows Newton's method applied to {{math|''f''(''x'') {{=}} ''x'' + ''x''<sup>2</sup>}}. 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,...).

{| class=wikitable style="border: none;"
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''x'' + ''x''{{su|p=4/3|b=''n''}}}}
| rowspan=9 style="border: none;"|
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''x'' + ''x''{{su|p=2|b=''n''}}}}
|-
| 1 || 2 || 1 || 2
|-
| 1.4286 × 10<sup>−1</sup> || 2.1754 × 10<sup>−1</sup> || 3.3333 × 10<sup>−1</sup> || 4.4444 × 10<sup>−1</sup>
|-
| 1.4669 × 10<sup>−2</sup> || 1.8260 × 10<sup>−2</sup> || 6.6666 × 10<sup>−2</sup> || 7.1111 × 10<sup>−2</sup>
|-
| 9.0241 × 10<sup>−4</sup> || 9.8961 × 10<sup>−4</sup> || 3.9216 × 10<sup>−3</sup> || 3.9369 × 10<sup>−3</sup>
|-
| 2.5750 × 10<sup>−5</sup> || 2.6511 × 10<sup>−5</sup> || 1.5259 × 10<sup>−5</sup> || 1.5259 × 10<sup>−5</sup>
|-
| 2.4386 × 10<sup>−7</sup> || 2.4539 × 10<sup>−7</sup> || 2.3283 × 10<sup>−10</sup> || 2.3283 × 10<sup>−10</sup>
|-
| 5.0366 × 10<sup>−10</sup> || 5.0406 × 10<sup>−10</sup> || 5.4210 × 10<sup>−20</sup> || 5.4210 × 10<sup>−20</sup>
|-
| 1.3344 × 10<sup>−13</sup> || 1.3344 × 10<sup>−13</sup> || 2.9387 × 10<sup>−39</sup> || 2.9387 × 10<sup>−39</sup>
|}

The rate of convergence is distinguished from the number of iterations required to reach a given accuracy. For example, the function {{math|''f''(''x'') {{=}} ''x''<sup>20</sup> − 1}} has a root at 1. Since {{math|''f'' ′(1) ≠ 0}} and {{mvar|f}} 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.<ref>Anthony Ralston and Philip Rabinowitz. A first course in numerical analysis, second edition</ref>

===Convergence dependent on initialization===
The function {{math|''f''(''x'') {{=}} ''x''(1 + ''x''<sup>2</sup>)<sup>−1/2</sup>}} has a root at 0. The Newton iteration is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=-x_n^3.</math>
From this, it can be seen that there are three possible phenomena for a Newton iteration. If initialized strictly between {{math|±1}}, the Newton iteration will converge (super-)quadratically to 0; if initialized exactly at {{math|1}} or {{math|−1}}, the Newton iteration will oscillate endlessly between {{math|±1}}; if initialized anywhere else, the Newton iteration will diverge.<ref>Yuri Nesterov. Lectures on convex optimization, second edition. Springer Optimization and its Applications, Volume 137.</ref> This same trichotomy occurs for {{math|''f''(''x'') {{=}} arctan ''x''}}.<ref name="dennis" />

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 function {{math|''f''(''x'') {{=}} ''x''(''x''<sup>2</sup> − 1)(''x'' − 3)e<sup>−(''x'' − 1)<sup>2</sup>/2</sup>}} has roots at −1, 0, 1, and 3.{{sfnm|1a1=Süli|1a2=Mayers|1y=2003}} If initialized at −1.488, the Newton iteration converges to 0; if initialized at −1.487, it diverges to {{math|∞}}; if initialized at −1.486, it converges to −1; if initialized at −1.485, it diverges to {{math|−∞}}; if initialized at −1.4843, it converges to 3; if initialized at −1.484, it converges to {{math|1}}. This kind of subtle dependence on initialization is not uncommon; it is frequently studied in the [[complex plane]] in the form of the [[Newton fractal]].

===Divergence even when initialization is close to the root===

Consider the problem of finding a root of {{math|''f''(''x'') {{=}} ''x''<sup>1/3</sup>}}. The Newton iteration is
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=x_n-\frac{x_n^{1/3}}{\frac{1}{3}x_n^{-2/3}}=-2x_n.</math>
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 case {{mvar|f}} is not differentiable at its root.

