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# Newton

[noot-n, nyoot-n]
Newton, Alfred, 1829-1907, English zoologist, b. Geneva. He studied (1854-65) ornithology in Lapland, Iceland, the West Indies, and North America and in 1866 became the first professor of zoology and comparative anatomy at Cambridge. In 1900 he received the Royal Medal of the Royal Society and the Gold Medal of the Linnaean Society. His writings include Zoology of Ancient Europe (1862) and Dictionary of Birds (1893-96). Newton edited (1865-70) the review Ibis.
Newton, Gilbert Stuart, 1794-1835, English genre and portrait painter, b. Halifax, N.S., studied in Boston with his uncle Gilbert Stuart, and later abroad. He was greatly influenced by the 17th-century Dutch masters. Representative examples of his art are Yorick and the Grisette; The Window (National Gall., London); and Deserted (Metropolitan Mus.).
Newton, Sir Isaac, 1642-1727, English mathematician and natural philosopher (physicist), who is considered by many the greatest scientist that ever lived.

## Early Life and Work

Newton studied at Cambridge and was professor there from 1669 to 1701, succeeding his teacher Isaac Barrow as Lucasian professor of mathematics. His most important discoveries were made during the two-year period from 1664 to 1666, when the university was closed and he retired to his hometown of Woolsthorpe. At that time he discovered the law of universal gravitation, began to develop the calculus, and discovered that white light is composed of all the colors of the spectrum. These findings enabled him to make fundamental contributions to mathematics, astronomy, and theoretical and experimental physics.

## The Principia

Newton summarized his discoveries in terrestrial and celestial mechanics in his Philosophiae naturalis principia mathematica [mathematical principles of natural philosophy] (1687), one of the greatest milestones in the history of science. In it he showed how his principle of universal gravitation provided an explanation both of falling bodies on the earth and of the motions of planets, comets, and other bodies in the heavens. The first part of the Principia is devoted to dynamics and includes Newton's three famous laws of motion; the second part to fluid motion and other topics; and the third part to the system of the world, i.e., the unification of terrestrial and celestial mechanics under the principle of gravitation and the explanation of Kepler's laws of planetary motion. Although Newton used the calculus to discover his results, he explained them in the Principia by use of older geometric methods.

## Later Work

Newton's discoveries in optics were presented in his Opticks (1704), in which he elaborated his theory that light is composed of corpuscles, or particles. His corpuscular theory dominated optics until the early 19th cent., when it was replaced by the wave theory of light. The two theories were combined in the modern quantum theory. Among his other accomplishments were his construction (1668) of a reflecting telescope and his anticipation of the calculus of variations, founded by Gottfried Leibniz and the Bernoullis. In later years Newton considered mathematics and physics a recreation and turned much of his energy toward alchemy, theology, and history, particularly problems of chronology.

## Later Life

Newton was his university's representative in Parliament (1689-90, 1701-2) and was president of the Royal Society from 1703 until his death. He was made warden of the mint in 1696 and master in 1699, being knighted in 1705 in recognition of his services at the mint as much as for his scientific accomplishments. Although Newton was known as an open and generous person, at various times in his life he became involved in quarrels and controversies. The most notable was his dispute with Leibniz over which of them had first invented calculus; today they are jointly ascribed the honor.

## Bibliography

An eight-volume edition of Newton's mathematical papers (ed. by D. H. Whiteside et al., 1967-81) has been published. See biographies by R. S. Westfall (1980), G. E. Christianson (1984), and J. Gleick (2003); J. Herivel, The Background to Newton's Principia (1965); A. Koyré, Newtonian Studies (1965); I. B. Cohen, Introduction to Newton's Principia (1971) and The Newtonian Revolution (1983); M. S. Stayer, ed., Newton's Dream (1988).

