In numerical analysis, Newton's method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. As such, it is an example of a root-finding algorithm. It can also be used to find a minimum or maximum of such a function, by finding a zero in the function's first derivative, see Newton's method as an optimization algorithm. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Bond signed by Joseph Raphson on his admittance to the Royal Society Joseph Raphson was an English mathematician known best for the Newton-Raphson method. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ... In mathematics, Newtons method is a well-known algorithm for finding roots of equations in one or more dimensions. ...

Description of the method

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), 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. In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... For other uses, see Calculus (disambiguation). ... It has been suggested that this article or section be merged with Guess value. ...

An illustration of one iteration of Newton's method (the function f is shown in blue and the tangent line is in red). We see that x_{n + 1} is a better approximation than x_n for the root x of the function f.

Suppose f : [a, b] → R is a differentiable function defined on the interval [a, b] with values in the real numbers 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. Image File history File links Newton_iteration. ... Image File history File links Newton_iteration. ... For a non-technical overview of the subject, see Calculus. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

That is

.

Here, f ' denotes the derivative of the function f. Then by simple algebra we can derive For a non-technical overview of the subject, see Calculus. ...

.

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.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that . Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic (see rate of convergence) in a neighbourhood 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. In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ... In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Example

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.

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.

History

Newton's method was described by Isaac Newton 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 in 1736 by John Colson). 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 x_n, 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 François Viète. The essence of Viète's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... // Events Samuel Pepys stopped writing his diary. ... 1711 (MDCCXI) was a common year starting on Thursday of the Gregorian calendar (or a common year starting on Monday of the 11-day slower Julian calendar). ... Sir William Jones (1675 - 3 July 1749) was a mathematician. ... Events May 9 - Thomas Blood, disguised as a clergyman, attempts to steal the Crown Jewels from the Tower of London. ... Method of Fluxions was a book by Isaac Newton. ... Events January 26 - Stanislaus I of Poland abdicates his throne. ... Johnathan John Colson was a Lucasian Professor of Mathematics at Cambridge University. ... François Viète. ... Anthem SorÅ«d-e MellÄ«-e Īrān Â² Capital (and largest city) Tehran Official languages Persian Demonym Iranian Government Islamic Republic  -  Supreme Leader  -  President Unification  -  Unified by Cyrus the Great 559 BCE   -  Parthian (Arsacid) dynastic empire (first reunification) 248 BCE-224 CE   -  Sassanid dynastic empire 224–651 CE   -  Safavid dynasty... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Sharafeddin Muzzafar-i Tusi (1135 - 1213) was a Persian mathematician of the Middle Ages. ...

Heron of Alexandria used essentially the same method in book 1, chapter 9, of his Metrica to determine the square root of 720. Heros aeolipile Hero (or Heron) of Alexandria (c. ...

Newton's method was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis. In 1690, Joseph Raphson 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 Simpson 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. Events February 6 - James Stuart, Duke of York becomes King James II of England and Ireland and King James VII of Scotland. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Events Giovanni Domenico Cassini observes differential rotation within Jupiters atmosphere. ... Bond signed by Joseph Raphson on his admittance to the Royal Society Joseph Raphson was an English mathematician known best for the Newton-Raphson method. ... Events May 31 - Friedrich II comes to power in Prussia upon the death of his father, Friedrich Wilhelm I. October 20 - Maria Theresia of Austria inherits the Habsburg hereditary dominions (Austria, Bohemia, Hungary and present-day Belgium). ... Thomas Simpson (August 20, 1710 – May 14, 1761) was a British mathematician, inventor and eponym of Simpsons rule to approximate definite integrals. ...

Arthur Cayley 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. Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... Year 1879 (MDCCCLXXIX) was a common year starting on Wednesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Monday of the 12-day slower Julian calendar). ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

Practical considerations

In general the convergence is quadratic: the error is essentially squared at each step (that is, the number of accurate digits doubles in each step). There are some caveats, however. First, Newton's method requires that the derivative be calculated directly. (If the derivative is approximated by the slope of a line through two points on the function, the secant method results; this can be more efficient depending on how one measures computational effort.) Second, if the initial value is too far from the true zero, Newton's method can fail to converge. Because of this, most practical implementations of Newton's method put an upper limit on the number of iterations and perhaps on the size of the iterates. Third, 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. In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ... This article is about the mathematical term. ... 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. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ...