In the above example, failure of convergence is reflected by the failure of {{math|''f''(''x''<sub>''n''</sub>)}} to get closer to zero as {{mvar|n}} increases, as well as by the fact that successive iterates are growing further and further apart. However, the function {{math|''f''(''x'') {{=}} ''x''<sup>1/3</sup>e<sup>−''x''<sup>2</sup></sup>}} also has a root at 0. The Newton iteration is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=x_n\left(1-\frac{3}{1-6x_n^2}\right).</math>
In this example, where again {{mvar|f}} is not differentiable at the root, any Newton iteration not starting exactly at the root will diverge, but with both {{math|''x''<sub>''n'' + 1</sub> − ''x''<sub>''n''</sub>}} and {{math|''f''(''x''<sub>''n''</sub>)}} converging to zero.<ref name="judd">Kenneth L. Judd. Numerical methods in economics. MIT Press</ref> This is seen in the following table showing the iterates with initialization 1:
{| class=wikitable style="border: none;"
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''f''(''x''<sub>''n''</sub>)}}
|-
| 1 || 0.36788
|-
| 1.6 || 9.0416 × 10<sup>−2</sup> 
|-
| 1.9342 || 2.9556 × 10<sup>−2</sup>
|-
| 2.2048 || 1.0076 × 10<sup>−2</sup>
|- 
| 2.4396 || 3.5015 × 10<sup>−3</sup>
|-
| 2.6505 || 1.2307 × 10<sup>−3</sup> 
|-
| 2.8437 || 4.3578 × 10<sup>−4</sup>
|-
| 3.0232 || 1.5513 × 10<sup>−4</sup>
|}

Although the convergence of {{math|''x''<sub>''n'' + 1</sub> − ''x''<sub>''n''</sub>}} 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 of {{math|''x''<sub>''n'' + 1</sub> − ''x''<sub>''n''</sub>}} and {{math|''f''(''x''<sub>''n''</sub>)}} might falsely identify a root.

===Oscillatory behavior===
[[Image:NewtonsMethodConvergenceFailure.svg|thumb|upright=1.4|The tangent lines of {{math|{{var|x}}{{sup|3}} − 2{{var|x}} + 2}} at 0 and 1 intersect the {{mvar|x}}-axis at 1 and 0 respectively, illustrating why Newton's method oscillates between these values for some starting points.]]

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 function {{mvar|f}} to oscillate between 0 and 1, it is only necessary that the tangent line to {{mvar|f}} at 0 intersects the {{mvar|x}}-axis at 1 and that the tangent line to {{mvar|f}} at 1 intersects the {{mvar|x}}-axis at 0.<ref name="judd" /> This is the case, for example, if {{math|''f''(''x'') {{=}} ''x''<sup>3</sup> − 2''x'' + 2}}. For this function, it is even the case that Newton's iteration as initialized sufficiently close to 0 or 1 will ''asymptotically'' 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 of {{mvar|f}} approximately equal to −1.76929.

===Undefinedness of Newton's method===
In some cases, it is not even possible to perform the Newton iteration. For example, if {{math|''f''(''x'') {{=}} ''x''<sup>2</sup> − 1}}, then the Newton iteration is defined by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} = \frac{x_n^2-1}{2x_n}.</math>
So Newton's method cannot be initialized at 0, since this would make {{math|''x''<sub>1</sub>}} undefined. Geometrically, this is because the tangent line to {{mvar|f}} at 0 is horizontal (i.e. {{math|''f'' ′(0) {{=}} 0}}), never intersecting the {{math|''x''}}-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.

If {{mvar|f}} 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.<ref name="judd" /> For example, the [[natural logarithm]] function {{math|''f''(''x'') {{=}} ln ''x''}} has a root at 1, and is defined only for positive {{mvar|x}}. Newton's iteration in this case is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=x_n(1- \ln x_n).</math>
So if the iteration is initialized at {{math|e}}, the next iterate is 0; if the iteration is initialized at a value larger than {{math|e}}, then the next iterate is negative. In either case, the method cannot be continued.

==Multidimensional formulations==
===Systems of equations===
===={{mvar|k}} variables, {{mvar|k}} functions{{anchor|multidimensional}}====
One may also use Newton's method to solve systems of {{mvar|k}} equations, which amounts to finding the (simultaneous) zeroes of {{mvar|k}} continuously differentiable functions <math>f:\R^k\to \R.</math> This is equivalent to finding the zeroes of a single vector-valued function <math>F:\R^k\to \R^k.</math> In the formulation given above, the scalars {{mvar|x<sub>n</sub>}} are replaced by vectors {{math|'''x'''{{sub|{{var|n}}}}}} and instead of dividing the function {{math|{{var|f}}({{var|x}}{{sub|{{var|n}}}})}} by its derivative {{math|{{var|{{prime|f}}}}({{var|x}}{{sub|{{var|n}}}})}} one instead has to left multiply the function {{math|{{var|F}}('''x'''{{sub|{{var|n}}}})}} by the inverse of its {{math|{{var|k}} × {{var|k}}}} [[Jacobian matrix]] {{math|{{var|J}}{{sub|{{var|F}}}}('''x'''{{sub|{{var|n}}}})}}.<ref name=":3">{{Cite book |last1=Burden |first1=Burton |url=https://archive.org/details/numericalanaly00burd/ |title=Numerical Analysis |last2=Fairs |first2=J. Douglas |last3=Reunolds |first3=Albert C |date=July 1981 |publisher=Prindle, Weber & Schmidt |isbn=0-87150-314-X |edition=2nd |location=Boston, MA, United States |oclc=1036752194 |pages=448–452 |language=en}}</ref><ref>{{Cite book |last= Evans |first=Gwynne A. |url-access= registration|url=https://archive.org/details/practicalnumeric0000evan/ |title=Practical Numerical Analysis |date=1995 |publisher=John Wiley & Sons|isbn=0471955353 |location= Chichester |publication-date=1995 |pages=30–33 |language=en | oclc=1319419671 }}</ref><ref>{{Cite book |last1=Demidovich |first1=Boris Pavlovich |url=https://archive.org/details/computational-mathematics/mode/2up |title=Computational Mathematics |last2=Maron |first2=Isaak Abramovich |date=1981 |publisher=MIR Publishers |isbn=9780828507042 |edition=Third |location=Moscow  |pages=460–478 |language=en}}</ref> This results in the expression

