In
numerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
,
Newton's method (also known as the
Newton–Raphson method), named after
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and
Joseph RaphsonJoseph Raphson was an English mathematician known best for the Newton–Raphson method. Little is known about his life, and even his exact years of birth and death are unknown, although the mathematical historian Florian Cajori provided the approximate dates 1648–1715. Raphson attended...
, is a method for finding successively better approximations to the roots (or zeroes) of a
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
valued
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. The algorithm is first in the class of
Householder's methodIn numerical analysis, the class of Householder's methods are rootfinding algorithms used for functions of one real variable with continuous derivatives up to some order d+1, where d will be the order of the Householder's method....
s, succeeded by
Halley's methodIn numerical analysis, Halley’s method is a rootfinding algorithm used for functions of one real variable with a continuous second derivative, i.e., C2 functions. It is named after its inventor Edmond Halley, who also discovered Halley's Comet....
. The method can also be extended to complex functions and to systems of equations.
The NewtonRaphson method in one variable is implemented as follows:
Given a function ƒ defined over the reals x, and its
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
ƒ
', we begin with a first guess x
_{0} for a root of the function f. Provided the function is reasonably wellbehaved a better approximation x
_{1} is
Geometrically, x
_{1} is the intersection with the xaxis of a line tangent to f at f (x
_{0}).
The process is repeated until a sufficiently accurate value is reached:
Description
The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of
calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
), and one computes the xintercept of this tangent line (which is easily done with elementary algebra). This xintercept will typically be a better approximation to the function's root than the original guess, and the method can be
iteratedIn computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
.
Suppose ƒ : [a, b] →
R is a
differentiableIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
function defined on the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[a, b] with values in the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s
R. The formula for converging on the root can be easily derived. Suppose we have some current approximation x
_{n}. Then we can derive the formula for a better approximation, x
_{n+1} by referring to the diagram on the right. We know from the definition of the derivative at a given point that it is the slope of a tangent at that point.
That is
Here, f ' denotes the
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the function f. Then by simple algebra we can derive
We start the process off with some arbitrary initial value x
_{0}. (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 theoremIn mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that ƒ(x
_{0}) ≠ 0. Furthermore, for a zero of
multiplicityIn mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....
1, the convergence is at least quadratic (see
rate of convergenceIn numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of...
) in a
neighbourhoodIn topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of the zero, which intuitively means that the number of correct digits roughly at least doubles in every step. More details can be found in the analysis section below.
The
Householder's methodIn numerical analysis, the class of Householder's methods are rootfinding algorithms used for functions of one real variable with continuous derivatives up to some order d+1, where d will be the order of the 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 f or its derivatives are computationally expensive to evaluate.
History
Newton's method was described by
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by
William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as
Method of FluxionsMethod of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
in 1736 by
John ColsonJohnathan "John" Colson was an English clergyman and mathematician, Lucasian Professor of Mathematics at Cambridge University.John Colson was educated at Lichfield School before becoming an undergraduate at Christ Church, Oxford, though he did not take a degree there...
). However, his description differs substantially from the modern description given above: Newton applies the method only to polynomials. He does not compute the successive approximations
, but computes a sequence of polynomials and only at the end, he arrives at an approximation for the root x. Finally, Newton views the method as purely algebraic and fails to notice the connection with calculus. Isaac Newton probably derived his method from a similar but less precise method by Vieta. The essence of Vieta's method can be found in the work of the Persian mathematician, Sharaf alDin alTusi, while his successor
Jamshīd alKāshīGhiyāth alDīn Jamshīd Masʾūd alKāshī was a Persian astronomer and mathematician.Biography:...
used a form of Newton's method to solve
to find roots of N (Ypma 1995). A special case of Newton's method for calculating square roots was known much earlier and is often called the Babylonian method.
Newton's method was used by 17th century Japanese mathematician Seki Kōwa to solve singlevariable equations, though the connection with calculus was missing.
Newton's method was first published in 1685 in A Treatise of Algebra both Historical and Practical by
John Wallis. In 1690,
Joseph RaphsonJoseph Raphson was an English mathematician known best for the Newton–Raphson method. Little is known about his life, and even his exact years of birth and death are unknown, although the mathematical historian Florian Cajori provided the approximate dates 1648–1715. Raphson attended...
published a simplified description in Analysis aequationum universalis. Raphson again viewed Newton's method purely as an algebraic method and restricted its use to polynomials, but he describes the method in terms of the successive approximations x
_{n} instead of the more complicated sequence of polynomials used by Newton. Finally, in 1740,
Thomas SimpsonThomas Simpson FRS was a British mathematician, inventor and eponym of Simpson's rule to approximate definite integrals...
described Newton's method as an iterative method for solving general nonlinear equations using fluxional 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 CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
in 1879 in The NewtonFourier imaginary problem was the first who noticed the difficulties in generalizing the 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 theory of iterations of rational functions.