Newton, John, 1725-1807, English clergyman and hymn writer, b. London. Until 1755, his life was spent chiefly at sea, where he eventually became the captain of a slave ship plying the waters between Liverpool and Sierra Leone. For the subsequent five years he was surveyor of tides at Liverpool, using his leisure time for the study of Latin, Greek, Hebrew, and theology. Newton, who was much influenced by the religious reformer George Whitefield, was ordained in the Church of England and appointed curate of Olney, Buckinghamshire, in 1764. When William Cowper made his home in the parish, friendship and literary sympathy between the two men resulted in their publishing jointly the Olney Hymns (1779 and later eds.). Among the best known of Newton's hymns are "Amazing Grace," by far his most famous work, "How Sweet the Name of Jesus Sounds", "Glorious Things of Thee Are Spoken," and "God Moves in a Mysterious Way." From 1779 he was rector of St. Mary Woolnoth, London where he became known as an abolitionist preacher. Newton's first-hand testimony before Parliament regarding the evils of slavery aided in the passage of legislation (1807) barring the British slave trade.

See his Cardiphonia, or the Utterance of the Heart (1795, repr. 1850, 1909); W. E. Phipps, Amazing Grace in John Newton (2001); S. Turner, Amazing Grace: The Story of America's Most Beloved Song (2002).

newton, abbr. N, unit of force in the mks system of units, which is based on the metric system; it is the force that produces an acceleration of 1 meter per second per second when exerted on a mass of 1 kilogram. The newton is named for Sir Isaac Newton.

Absolute unit of force, abbreviated N, in the metre-kilogram-second (MKS) system of physical units (see International System of Units). It is defined as the force necessary to provide a mass of 1 kg with an acceleration of 1 m per second per second. One newton is equal to a force of 100,000 dynes in the centimetre-gram-second (CGS) system, or a force of about 0.2248 lb in the foot-pound-second (English or U.S.) system. It is named for Isaac Newton, whose second law of motion describes the changes a force can produce in the motion of a body.

(born July 29, 1869, Indianapolis, Ind., U.S.—died May 19, 1946, Indianapolis) U.S. novelist and dramatist. He became known for satirical and sometimes romanticized pictures of Midwesterners in humorous portrayals of boyhood and adolescence that include the young-people's classics Penrod (1914), Seventeen (1916), and Gentle Julia (1922). The trilogy Growth (1927) includes The Magnificent Ambersons (1918, Pulitzer Prize; film, 1942), which traces the decline of a once-powerful and prominent family. Alice Adams (1921; film, 1923, 1935), a searching character study, is perhaps his most finished novel.

(born Jan. 4, 1643, Woolsthorpe, Lincolnshire, Eng.—died March 31, 1727, London) English physicist and mathematician. The son of a yeoman, he was raised by his grandmother. He was educated at Cambridge University (1661–65), where he discovered the work of René Descartes. His experiments passing sunlight through a prism led to the discovery of the heterogeneous, corpuscular nature of white light and laid the foundation of physical optics. He built the first reflecting telescope in 1668 and became a professor of mathematics at Cambridge in 1669. He worked out the fundamentals of calculus, though this work went unpublished for more than 30 years. His most famous publication, Principia Mathematica (1687), grew out of correspondence with Edmond Halley. Describing his works on the laws of motion (see Newton's laws of motion), orbital dynamics, tidal theory, and the theory of universal gravitation, it is regarded as the seminal work of modern science. He was elected president of the Royal Society of London in 1703 and became the first scientist ever to be knighted in 1705. During his career he engaged in heated arguments with several of his colleagues, including Robert Hooke (over authorship of the inverse square relation of gravitation) and G.W. Leibniz (over the authorship of calculus). The battle with Leibniz dominated the last 25 years of his life; it is now well established that Newton developed calculus first, but that Leibniz was the first to publish on the subject. Newton is regarded as one of the greatest scientists of all time.