<math display="block">\mathbf{x}_{n+1} = \mathbf{x}_{n} - J_F(\mathbf{x}_n)^{-1} F(\mathbf{x}_n) .</math>

or, by solving the [[system of linear equations]]

<math display="block">J_F(\mathbf{x}_n) (\mathbf{x}_{n+1} - \mathbf{x}_n) = -F(\mathbf{x}_n)</math>

for the unknown {{math|'''x'''{{sub|{{var|n}} + 1}} − '''x'''{{sub|{{var|n}}}}}}.<ref>{{cite book |last1=Kiusalaas |first1=Jaan |title=Numerical Methods in Engineering with Python 3 |date=March 2013 |publisher=Cambridge University Press |location=New York |isbn=978-1-107-03385-6 |pages=175–176 |edition=3rd |url=https://www.cambridge.org/9781107033856}}</ref>

===={{mvar|k}} variables, {{mvar|m}} equations, with {{math|{{var|m}} > {{var|k}}}}====
The {{mvar|k}}-dimensional variant of Newton's method can be used to solve systems of greater than {{mvar|k}} (nonlinear) equations as well if the algorithm uses the [[generalized inverse]] of the non-square [[Jacobian matrix and determinant|Jacobian]] matrix {{math|{{var|J}}{{isup|+}} {{=}} ({{var|J}}{{isup|T}}{{var|J}}){{sup|−1}}{{var|J}}{{isup|T}}}} instead of the inverse of {{mvar|J}}. If the [[system of nonlinear equations|nonlinear system]] has no solution, the method attempts to find a solution in the [[non-linear least squares]] sense. See [[Gauss–Newton algorithm]] for more information.

==== Example ====
For example, the following set of equations needs to be solved for vector of points <math>\ [\ x_1, x_2\ ]\ ,</math> given the vector of known values <math>\ [\ 2, 3\ ] ~.</math>{{refn | This example is similar to one in reference,<ref name=":3" /> pages 451 and 452, but simplified to two equations instead of three.}}

<math>
\begin{array}{lcr} 
5\ x_1^2 + x_1\ x_2^2 + \sin^2( 2\ x_2 ) &= \quad 2  \\ 
e^{ 2\ x_1 - x_2 } + 4\ x_2              &= \quad 3 
\end{array}</math>

the function vector, <math>\ F (X_k)\ ,</math> and Jacobian Matrix, <math>\ J(X_k)\ </math> for iteration k, and the vector of known values, <math>\ Y\ ,</math> are defined below.

<math>\begin{align}
~ & F(X_k) ~ = ~
\begin{bmatrix} 
\begin{align}
~ & f_{1}(X_{k}) \\ 
~ & f_{2}(X_{k}) 
\end{align}
\end{bmatrix} ~ = ~
\begin{bmatrix} 
\begin{align}
~ & 5\ x_{1}^2 + x_{1}\ x^2_{2} + \sin^2( 2\ x_{2} ) \\ 
~ & e^{ 2\ x_{1}-x_{2} } + 4\ x_{2} 
\end{align}
\end{bmatrix}_k
\\
~ & J(X_k) = 
\begin{bmatrix} 
~ \frac{\ \partial{ f_{1}(X) }\ }{ \partial{x_{1}} }\ , & ~ \frac{\ \partial{ f_{1}(X) }\ }{ \partial{x_{2}} } ~\\
~ \frac{\ \partial{ f_{2}(X) }\ }{ \partial{x_{1}} }\ , & ~ \frac{\ \partial{ f_{2}(X) }\ }{ \partial{x_{2}} } ~
\end{bmatrix}_k
~ = ~
\begin{bmatrix} 
\begin{align}
~ & 10\ x_{1} + x^2_{2}\ ,  & & 2\ x_1\ x_2+4\ \sin( 2\ x_{2} )\ \cos( 2\ x_{2} ) \\ 
~ & 2\ e^{ 2\ x_{1} - x_{2} }\ ,  & &-e^{ 2\ x_{1} - x_{2}} + 4
\end{align}
\end{bmatrix}_k
\\
~ & Y = 
 \begin{bmatrix}~ 2 ~\\~ 3 ~\end{bmatrix}

\end{align} </math>

Note that <math>\ F(X_k)\ </math> could have been rewritten to absorb <math>\ Y\ ,</math> and thus eliminate  <math>Y</math> from the equations. The equation to solve for each iteration are

