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# Newton's method

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. [1]

82 relations: Aitken's delta-squared process, Almost all, Arthur Cayley, Attractor, Banach space, Bessel function, Bisection method, Calculus, Chaos theory, Complex analysis, Complex-valued function, De analysi per aequationes numero terminorum infinitas, Derivative, Division by zero, Elementary algebra, Endre Süli, Euler method, Fast inverse square root, Fractal, François Viète, Fréchet derivative, Function (mathematics), Functional (mathematics), Gauss–Newton algorithm, Generalized inverse, Gradient descent, Graph of a function, Halley's method, Hensel's lemma, Hessian matrix, Householder's method, Integer square root, Intermediate value theorem, Interval (mathematics), Isaac Newton, Iterative method, Jacobian matrix and determinant, Jamshīd al-Kāshī, John Colson, John Wallis, Joseph Fourier, Joseph Raphson, Kantorovich theorem, Laguerre's method, Leonid Kantorovich, Limit of a sequence, Mathematics in medieval Islam, Method of Fluxions, Methods of computing square roots, Multiplicative inverse, ... Expand index (32 more) »

## Aitken's delta-squared process

In numerical analysis, Aitken's delta-squared process or Aitken Extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence.

## Almost all

In mathematics, the term "almost all" means "all but a negligible amount".

## Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

## Attractor

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions of Bessel's differential equation for an arbitrary complex number, the order of the Bessel function.

## Bisection method

The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing.

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Chaos theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

## 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.

## Complex-valued function

In mathematics, a complex-valued function (not to be confused with complex variable function) is a function whose values are complex numbers.

## De analysi per aequationes numero terminorum infinitas

De analysi per aequationes numero terminorum infinitas (or On analysis by infinite series, On Analysis by Equations with an infinite number of terms, On the Analysis by means of equations of an infinite number of terms,About completely loosening infinity by way of number equalisations limits) cf. (aequatio, analysi.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Division by zero

In mathematics, division by zero is division where the divisor (denominator) is zero.

## Elementary algebra

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics.

## Endre Süli

Endre Süli (also, Endre Suli) is Professor of Numerical Analysis in the Mathematical Institute, University of Oxford, Fellow and Tutor in Mathematics at Worcester College, Oxford and Supernumerary Fellow of Linacre College, Oxford.

## Euler method

In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

## Fast inverse square root

Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates, the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number in IEEE 754 floating-point format.

## Fractal

In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.

## François Viète

François Viète (Franciscus Vieta; 1540 – 23 February 1603), Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations.

## Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Functional (mathematics)

In mathematics, the term functional (as a noun) has at least two meanings.

## Gauss–Newton algorithm

The Gauss–Newton algorithm is used to solve non-linear least squares problems.

## Generalized inverse

In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them.

Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## 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.

## 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 of, which can be found by iteratively "lifting" the solution modulo successive powers of.

## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

## Householder's method

In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order.

## Integer square root

In number theory, the integer square root (isqrt) 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, For example, \mbox(27).

## Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

## Interval (mathematics)

In mathematics, a (real) 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.

## Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

## Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.

## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

## Jamshīd al-Kāshī

Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (غیاث الدین جمشید کاشانی Ghiyās-ud-dīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician.

## John Colson

John Colson (1680–1760) was an English clergyman and mathematician, Lucasian Professor of Mathematics at Cambridge University.

## John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

## Joseph Fourier

Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

## Joseph Raphson

Joseph Raphson (c. 1648 – c. 1715) was an English mathematician known best for the Newton–Raphson method.

## Kantorovich theorem

The Kantorovich theorem (Newton-Kantorovich theorem) is a mathematical statement on the convergence of Newton's method.

## Laguerre's method

In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials.

## Leonid Kantorovich

Leonid Vitaliyevich Kantorovich (a) (19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources.

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

## Mathematics in medieval Islam

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta).

## Method of Fluxions

Method of Fluxions is a book by Isaac Newton.

## Methods of computing square roots

In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number.

## Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x&minus;1, is a number which when multiplied by x yields the multiplicative identity, 1.

## Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.

## Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

## Newton fractal

The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(Z)\in\mathbb or transcendental function.

## Newton's method in optimization

In calculus, Newton's method is an iterative method for finding the roots of a differentiable function (i.e. solutions to the equation). In optimization, Newton's method is applied to the derivative of a twice-differentiable function to find the roots of the derivative (solutions to), also known as the stationary points of.

## Non-linear least squares

Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m > n).

## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

## Peter Henrici (mathematician)

Peter Karl Henrici (13 September 1923 – 13 March 1987) was a Swiss mathematician best known for his contributions to the field of numerical analysis.

## Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

## Q-analog

In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as.

## Quasi-Newton method

Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method.

## Rate of convergence

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Richardson extrapolation

In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence.

## Root-finding algorithm

In mathematics and computing, a root-finding algorithm is an algorithm for finding roots of continuous functions.

## 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.

## Second derivative

In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of.

## Seki Takakazu

, also known as,Selin, was a Japanese mathematician and author of the Edo period.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Sharaf al-Dīn al-Ṭūsī

(c. 1135 – c. 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages).

## Stationary point

In mathematics, particularly in calculus, a stationary point or critical point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.

## Steffensen's method

In numerical analysis, Steffensen's method is a root-finding technique similar to Newton's method, named after Johan Frederik Steffensen.

Subgradient methods are iterative methods for solving convex minimization problems.

## System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables.

## Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

## 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.

## The College Mathematics Journal

The College Mathematics Journal, published by the Mathematical Association of America, is an expository journal aimed at teachers of college mathematics, particular those teaching the first two years.

## The Mathematical Gazette

The Mathematical Gazette is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching.

## Thomas Simpson

Thomas Simpson FRS (20 August 1710 – 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals.

## Transcendental equation

A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for.

## Transcendental function

A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.

## William Jones (mathematician)

William Jones, FRS (1675 &ndash; 3 July 1749) was a Welsh mathematician, most noted for his use of the symbol (the Greek letter pi) to represent the ratio of the circumference of a circle to its diameter.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

## References

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