(born Jan. 4, 1643, Woolsthorpe, Lincolnshire, Eng.—died March 31, 1727, London) English physicist and mathematician. The son of a yeoman, he was raised by his grandmother. He was educated at Cambridge University (1661–65), where he discovered the work of René Descartes. His experiments passing sunlight through a prism led to the discovery of the heterogeneous, corpuscular nature of white light and laid the foundation of physical optics. He built the first reflecting telescope in 1668 and became a professor of mathematics at Cambridge in 1669. He worked out the fundamentals of calculus, though this work went unpublished for more than 30 years. His most famous publication, Principia Mathematica (1687), grew out of correspondence with Edmond Halley. Describing his works on the laws of motion (see Newton's laws of motion), orbital dynamics, tidal theory, and the theory of universal gravitation, it is regarded as the seminal work of modern science. He was elected president of the Royal Society of London in 1703 and became the first scientist ever to be knighted in 1705. During his career he engaged in heated arguments with several of his colleagues, including Robert Hooke (over authorship of the inverse square relation of gravitation) and G.W. Leibniz (over the authorship of calculus). The battle with Leibniz dominated the last 25 years of his life; it is now well established that Newton developed calculus first, but that Leibniz was the first to publish on the subject. Newton is regarded as one of the greatest scientists of all time.

(born Dec. 3, 1871, Martinsburg, W.Va., U.S.—died Dec. 25, 1937, Cleveland, Ohio) U.S. secretary of war. He practiced law in Martinsburg from 1897. After moving to Cleveland, he was elected mayor (1912–16). He helped obtain the 1912 Democratic presidential nomination for Woodrow Wilson, who appointed him secretary of war (1916–21). Although he was a pacifist, Baker developed a plan for the military draft and oversaw the mobilization of more than four million men during World War I. In 1928 he was appointed to the International Court of Arbitration at The Hague.

(born Sept. 15, 1913, Detroit, Mich., U.S.—died Nov. 9, 1988, Washington, D.C.) U.S. public official. A prominent attorney in New York City, he practiced with Richard Nixon after their firms merged in 1967. In 1968 he managed Nixon's successful presidential campaign. As U.S. attorney general (1969–72), he was criticized for prosecuting war protesters, approving wiretaps without court authorization, and attempting to block publication of the Pentagon Papers. He resigned to direct Nixon's reelection campaign but was soon caught up in the Watergate scandal. Convicted of conspiracy, obstruction of justice, and perjury, he served 19 months in prison.

(born Sept. 15, 1913, Detroit, Mich., U.S.—died Nov. 9, 1988, Washington, D.C.) U.S. public official. A prominent attorney in New York City, he practiced with Richard Nixon after their firms merged in 1967. In 1968 he managed Nixon's successful presidential campaign. As U.S. attorney general (1969–72), he was criticized for prosecuting war protesters, approving wiretaps without court authorization, and attempting to block publication of the Pentagon Papers. He resigned to direct Nixon's reelection campaign but was soon caught up in the Watergate scandal. Convicted of conspiracy, obstruction of justice, and perjury, he served 19 months in prison.

(born Dec. 3, 1871, Martinsburg, W.Va., U.S.—died Dec. 25, 1937, Cleveland, Ohio) U.S. secretary of war. He practiced law in Martinsburg from 1897. After moving to Cleveland, he was elected mayor (1912–16). He helped obtain the 1912 Democratic presidential nomination for Woodrow Wilson, who appointed him secretary of war (1916–21). Although he was a pacifist, Baker developed a plan for the military draft and oversaw the mobilization of more than four million men during World War I. In 1928 he was appointed to the International Court of Arbitration at The Hague.

In numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the best known method for finding successively better approximations to the zeros (or roots) of a real-valued function. Newton's method can often converge remarkably quickly, especially if the iteration begins "sufficiently near" the desired root. Just how near "sufficiently near" needs to be, and just how quickly "remarkably quickly" can be, depends on the problem. This is discussed in detail below. Unfortunately, far from the desired root, Newton's method can easily lead an unwary user astray with little warning. Thus, good implementations of the method embed it in a routine that also detects and perhaps overcomes possible convergence failures.

Newton's method 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.

The algorithm is first in the class of Householder's methods, succeeded by Halley's method.

## 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.

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 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

$f\text{'}\left(x_\left\{n\right\}\right) = frac\left\{ mathrm\left\{rise\right\} \right\}\left\{ mathrm\left\{run\right\} \right\} = frac\left\{ mathrm\left\{Delta y\right\} \right\}\left\{ mathrm\left\{Delta x\right\} \right\} = frac\left\{ f\left(x_\left\{n\right\} \right) - 0 \right\}\left\{ x_\left\{n\right\} - x_\left\{n+1\right\} \right\}.,!$

Here, f ' denotes the derivative of the function f. Then by simple algebra we can derive

$x_\left\{n+1\right\} = x_n - frac\left\{f\left(x_n\right)\right\}\left\{f\text{'}\left(x_n\right)\right\},!$.
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 $f\text{'}\left(x_0\right) neq 0$. 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.