<math>\begin{align}
\begin{bmatrix}
\begin{align}
~ & ~ 10\ x_{1} + x^2_{2 }\ , & & 2 x_1 x_2 + 4\ \sin( 2\ x_{2} )\ \cos( 2\ x_{2} ) ~\\
~ & ~ 2\ e^{ 2\ x_{1} - x_{2} }\ ,  & & -e^{ 2\ x_{1} - x_{2} } + 4 ~
\end{align}
\end{bmatrix}_k

\begin{bmatrix}
~ c_{1} ~\\
~ c_{2} ~
\end{bmatrix}_{k+1}
=
\begin{bmatrix}
~ 5\ x_{1}^2 + x_{1}\ x^2_{2} + \sin^2( 2\ x_{2} ) - 2 ~\\
~ e^{ 2\ x_{1} - x_{2} } + 4\ x_{2} - 3 ~
\end{bmatrix}_k

\end{align}</math>

and

<math> X_{ k+1 } ~=~ X_k - C_{ k+1 } </math>

The iterations should be repeated until <math>\ \Bigg[ \sum_{i=1}^{i=2} \Bigl| f(x_i)_k - (y_i)_k \Bigr|\Bigg] < E\ ,</math> where <math>\ E\ </math> is a value acceptably small enough to meet application requirements.

If vector <math>\ X_0\ </math> is initially chosen to be <math>\ \begin{bmatrix}~ 1 ~&~ 1 ~\end{bmatrix}\ ,</math> that is, <math>\ x_1 = 1\ ,</math> and <math>\ x_2=1\ ,</math> and <math>\ E\ ,</math> is chosen to be 1.{{10^|−3}}, then the example converges after four iterations to a value of <math>\ X_4 = \left[~ 0.567297,\ -0.309442 ~\right] ~.</math>

==== Iterations ====
The following iterations were made during the course of the solution.
:{| class="wikitable"
|+ Converging iteration sequence
|- style="vertical-align:bottom;" 
! Step
! Variable
! {{left|Value}}
|-
|rowspan="2"; align="center";| {{math| 0 }}
|align="right";| {{mvar| x {{=}} }}
| <math> \begin{bmatrix}\ 1\ , & 1 \end{bmatrix} </math>
|-
|align="right";| {{math| ''f''(''x'') {{=}} }}
|<math> \begin{bmatrix}\ 6.82682\ , & 6.71828\ \end{bmatrix} </math>
|-
|colspan="3" ; style="background:white;"| 
|-
|rowspan="4"; align="center";| {{math| 1 }}
|align="right";| {{mvar| J  {{=}} }}
| <math> \begin{bmatrix}\ 11 ~,      & \quad 0.486395 \\
                       \  5.43656\ , & 1.28172 \end{bmatrix} </math>
|-
|align="right";| {{mvar| c {{=}} }}
| <math>\begin{bmatrix}\ 0.382211\ , & 1.27982\ \end{bmatrix} </math>
|-
|align="right";| {{mvar| x {{=}} }}
| <math>\begin{bmatrix}\ 0.617789\ , & -0.279818\ \end{bmatrix} </math>
|-
|align="right";| {{math| ''f''(''x'')  {{=}} }}
| <math>\begin{bmatrix}\ 2.23852\ , & 3.43195\ \end{bmatrix} </math>
|-
|colspan="3" ; style="background:white;"|
|-
|rowspan="4"; align="center";| {{math| 2 }}
|align="right";| {{mvar| J {{=}} }}
| <math>\begin{bmatrix}\ 6.25618\ , & -2.1453 \\
                      \  9.10244\ , &\quad  -0.551218 \end{bmatrix} </math>
|-
|align="right";| {{mvar| c {{=}} }}
| <math>\begin{bmatrix} 0.0494549\ , & 0.0330411\ \end{bmatrix} </math>
|-
|align="right";| {{mvar| x {{=}} }}
| <math>\begin{bmatrix}\ 0.568334\ , & -0.312859\ \end{bmatrix} </math>
|-
|align="right";| {{math| ''f''(''x'')  {{=}} }}
| <math>\begin{bmatrix}\ 2.01366\ , & 3.00966\ \end{bmatrix} </math>
|-
|colspan="3" ; style="background:white;"|
|-
|rowspan="4"; align="center";| {{math| 3 }}
|align="right";| {{mvar| J {{=}} }}
| <math>\begin{bmatrix}\ 5.78122\ , & -2.25449 \\
                      \  8.52219\ , &\quad  -0.261095\ \end{bmatrix} </math>
|-
|align="right";| {{mvar| c {{=}} }}
| <math>\begin{bmatrix} 0.00102862\ , & -0.00342339\ \end{bmatrix} </math>
|-
|align="right";| {{mvar| x {{=}} }}
| <math>\begin{bmatrix}\ 0.567305\ , & -0.309435\ \end{bmatrix} </math>
|-
|align="right";| {{math| ''f''(''x'')  {{=}} }}
| <math>\begin{bmatrix}\ 2.00003\ , & 3.00006\ \end{bmatrix} </math>
|-
|colspan="3" ; style="background:white;"|
|-
|rowspan="4"; align="center";| {{math| 4 }}
|align="right";| {{mvar| J {{=}} }}
| <math>\begin{bmatrix}\ 5.7688~ ,  & ~ -2.24118 \\
                      \  8.47561\ , &\quad -0.237805 \end{bmatrix}\ </math>
|-
|align="right";| {{mvar| c  {{=}} }}
| <math>\begin{bmatrix}\ 7.73132\!\times\!10^{-6} ~, & ~ 6.93265\!\times\!10^{-6}\ \end{bmatrix} </math>
|-
|align="right";| {{mvar| x  {{=}} }}
| <math>\begin{bmatrix}\ 0.567297\ , & -0.309442\ \end{bmatrix} </math>
|-
|align="right";| {{math| ''f''(''x'')  {{=}} }}
| <math>\begin{bmatrix} ~ 2\ ,~ & ~ 3 ~ \end{bmatrix} </math>
|}