Practical considerations
Newton's method is an extremely powerful technique—in general the
convergenceIn numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of...
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 derivative of a function
Newton's method requires that the derivative be calculated directly. An analytical expression for the derivative may not be easily obtainable and 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 methodIn numerical analysis, the secant method is a rootfinding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method. However, the method was developed...
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 of Newton's Method before implementing it. Specifically, one should review the assumptions made in the proof. For situations where the method fails to converge, it is because the assumptions made in this proof are not met.
Overshoot
If the first derivative is not well behaved in the neighborhood of the root, the method may overshoot, and diverge from the desired root. Furthermore, if a
stationary pointIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
of the function is encountered, the derivative is zero and the method will terminate due to
division by zeroIn mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a welldefined value depends upon the mathematical setting...
.
Poor initial estimate
A large error in the initial estimate can contribute to nonconvergence of the algorithm.
Mitigation of nonconvergence
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 > 1
If the root being sought has 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
of the root is known, one can use the following modified algorithm that preserves the quadratic convergence rate:
Analysis
Suppose that the function ƒ has a zero at α, i.e., ƒ(α) = 0.
If f is continuously differentiable and its derivative is nonzero at α, then there exists a neighborhood of α such that for all starting values x
_{0} in that neighborhood, the
sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
{x
_{n}} will
convergeThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
to α.
If the function is continuously differentiable and its derivative is not 0 at α and it has a
second derivativeIn calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
at α then the convergence is quadratic or faster. If the second derivative is not 0 at α then the convergence is merely quadratic. If the third derivative exists and is bounded in a neighborhood of α, then:
where
If the derivative is 0 at α, then the convergence is usually only linear. Specifically, if ƒ is twice continuously differentiable, ƒ
'(α) = 0 and ƒ
(α) ≠ 0, then there exists a neighborhood of α such that for all starting values x
_{0} in that neighborhood, the sequence of iterates converges linearly, with
rateIn numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of...
log
_{10} 2 (Süli & Mayers, Exercise 1.6). Alternatively if ƒ
'(α) = 0 and ƒ
'(x) ≠ 0 for x ≠ 0, x in a neighborhood U of α, α being a zero of
multiplicityIn mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....
r, and if ƒ ∈ C
^{r}(U) then there exists a neighborhood of α such that for all starting values x
_{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 are not known a priori, but there are also some results on global convergence, for instance, given a right neighborhood U
_{+} of α, if f is twice differentiable in U
_{+} and if
,
in U
_{+}, then, for each x
_{0} in U
_{+} the sequence x
_{k} is monotonically decreasing to α.
Proof of quadratic convergence for Newton's iterative method
According to
Taylor's theoremIn calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a kth order Taylorpolynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...
, any function f(x) which has a continuous second derivative can be represented by an expansion about a point that is close to a root of f(x). Suppose this root is
Then the expansion of f(α) about x
_{n} is:
where the Lagrange form of the Taylor series expansion remainder is
where ξ
_{n} is in between x
_{n} and
Since
is the root, becomes:
Dividing equation by
and rearranging gives
Remembering that x
_{n+1} is defined by
one finds that
That is,
Taking absolute value of both sides gives
Equation shows that the
rate of convergenceIn numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of...
is quadratic if following conditions are satisfied:


 sufficiently close to the root
The term sufficiently close in this context means the following:
(a) Taylor approximation is accurate enough such that we can ignore higher order terms,
(b)
(c)
Finally, can be expressed in the following way:
where M is the
supremumIn mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of the variable coefficient of
on the interval
defined in the condition 1, that is:
The initial point
has to be chosen such that conditions 1 through 3 are satisfied, where the third condition requires that
Failure analysis
Newton's method is only guaranteed to converge if certain conditions are satisfied. If the assumptions made in the proof of Quadratic Convergence are met, the method will converge. For the following subsections, failure of the method to converge indicates that the assumptions made in the proof were not met.
Bad starting points
In some cases the conditions on function necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. In such cases a different method, such as
bisectionThe bisection method in mathematics is a rootfinding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow...
, should be used to obtain a better estimate for the zero to use as an initial point.
Iteration point is stationary
Consider the function:
It has a maximum at x=0 and solutions of f(x) = 0 at x = ±1. If we start iterating from the
stationary pointIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
x
_{0}=0 (where the derivative is zero), x
_{1} will be undefined, since the tangent at (0,1) is parallel to the xaxis:
The same issue occurs if, instead of the starting point, any iteration point is stationary. Even if the derivative is small but not zero, the next iteration will be a far worse approximation.