## Examples

### Example 1

Consider the problem of finding the square root of a number. There are many Methods of computing square roots, 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

$x^2 = 612,$
The function to use in Newton's Method is then,
$f\left(x\right) = x^2 - 612 ,$
with derivative,
$f\text{'}\left(x\right) = 2x ,$

With an initial guess of 10, the sequence given by newtons method is

$begin\left\{matrix\right\}$
x_1 & = & x_0 - frac{f(x_0)}{f'(x_0)} & = & 10 - frac{10^2 - 612}{2 cdot 10} & = & 35.6 x_2 & = & x_1 - frac{f(x_1)}{f'(x_1)} & = & 35.6 - frac{35.6^2 - 612}{2 cdot 35.6} & = & underline{2}6.3955056 x_3 & = & vdots & = & vdots & = & underline{24.7}906355 x_4 & = & vdots & = & vdots & = & underline{24.7386}883 x_5 & = & vdots & = & vdots & = & underline{24.7386338} end{matrix} Where the correct digits are underlined. With only a few iterations one can obtain a solution accurate to a few decimal places.

### Example 2

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.

$begin\left\{matrix\right\}$
x_1 & = & x_0 - frac{f(x_0)}{f'(x_0)} & = & 0.5 - frac{cos(0.5) - 0.5^3}{-sin(0.5) - 3 times 0.5^2} & = & 1.112141637097 x_2 & = & x_1 - frac{f(x_1)}{f'(x_1)} & = & vdots & = & underline{0.}909672693736 x_3 & = & vdots & = & vdots & = & underline{0.86}7263818209 x_4 & = & vdots & = & vdots & = & underline{0.86547}7135298 x_5 & = & vdots & = & vdots & = & underline{0.8654740331}11 x_6 & = & vdots &= & vdots & = & underline{0.865474033102} end{matrix}

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, while his successor Jamshīd al-Kāshī used a form of Newton's method to solve $x^P - N = 0$ 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 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 xn 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.

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.

## 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.

Since the most serious of the problems above is the possibility of a failure of convergence, Press et al. (1992) present a version of Newton's method that starts at the midpoint of an interval in which the root is known to lie and stops the iteration if an iterate is generated that lies outside the interval.

Developers of large scale computer systems involving root finding tend to prefer the secant method over Newton's method because the use of a difference quotient in place of the derivative in Newton's method implies that the additional code to compute the derivative need not be maintained. In practice, the advantages of maintaining a smaller code base usually outweigh the superior convergence characteristics of Newton's method.

## Counter examples

Newton's method makes no guarantee on convergence, and depending on the shape of the function and the starting point it may or may not converge.

#### Starting point at maximum or minimum

Consider the function:
$f\left(x\right) = 1-x^2 !$
It has a maximum at x=0 and solutions of f(x)=0 at x=±1. However, if we start iterating from x0=0, where the derivative is zero, x1 will be undefined:

$x_1 = x_0 - frac\left\{f\left(x_0\right)\right\}\left\{f\text{'}\left(x_0\right)\right\} = 0 - frac\left\{1\right\}\left\{0\right\} = infty$

#### Starting point enters a cycle

For some functions, some starting points may enter an infinite cycle, preventing convergence. Let

$f\left(x\right) = x^3 - 2x + 2 !$
and take 0 as the starting point. The first iteration produces 1 and the second iteration returns to 0 so the sequence will oscillate between the two without converging to a root. In general, the behavior of the sequence can be very complex. (See Newton fractal.)