===Complex functions===
{{main|Newton fractal}}
[[Image:newtroot 1 0 0 0 0 m1.png|thumb|Basins of attraction for {{math|{{var|x}}{{sup|5}} − 1 {{=}} 0}}; darker means more iterations to converge.]]
When dealing with [[complex analysis|complex functions]], Newton's method can be directly applied to find their zeroes.<ref>{{cite book |last=Henrici|author-link=Peter Henrici (mathematician)|first=Peter |title= Applied and Computational Complex Analysis |volume=1 |date=1974 |publisher=Wiley | isbn =9780471598923  }}</ref> Each zero has a [[basin 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 are [[fractal]]s.

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,<ref>{{cite journal|last=Strang |first=Gilbert |title=A chaotic search for {{mvar|i}} |journal=[[The College Mathematics Journal]] |volume=22 |date=January 1991 |issue=1 |pages=3–12 |doi=10.2307/2686733|jstor=2686733 }}</ref> if one uses a real initial condition to seek a root of {{math|{{var|x}}{{sup|2}} + 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 case [[almost all]] real initial conditions lead to [[chaos theory|chaotic 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.<ref>{{cite journal|last=McMullen |first=Curt |title=Families of rational maps and iterative root-finding algorithms |journal=Annals of Mathematics |series=Second Series |volume=125 |date=1987 |issue=3 |pages=467–493 |doi=10.2307/1971408|jstor=1971408 |url=https://dash.harvard.edu/bitstream/handle/1/9876064/McMullen_FamiliesRationalMap.pdf?sequence=1 }}</ref> 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.<ref>{{Cite journal |last1=Hubbard |first1=John |last2=Schleicher |first2=Dierk |last3=Sutherland |first3=Scott |date=October 2001 |title=How to find all roots of complex polynomials by Newton's method |url=http://dx.doi.org/10.1007/s002220100149 |journal=Inventiones Mathematicae |volume=146 |issue=1 |pages=1–33 |doi=10.1007/s002220100149 |bibcode=2001InMat.146....1H |s2cid=12603806 |issn=0020-9910}}</ref>

===In a Banach space===
Another generalization is Newton's method to find a root of a [[Functional (mathematics)|functional]] {{mvar|F}} defined in a [[Banach space]]. In this case the formulation is

<math display="block">X_{n+1}=X_n-\bigl(F'(X_n)\bigr)^{-1}F(X_n),\,</math>

where {{math|{{var|{{prime|F}}}}({{var|X}}{{sub|{{var|n}}}})}} is the [[Fréchet derivative]] computed at {{math|{{var|X}}{{sub|{{var|n}}}}}}. One needs the Fréchet derivative to be boundedly invertible at each {{math|{{var|X}}{{sub|{{var|n}}}}}} in order for the method to be applicable. A condition for existence of and convergence to a root is given by the [[Kantorovich theorem|Newton–Kantorovich theorem]].<ref>{{cite book |first=Tetsuro |last=Yamamoto |chapter=Historical Developments in Convergence Analysis for Newton's and Newton-like Methods |pages=241–263 |editor-first=C. |editor-last=Brezinski |editor2-first=L. |editor2-last=Wuytack |title=Numerical Analysis: Historical Developments in the 20th Century |publisher=North-Holland |year=2001 |isbn=0-444-50617-9 }}</ref>