Starting point enters a cycle
For some functions, some starting points may enter an infinite cycle, preventing convergence. Let
and take 0 as the starting point. The first iteration produces 1 and the second iteration returns to 0 so the sequence will alternate between the two without converging to a root. In general, the behavior of the sequence can be very complex. (See
Newton fractal.)
Derivative issues
If the function is not continuously differentiable in a neighborhood of the root then it is possible that Newton's method will always diverge and fail, unless the solution is guessed on the first try.
Derivative does not exist at root
A simple example of a function where Newton's method diverges is the cube root, which is continuous and infinitely differentiable, except for x = 0, where its derivative is undefined (this, however, does not affect the algorithm, since it will never require the derivative if the solution is already found):
For any iteration point x
_{n}, the next iteration point will be:
The algorithm overshoots the solution and lands on the other side of the yaxis, farther away than it initially was; applying Newton's method actually doubles the distances from the solution at each iteration.
In fact, the iterations diverge to infinity for every
, where
. In the limiting case of
(square root), the iterations will alternate indefinitely between points x
_{0} and −x
_{0}, so they do not converge in this case either.
Discontinuous derivative
If the derivative is not continuous at the root, then convergence may fail to occur in any neighborhood of the root. Consider the function
Its derivative is:
Within any neighborhood of the root, this derivative keeps changing sign as x approaches 0 from the right (or from the left) while f(x) ≥ x − x
^{2} > 0 for 0 < x < 1.
So f(x)/f
'(x) is unbounded near the root, and Newton's method will diverge almost everywhere in any neighborhood of it, even though:
 the function is differentiable (and thus continuous) everywhere;
 the derivative at the root is nonzero;
 f is infinitely differentiable except at the root; and
 the derivative is bounded in a neighborhood of the root (unlike f(x)/f'(x)).
Nonquadratic convergence
In some cases the iterates converge but do not converge as quickly as promised. In these cases simpler methods converge just as quickly as Newton's method.
Zero derivative
If the first derivative is zero at the root, then convergence will not be quadratic. Indeed, let
then
and consequently
. So convergence is not quadratic, even though the function is infinitely differentiable everywhere.
Similar problems occur even when the root is only "nearly" double. For example, let
Then the first few iterates starting at x
_{0} = 1 are
1, 0.500250376, 0.251062828, 0.127507934, 0.067671976, 0.041224176, 0.032741218, 0.031642362; it takes six iterations to reach a point where the convergence appears to be quadratic.
No second derivative
If there is no second derivative at the root, then convergence may fail to be quadratic. Indeed, let
Then
And
except when
where it is undefined. Given
,
which has approximately 4/3 times as many bits of precision as
has. This is less than the 2 times as many which would be required for quadratic convergence. So the convergence of Newton's method (in this case) is not quadratic, even though: the function is continuously differentiable everywhere; the derivative is not zero at the root; and
is infinitely differentiable except at the desired root.
Complex functions
When dealing with
complex functionsComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, Newton's method can be directly applied to find their zeroes. Each zero has a basin of attraction, 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 boundary of the basins of attraction is a
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reducedsize copy of the whole," a property called selfsimilarity...
. 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.
k variables, k functions
One may use Newton's method also to solve systems of k (nonlinear) equations, which amounts to finding the zeroes of continuously differentiable functions F : R
^{k} → R
^{k}. In the formulation given above, one then has to left multiply with the inverse of the kbyk Jacobian matrix J
_{F}(x
_{n}) instead of dividing by f '(x
_{n}).
Rather than actually computing the inverse of this matrix, one can save time by solving the
system of linear equations
for the unknown x
_{n+1} − x
_{n}.
k variables, > k equations
The кdimensional Newton's method can be used to solve systems of >k (nonlinear) equations as well if the algorithm uses the
generalized inverseIn mathematics, a generalized inverse or pseudoinverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them...
of the nonsquare Jacobian matrix J
^{+} = ((J
^{T}J)
^{−1})J
^{T} instead of the inverse of J. If the nonlinear system has no solution, the methods attempts to find a solution in the
nonlinear least squaresNonlinear least squares is the form of least squares analysis which is used to fit a set of m observations with a model that is nonlinear in n unknown parameters . It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to...
sense.
Nonlinear equations in a Banach space
Another generalization is Newton's method to find a root of a
functionalIn mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
F defined in a
Banach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm · such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
. In this case the formulation is
where
is the
Fréchet derivativeIn mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
computed at
. One needs the Fréchet derivative to be boundedly invertible at each
in order for the method to be applicable. A condition for existence of and convergence to a root is given by the
Newton–Kantorovich theoremThe Kantorovich theorem is a mathematical statement on the convergence of Newton's method. It was first stated by Leonid Kantorovich in 1940.Newton's method constructs a sequence of points that—with good luck—will converge to a solution x of an equation f = 0 or a vector solution of a...