#### No feasible starting points

Some functions can be constructed on which Newton's method will always diverge, unless the solution is guessed on the first try; a simple one is the cubic root, which is continuous and infinitely differentiable with continuity (except for x=0, where its derivatives are infinite, but that is the one point that Newton's method will never use as it is the solution):
$f\left(x\right) = sqrt\left[3\right]\left\{x\right\}$
For any iteration point xn, the next iteration point will be:
$x_\left\{n+1\right\} = x_n - frac\left\{f\left(x_n\right)\right\}\left\{f\text{'}\left(x_n\right)\right\} = x_n - frac\left\{x_n^\left\{1/3\right\}\right\}\left\{1/3,x_n^\left\{1/3-1\right\}\right\} = x_n - 3,x_n = -2,x_n$
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.

This consideration holds true for every $f\left(x\right) = sqrt\left[n\right]$

>, where n>2. In the limiting case of n=2 (square root), the iterations will oscillate indefinitely between points x0 and -x0.

### Derivative issues

#### 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

$f\left(x\right) = begin\left\{cases\right\}$
0 & text{if } x = 0, x + x^2sinleft(frac{2}{x}right) & text{if } x neq 0. end{cases}

Its derivative is:

$f\text{'}\left(x\right) = begin\left\{cases\right\}$
nexists & text{if } x = 0, 1 + 2,x,sinleft(frac{2}{x}right) + 2,cosleft(frac{2}{x}right) & text{if } x neq 0. end{cases}

The derivative does not exist at x=0 since both f'(0-) and f'(0+) are undefined.

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->0 for 0<x<1.

So f(x)/f'(x) is unbounded near the root, and Newton's method will not converge, even though:

• the function is everywhere differentiable with continuity, except at the root;
• f is infinitely differentiable except at the root; and
• the derivative is bounded in a neighborhood of the root (unlike f(x)/f'(x)).

#### No second derivative

If there is no second derivative at the root, then convergence may fail to be quadratic. Indeed, let
$f\left(x\right) = x + x^\left\{4/3\right\} !$
Then
$f\text{'}\left(x\right) = 1 + \left(4/3\right)x^\left\{1/3\right\} !$
And
$f\left(x\right) = \left(4/9\right)x^\left\{-2/3\right\} !$
except when $x = 0 !$ where it is undefined. Given $x_n !$,

$x_\left\{n+1\right\} = x_n - frac\left\{f\left(x_n\right)\right\}\left\{f \text{'}\left(x_n\right)\right\} = frac\left\{\left(1/3\right)x_n^\left\{4/3\right\}\right\}\left\{\left(1 + \left(4/3\right)x_n^\left\{1/3\right\}\right)\right\} !$

which has approximately 4/3 times as many bits of precision as $x_n !$ 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 $f !$ is infinitely differentiable except at the desired root.

#### Zero derivative

If the first derivative is zero at the root, then convergence will not be quadratic. Indeed, let
$f\left(x\right) = x^2 !$
Then $f\text{'}\left(x\right) = 2x !$ and consequently $x - f\left(x\right)/f\text{'}\left(x\right) = x/2 !$. 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 {xn} will converge to α.

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, $f\text{'}\left(alpha\right) = 0 !$ and $f$(alpha) ne 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 log10 2 (Süli & Mayers, Exercise 1.6). Alternatively if $f\text{'}\left(alpha\right) = 0 !$ and $f\text{'}\left(x\right) ne 0 !$ elsewhere, in a neighborhood U of α, α being a zero of multiplicity r and if $f in C^r\left(U\right)$ 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 $f\text{'} ne 0 !$, $f cdot f$ > 0 ! in U+, then, for each x0 in U+ the sequence xk'' 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 : 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

$J_F\left(x_n\right) \left(x_\left\{n+1\right\} - x_n\right) = -F\left(x_n\right),!$

for the unknown xn+1xn. 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.

### 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
$X_\left\{n+1\right\}=X_n-\left(F\text{'}_\left\{X_n\right\}\right)^\left\{-1\right\}\left[F\left(X_n\right)\right]$,
where $F\text{'}_\left\{X_n\right\}$ 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.

### Complex functions

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.

## 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 (available free online, with code samples: ), sections 9.4 and 9.6
• W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007. ISBN 0-521-88068-8 (available for a fee online, with code samples ).
• 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: )
• Endre Süli and David Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003. ISBN 0-521-00794-1.