Newton's method

Newton's method

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In numerical analysis
Numerical analysis
Numerical 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 Newton
Isaac Newton
Sir 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 Raphson
Joseph Raphson
Joseph 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 real
Real number
In 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 function
Function (mathematics)
In 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 method
Householder's method
In numerical analysis, the class of Householder's methods are root-finding 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 method
Halley's method
In numerical analysis, Halley’s method is a root-finding 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 Newton-Raphson method in one variable is implemented as follows:

Given a function ƒ defined over the reals x, and its derivative
Derivative
In 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 x0 for a root of the function f. Provided the function is reasonably well-behaved a better approximation x1 is


Geometrically, x1 is the intersection with the x-axis of a line tangent to f at f (x0).

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 calculus
Calculus
Calculus 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 x-intercept of this tangent line (which is easily done with elementary algebra). This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated
Iterative method
In 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 differentiable
Derivative
In 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 interval
Interval (mathematics)
In 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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

R. The formula for converging on the root can be easily derived. Suppose we have some current approximation xn. Then we can derive the formula for a better approximation, xn+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 derivative
Derivative
In 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 x0. (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
Intermediate value theorem
In 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(x0) ≠ 0. Furthermore, for a zero of multiplicity
Multiplicity (mathematics)
In 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 convergence
Rate of convergence
In 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 neighbourhood
Neighbourhood (mathematics)
In 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 method
Householder's method
In numerical analysis, the class of Householder's methods are root-finding 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 Newton
Isaac Newton
Sir 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 Fluxions
Method of Fluxions
Method 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 Colson
John Colson
Johnathan "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 al-Din al-Tusi, while his successor Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī 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 single-variable 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 Raphson
Joseph Raphson
Joseph 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 xn instead of the more complicated sequence of polynomials used by Newton. Finally, in 1740, Thomas Simpson
Thomas Simpson
Thomas 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 Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

 in 1879 in The Newton-Fourier 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 convergence
Rate of convergence
In 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 method
Secant method
In numerical analysis, the secant method is a root-finding 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 point
Stationary point
In 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 zero
Division by zero
In 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 well-defined value depends upon the mathematical setting...

.

Poor initial estimate


A large error in the initial estimate can contribute to non-convergence of the algorithm.

Mitigation of non-convergence


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 x0 in that neighborhood, the sequence
Sequence
In 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...

 {xn} will converge
Limit of a sequence
The 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 derivative
Second derivative
In 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 x0 in that neighborhood, the sequence of iterates converges linearly, with rate
Rate of convergence
In 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...

 log10 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 multiplicity
Multiplicity (mathematics)
In 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 ∈ Cr(U) then there exists a neighborhood of α such that for all starting values x0 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 x0 in U+ the sequence xk is monotonically decreasing to α.

Proof of quadratic convergence for Newton's iterative method


According to Taylor's theorem
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. 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 xn is:
where the Lagrange form of the Taylor series expansion remainder is

where ξn is in between xn and

Since is the root, becomes:
Dividing equation by and rearranging gives
Remembering that xn+1 is defined by
one finds that

That is,
Taking absolute value of both sides gives
Equation shows that the rate of convergence
Rate of convergence
In 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:
  1. 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 supremum
Supremum
In 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 bisection
Bisection method
The bisection method in mathematics is a root-finding 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 point
Stationary point
In 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 ....

 x0=0 (where the derivative is zero), x1 will be undefined, since the tangent at (0,1) is parallel to the x-axis:


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 xn, the next iteration point will be:


The algorithm overshoots the solution and lands on the other side of the y-axis, 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 x0 and −x0, 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 − x2 > 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)).

Non-quadratic 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 x0 = 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 functions
Complex analysis
Complex 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 fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

. 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 (non-linear) equations, which amounts to finding the zeroes of continuously differentiable functions F : Rk Rk. In the formulation given above, one then has to left multiply with the inverse of the k-by-k Jacobian matrix JF(xn) instead of dividing by f '(xn).

Rather than actually computing the inverse of this matrix, one can save time by solving the system of linear equations


for the unknown xn+1 − xn.

k variables, > k equations


The к-dimensional Newton's method can be used to solve systems of >k (non-linear) equations as well if the algorithm uses the generalized inverse
Generalized inverse
In 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 non-square Jacobian matrix J+ = ((JTJ)−1)JT instead of the inverse of J. If the nonlinear system has no solution, the methods attempts to find a solution in the non-linear least squares
Non-linear least squares
Non-linear least squares is the form of least squares analysis which is used to fit a set of m observations with a model that is non-linear in n unknown parameters . It is used in some forms of non-linear 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 functional
Functional (mathematics)
In 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 space
Banach space
In 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 derivative
Fréchet derivative
In 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 theorem
Kantorovich theorem
The 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 p-adic numbers


In p-adic analysis, the standard method to show a polynomial equation in one variable has a p-adic root is Hensel's lemma
Hensel's lemma
In 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 p-adic numbers. Because of the more stable behavior of addition and multiplication in the p-adic numbers compared to the real numbers (specifically, the unit ball in the 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.

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 reciprocal
Multiplicative inverse
In 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 function
Transcendental function
A 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 roots
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 :...

, 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) = x3


Consider the problem of finding the positive number x with cos(x) = x3. We can rephrase that as finding the zero of f(x) = cos(x) − x3. We have f(x) = −sin(x) − 3x2. Since cos(x) ≤ 1 for all x and x3 > 1 for x > 1, we know that our zero lies between 0 and 1. We try a starting value of x0 = 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, x6 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 x3) to 5 and 10, illustrating the quadratic convergence.

See also

  • Bisection method
    Bisection method
    The bisection method in mathematics is a root-finding 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
    Fast inverse square root is a method of calculating x-½, the reciprocal of a square root for a 32-bit floating point number in IEEE 754 floating point format...

  • Gradient descent
    Gradient descent
    Gradient descent is a first-order 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
    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 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
    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
    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:...

  • Root-finding algorithm
    Root-finding algorithm
    A root-finding 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
    Secant method
    In numerical analysis, the secant method is a root-finding 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
    Steffensen's method
    In numerical analysis, Steffensen's method is a root-finding 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 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 non-differentiable objective function...


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