.
Nonlinear equations over padic numbers
In padic analysis, the standard method to show a polynomial equation in one variable has a padic root is
Hensel's lemmaIn mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power...
, which uses the recursion from Newton's method on the padic numbers. Because of the more stable behavior of addition and multiplication in the padic numbers compared to the real numbers (specifically, the unit ball in the padics 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.
Minimization and maximization problems
Newton's method can be used to find a minimum or maximum of a function. The derivative is zero at a minimum or maximum, so minima and maxima can be found by applying Newton's method to the derivative. The iteration becomes:
Digital division
An important and somewhat surprising application is Newton–Raphson division, which can be used to quickly find the
reciprocalIn mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of a number using only multiplication and subtraction.
Solving transcendental equations
Many
transcendental equations can be solved using Newton's method. Given the equation
with g(x) and/or h(x) a
transcendental functionA transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
, one writes
The values of x that solves the original equation are then the roots of f(x), which may be found via Newton's method.
Square root of a number
Consider the problem of finding the square root of a number. There are many
methods of computing square rootsThere are several methods for calculating the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below. Rough estimation :...
, and Newton's method is one.
For example, if one wishes to find the square root of 612, this is equivalent to finding the solution to
The function to use in Newton's method is then,
with derivative,
With an initial guess of 10, the sequence given by Newton's method is
Where the correct digits are underlined. With only a few iterations one can obtain a solution accurate to many decimal places.
Solution of cos(x) = x^{3}
Consider the problem of finding the positive number x with cos(x) = x
^{3}. We can rephrase that as finding the zero of f(x) = cos(x) − x
^{3}. We have f(x) = −sin(x) − 3x
^{2}. Since cos(x) ≤ 1 for all x and x
^{3} > 1 for x > 1, we know that our zero lies between 0 and 1. We try a starting value of x
_{0} = 0.5. (Note that a starting value of 0 will lead to an undefined result, showing the importance of using a starting point that is close to the zero.)
The correct digits are underlined in the above example. In particular, x
_{6} is correct to the number of decimal places given. We see that the number of correct digits after the decimal point increases from 2 (for x
_{3}) to 5 and 10, illustrating the quadratic convergence.
See also
 Bisection method
The bisection method in mathematics is a rootfinding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow...
 Euler method
 Fast inverse square root
Fast inverse square root is a method of calculating x½, the reciprocal of a square root for a 32bit floating point number in IEEE 754 floating point format...
 Gradient descent
Gradient descent is a firstorder optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point...
 Integer square root
In number theory, the integer square root of a positive integer n is the positive integer m which is the greatest integer less than or equal to the square root of n,...
 Leonid Kantorovich
Leonid Vitaliyevich Kantorovich was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources...
, who initiated the convergence analysis of Newton's method in Banach spaces.
 Methods of computing square roots
There are several methods for calculating the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below. Rough estimation :...
 Newton's method in optimization
In mathematics, Newton's method is an iterative method for finding roots of equations. More generally, Newton's method is used to find critical points of differentiable functions, which are the zeros of the derivative function.Method:...
 Rootfinding algorithm
A rootfinding algorithm is a numerical method, or algorithm, for finding a value x such that f = 0, for a given function f. Such an x is called a root of the function f....
 Secant method
In numerical analysis, the secant method is a rootfinding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method. However, the method was developed...
 Steffensen's method
In numerical analysis, Steffensen's method is a rootfinding method, similar to Newton's method, named after Johan Frederik Steffensen. Steffensen's method also achieves quadratic convergence, but without using derivatives as Newton's method does....
 Subgradient method
Subgradient methods are iterative methods for solving convex minimization problems. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient methods are convergent when applied even to a nondifferentiable objective function...
External links
 Newton's method  section from an online textbook
 NewtonRaphson online calculator
 Animations for Newton's method by Prof. John H. Mathews
 Animations for Newton's method by Yihui Xie using the R
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
package animation
 NewtonRaphson Method Notes, PPT, Mathcad, Maple, Matlab, Mathematica at Holistic Numerical Methods Institute
 Module for Newton’s Method by John H. Mathews
 Worked example
 The NewtonRaphson algorithm coded in C++ as a template class
Templates are a feature of the C++ programming language that allow functions and classes to operate with generic types. This allows a function or class to work on many different data types without being rewritten for each one....
which takes a function objectA function object, also called a functor, functional, or functionoid, is a computer programming construct allowing an object to be invoked or called as though it were an ordinary function, usually with the same syntax.Description:...
 Newton's Method for finding roots  Source provides for C++ function and examples
 Java code by Behzad Torkian
 Matlab implementation of Newton's method
 Implementation in Ruby