Counter examples

In the following examples, the desired root is at zero for simplicity — it could have been placed elsewhere.

• If the derivative is not continuous at the root, then convergence may fail to occur.

Consider the function

Then and elsewhere.

Within any neighborhood of the root, this derivative keeps changing sign as x approaches 0 from the right (or from the left) while for .

So is unbounded near the root and Newton's method will not converge, even though: the function is differentiable everywhere; the derivative is not zero at the root; is infinitely differentiable except at the root; and the derivative is bounded in a neighborhood of the root (unlike its reciprocal).

• 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.

• 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.

Analysis

Suppose that the function f has a zero at α, i.e., f(α) = 0.

If f  is continuously differentiable and its derivative does not vanish at α, then there exists a neighborhood of α such that for all starting values x₀ in that neighborhood, the sequence {x_n} will converge to α. This is a glossary of some terms used in the branch of mathematics known as topology. ... For other senses of this word, see sequence (disambiguation). ... The limit of a sequence is one of the oldest concepts in mathematical analysis. ...

If the function is continuously differentiable and its derivative does not vanish at α and it has a second derivative at α then the convergence is quadratic or faster. If the second derivative does not vanish at α then the convergence is merely quadratic.

If the derivative does vanish at α, then the convergence is usually only linear. Specifically, if f is twice continuously differentiable, and , 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 log₁₀ 2 (Süli & Mayers, Exercise 1.6). Alternatively if and elsewhere, in a neighborhood U of α, α being a zero of multiplicity r and if then there exists a neighborhood of α such that for all starting values x₀ in that neighborhood, the sequence of iterates converges linearly. In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ...

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 x_k is monotonically decreasing to α.

Generalizations

Nonlinear systems of equations

One may use Newton's method also to solve systems of k (non-linear) equations, which amounts to finding the zeros 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 k-by-k 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 In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ...

for the unknown x_n+1 − x_n. Again, this method only works if the initial value x₀ is close enough to the true zero. Typically, a region which is well-behaved is located first with some other method and Newton's method is then used to "polish" a root which is already known approximately. Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...

Nonlinear equations in a Banach space

Another generalization is the Newton's method to find a zero of a function F defined in a Banach space. In this case the formulation is In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

,

where is the Fréchet derivative applied at the point X_n. One needs the Fréchet derivative to be invertible at each X_n in order for the method to be applicable. In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. ...

Complex functions

Basins of attraction for x⁵ − 1 = 0; darker means more iterations to converge.

Main article: Newton fractal

When dealing with complex functions, however, Newton's method can be directly applied to find their zeros. For many complex functions, the boundary of the set (also known as the basin of attraction) of all starting values that cause the method to converge to a particular zero is a fractal. Image File history File links Download high resolution version (1111x1111, 514 KB)// #include <stdio. ... Image File history File links Download high resolution version (1111x1111, 514 KB)// #include <stdio. ... The Newton fractal is a boundary set in the complex plane which is characterized by Newtons method applied to a fixed polynomial p(Z)∈ℂ[Z]. It divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, . In this way the... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...

References

• Tjalling J. Ypma, Historical development of the Newton-Raphson method, SIAM Review 37 (4), 531–551, 1995.
• 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.
• W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, 1992. ISBN 0-521-43108-5 (online, with code samples: [1]), sections 9.4 [2] and 9.6 [3].
• W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran, Cambridge University Press, 1992. ISBN 0-521-43064-X (online, with code samples: [4])
• Endre Süli and David Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003. ISBN 0-521-00794-1.

Numerical Recipes is the generic term for the following books on algorithms and numerical analysis, all by William Press, Saul Teukolsky, William Vetterling and Brian Flannery: Numerical Recipes in C++. The Art of Scientific Computing, ISBN 0-521-75033-4. ... Numerical Recipes is the generic term for the following books on algorithms and numerical analysis, all by William Press, Saul Teukolsky, William Vetterling and Brian Flannery: Numerical Recipes in C++. The Art of Scientific Computing, ISBN 0-521-75033-4. ...

External links

• Newton's method on Wolfram.com
• Animations for Newton's method
• Newton's method on the Mathcad Application Server (with animation)
• Newton-Raphson Method Notes, PPT, Mathcad, Maple, Matlab, Mathematica at Holistic Numerical Methods Institute
• Module for Newton’s Method by John H. Mathews
• worked example

See also

• Euler method
• Integer square root
• Methods of computing square roots