====Nash–Moser iteration====
{{details|Nash–Moser theorem}}
In the 1950s, [[John Forbes Nash Jr.|John Nash]] developed a version of the Newton's method to apply to the problem of constructing [[isometric embedding]]s of general [[Riemannian manifold]]s in [[Euclidean space]]. The ''loss 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 of [[smoothing]] operators into the iteration. He was able to prove the convergence of his smoothed Newton method, for the purpose of proving an [[implicit 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 in [[celestial mechanics]]. Since then, a number of mathematicians, including [[Mikhael Gromov (mathematician)|Mikhael Gromov]] and [[Richard S. Hamilton|Richard Hamilton]], have found generalized abstract versions of the Nash–Moser theory.<ref>{{cite journal|first=Richard S.|last=Hamilton|mr=0656198|title=The inverse function theorem of Nash and Moser|journal=[[Bulletin of the American Mathematical Society]] |series=New Series |volume=7|year=1982|issue=1|pages=65–222|doi-access=free|doi=10.1090/s0273-0979-1982-15004-2|zbl=0499.58003|author-link1=Richard S. Hamilton}}</ref><ref>{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|author-link1=Mikhael Gromov (mathematician)|doi=10.1007/978-3-662-02267-2}}</ref> In Hamilton's formulation, the Nash–Moser theorem forms a generalization of the Banach space Newton method which takes place in certain [[Fréchet space]]s.

==Modifications==
===Quasi-Newton methods===
When the Jacobian is unavailable or too expensive to compute at every iteration, a [[quasi-Newton method]] can be used.

===Chebyshev's third-order method===
{{expand section|date=February 2019}}
{{Main|Chebyshev iteration}}

===Over {{mvar|p}}-adic numbers===
In {{mvar|p}}-adic analysis, the standard method to show a polynomial equation in one variable has a {{mvar|p}}-adic root is [[Hensel's lemma]], which uses the recursion from Newton's method on the {{mvar|p}}-adic numbers. Because of the more stable behavior of addition and multiplication in the {{mvar|p}}-adic numbers compared to the real numbers (specifically, the unit ball in the {{mvar|p}}-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.

==={{mvar|q}}-analog===
Newton's method can be generalized with the [[q-analog|{{mvar|q}}-analog]] of the usual derivative.<ref>{{ cite journal| journal=Matematicki Vesnik| volume=54|number=3–4| pages=171–178 |date=2002| title=Mean value theorems in $q$-calculus| first1=Predrag M. | last1= Rajković|first2= Miomir S. | last2= Stanković| first3= Slađana D. | last3=  Marinković |url=https://www.emis.de/journals/MV/0234/16.html}}</ref>

===Modified Newton methods===
====Maehly's procedure====
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 found {{mvar|N}} solutions of <math>f(x)=0</math>, then the next root can be found by applying Newton's method to the next equation:<ref>{{harvnb|Press|Teukolsky|Vetterling|Flannery|2007}}</ref><ref>{{cite book | url=https://archive.org/details/introductiontonu0000stoe/mode/2up | page=279 | last1=Stoer|last2=Bulirsch|date=1980| first1= Josef | first2= Roland | oclc= 1244842246 | title= Introduction to numerical analysis | url-access= registration}}</ref>

<math display="block">F(x) = \frac{f(x)}{\prod_{i=1}^N(x-x_i)} = 0 .</math>

This method is applied to obtain zeros of the [[Bessel function]] of the second kind.<ref>{{cite book <!-- Citation bot: don't add identifiers. there is a bogus doi at the s2cid for this title that actually points to a review of the book, and i know from experience that bots will conflate the two and bugger up the reference -->| last1= Zhang | first1= Shanjie | last2= Jin | first2= Jianming | date= 1996 | publisher= Wiley | title= Computation of Special Functions|isbn=9780471119630}}{{page needed |date=June 2024}}</ref>

====Hirano's modified Newton method====
Hirano's modified Newton method is a modification conserving the convergence of Newton method and avoiding unstableness.<ref>{{cite journal |first=Kazuo |last=Murota |date=1982 |doi=10.1137/0719055 |title=Global Convergence of a Modified Newton Iteration for Algebraic Equations |journal=SIAM Journal on Numerical Analysis |volume=19 |issue=4 |pages=793–799|bibcode=1982SJNA...19..793M }}</ref> It is developed to solve complex polynomials.

====Interval Newton's method====
{{cite check|section|date=February 2019}}
Combining Newton's method with [[interval 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 insufficient [[floating-point arithmetic|floating-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; see [[Wilkinson's polynomial]]).<ref>Moore, R. E. (1979). ''Methods and applications of interval analysis'' (Vol. 2). Siam.</ref><ref>Hansen, E. (1978). Interval forms of Newtons method. ''Computing'', 20(2), 153–163.</ref>

Consider {{math|{{var|f}} → {{mathcal|C}}{{sup|1}}({{var|X}})}}, where {{mvar|X}} is a real interval, and suppose that we have an interval extension {{mvar|{{prime|F}}}} of {{mvar|{{prime|f}}}}, meaning that {{mvar|{{prime|F}}}} takes as input an interval {{math|{{var|Y}} ⊆ {{var|X}}}} and outputs an interval {{math|{{var|{{prime|F}}}}({{var|Y}})}} such that:

<math display="block">\begin{align}
    F'([y,y]) &= \{f'(y)\}\\[5pt]
    F'(Y) &\supseteq \{f'(y)\mid y \in Y\}.
\end{align}</math>

We also assume that {{math|0 ∉ {{var|{{prime|F}}}}({{var|X}})}}, so in particular {{mvar|f}} has at most one root in {{mvar|X}}.
We then define the interval Newton operator by:

<math display="block">N(Y) = m - \frac{f(m)}{F'(Y)} = \left\{\left.m - \frac{f(m)}{z} ~\right|~ z \in F'(Y)\right\}</math>

where {{math|{{var|m}} ∈ {{var|Y}}}}. Note that the hypothesis on {{mvar|{{prime|F}}}} implies that {{math|{{var|N}}({{var|Y}})}} is well defined and is an interval (see [[interval arithmetic]] for further details on interval operations). This naturally leads to the following sequence:

<math display="block">
\begin{align}
X_0 &= X\\
X_{k+1} &= N(X_k) \cap X_k.
\end{align}
</math>

The [[mean value theorem]] ensures that if there is a root of {{mvar|f}} in {{math|{{var|X}}{{sub|{{var|k}}}}}}, then it is also in {{math|{{var|X}}{{sub|{{var|k}} + 1}}}}. Moreover, the hypothesis on {{mvar|F′}} ensures that {{math|{{var|X}}{{sub|{{var|k}} + 1}}}} is at most half the size of {{math|{{var|X}}{{sub|{{var|k}}}}}} when {{mvar|m}} is the midpoint of {{mvar|Y}}, so this sequence converges towards {{math|[{{var|x*}}, {{var|x*}}]}}, where {{mvar|x*}} is the root of {{mvar|f}} in {{mvar|X}}.

If {{math|{{var|{{prime|F}}}}({{var|X}})}} strictly contains 0, the use of extended interval division produces a union of two intervals for {{math|{{var|N}}({{var|X}})}}; multiple roots are therefore automatically separated and bounded.

==Applications==
===Minimization and maximization problems===
{{main|Newton's method in optimization}}
Newton's method can be used to find a minimum or maximum of a function {{math|{{var|f}}({{var|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.<ref>{{cite book|mr=2061575|title=Convex optimization|last1=Boyd|first1=Stephen|last2=Vandenberghe|first2=Lieven|publisher=[[Cambridge University Press]]|location=Cambridge|year=2004|isbn=0-521-83378-7|author-link1=Stephen P. Boyd|doi=10.1017/CBO9780511804441|zbl=1058.90049}}</ref> The iteration becomes:

<math display="block">x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)}. </math>

===Multiplicative inverses of numbers and power series===
An important application is [[Division algorithm#Newton–Raphson division|Newton–Raphson division]], which can be used to quickly find the [[Multiplicative inverse|reciprocal]] of a number {{mvar|a}}, using only multiplication and subtraction, that is to say the number {{mvar|x}} such that {{math|1={{sfrac|1|{{var|x}}}} = {{var|a}}}}. We can rephrase that as finding the zero of {{math|1={{var|f}}({{var|x}}) = {{sfrac|1|{{var|x}}}} − {{var|a}}}}. We have {{math|1={{var|{{prime|f}}}}({{var|x}}) = −{{sfrac|1|{{var|x}}{{sup|2}}}}}}.

Newton's iteration is

<math display="block">x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)} = x_n+\frac{\frac{1}{x_n}-a}{\frac{1}{x_n^2}} = x_n(2-ax_n).
</math>

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

This method is also very efficient to compute the multiplicative inverse of a [[power series]].

===Solving transcendental equations===
Many [[transcendental equation]]s can be solved up to an arbitrary precision by using Newton's method. For example, finding the cumulative [[probability density function]], such as a [[Normal 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 the [[Normal distribution#Using the Taylor series and Newton's method for the inverse function|inverse Normal cumulative distribution]].

===Numerical verification for solutions of nonlinear equations===
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.{{cn|date=June 2024}}

==Code==
The following is an example of a possible implementation of Newton's method in the [[Python (programming language)|Python]] (version 3.x) programming language for finding a root of a function <code>f</code> which has derivative <code>f_prime</code>.

The initial guess will be {{math|1={{var|x}}{{sub|0}} = 1}} and the function will be {{math|1={{var|f}}({{var|x}}) = {{var|x}}{{sup|2}} − 2}} so that {{math|1={{var|{{prime|f}}}}({{var|x}}) = 2{{var|x}}}}.

Each new iteration of Newton's method will be denoted by <code>x1</code>. We will check during the computation whether the denominator (<code>yprime</code>) becomes too small (smaller than <code>epsilon</code>), which would be the case if {{math|{{var|{{prime|f}}}}({{var|x}}{{sub|{{var|n}}}}) ≈ 0}}, since otherwise a large amount of error could be introduced.

<syntaxhighlight lang="python3" line="1">
def f(x):             
	return x**2 - 2   # f(x) = x^2 - 2

def f_prime(x):
	return 2*x        # f'(x) = 2x

def newtons_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 _ in range(max_iterations):
        y = f(x0)
        yprime = f_prime(x0)

        if abs(yprime) < epsilon:       # Give up if the denominator is too small
            break

        x1 = x0 - y / yprime            # Do Newton's computation

        if abs(x1 - x0) <= tolerance:   # Stop when the result is within the desired tolerance
            return x1                   # x1 is a solution within tolerance and maximum number of iterations

        x0 = x1                         # Update x0 to start the process again

    return None                         # Newton's method did not converge
</syntaxhighlight>

==See also==
{{Div col|colwidth=20em}}
* [[Aitken's delta-squared process]]
* [[Bisection method]]
* [[Euler method]]
* [[Fast inverse square root]]
* [[Scoring algorithm#Fisher scoring|Fisher scoring]]
* [[Gradient descent]]
* [[Integer square root]]
* [[Kantorovich theorem]]
* [[Laguerre's method]]
* [[Methods of computing square roots]]
* [[Newton's method in optimization]]
* [[Richardson extrapolation]]
* [[Root-finding algorithm]]
* [[Secant method]]
* [[Steffensen's method]]
* [[Subgradient method]]
{{Div col end}}

==Notes==

{{reflist}}

==References==
* {{cite book |last1=Gil |first1=A. |last2=Segura |first2=J. |last3=Temme |first3=N. M. |url=https://www.researchgate.net/publication/220693008 |title=Numerical methods for special functions |isbn=978-0-89871-634-4 |date=2007 |publisher=Society for Industrial and Applied Mathematics }}
* {{cite book |author1-link=Endre Süli |first1=Endre |last1=Süli |first2=David |last2=Mayers |title=An Introduction to Numerical Analysis |publisher=Cambridge University Press |date=2003 |isbn=0-521-00794-1 }}

==Further reading==
* Kendall E. Atkinson, ''An Introduction to Numerical Analysis'', (1989) John Wiley & Sons, Inc, {{isbn|0-471-62489-6}}
* Tjalling J. Ypma, Historical development of the Newton–Raphson method, ''SIAM Review'' '''37''' (4), 531–551, 1995. {{doi|10.1137/1037125}}.
* {{cite book|last1=Bonnans|first1=J.&nbsp;Frédéric|last2=Gilbert|first2=J.&nbsp;Charles|last3=Lemaréchal|first3=Claude| author-link3=Claude Lemaréchal|last4=Sagastizábal|first4=Claudia&nbsp;A.|author4-link= Claudia Sagastizábal |title=Numerical optimization: Theoretical and practical aspects|url=https://www.springer.com/mathematics/applications/book/978-3-540-35445-1|edition=Second revised ed. of translation of 1997 <!-- ''Optimisation numérique: Aspects théoriques et pratiques'' --> French| series=Universitext|publisher=Springer-Verlag|location=Berlin|year=2006|pages=xiv+490|isbn=3-540-35445-X|doi=10.1007/978-3-540-35447-5|mr=2265882}}
* P. Deuflhard, ''Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms.'' Springer Series in Computational Mathematics, Vol. 35. Springer, Berlin, 2004. {{isbn|3-540-21099-7}}.
* C. T. Kelley, ''Solving Nonlinear Equations with Newton's Method'', no 1 in Fundamentals of Algorithms, SIAM, 2003. {{isbn|0-89871-546-6}}.
* J. M. Ortega, W. C. Rheinboldt, ''Iterative Solution of Nonlinear Equations in Several Variables.'' Classics in Applied Mathematics, SIAM, 2000. {{isbn|0-89871-461-3}}.
*{{Cite book |last1=Press |first1=W. H. |last2=Teukolsky |first2=S. A. |last3=Vetterling |first3=W. T. |last4=Flannery |first4=B. P. |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=Cambridge University Press |location=New York |isbn=978-0-521-88068-8 |chapter=Chapter 9. Root Finding and Nonlinear Sets of Equations Importance Sampling |chapter-url=http://apps.nrbook.com/empanel/index.html#pg=442}}. See especially Sections [http://apps.nrbook.com/empanel/index.html#pg=456 9.4], [http://apps.nrbook.com/empanel/index.html#pg=473 9.6], and [http://apps.nrbook.com/empanel/index.html#pg=477 9.7].
* {{cite book |first=Mordecai |last=Avriel |title=Nonlinear Programming: Analysis and Methods |publisher=Prentice Hall |year=1976 |isbn=0-13-623603-0 |pages=216–221 }}

==External links==
{{Commons category|Newton Method}}
{{Wikibooks category|Newton's method}}
* {{springer|title=Newton method|id=p/n066560}}
*{{MathWorld|title=Newton's Method|urlname=NewtonsMethod}}
*[http://en.citizendium.org/wiki/Newton%27s_method Newton's method, Citizendium.]
*[https://web.archive.org/web/20190524083302/http://mathfaculty.fullerton.edu/mathews/n2003/NewtonAccelerateMod.html Mathews, J., The Accelerated and Modified Newton Methods, Course notes.]
*[http://www.ece.mcmaster.ca/~xwu/part2.pdf Wu, X., Roots of Equations, Course notes.]

{{Isaac Newton}}
{{Optimization algorithms}}
{{Root-finding algorithms}}
{{Authority control}}

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