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Numerical analysis is the study of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
s that use numerical approximation (as opposed to symbolic manipulations) for the problems of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
(as distinguished from
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include:
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s as found in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
(predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and
Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
s for simulating living cells in medicine and biology. Before modern computers, numerical methods often relied on hand
interpolation In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
formulas, using data from large printed tables. Since the mid-20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms. The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection ( YBC 7289), gives a
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
numerical approximation of the
square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, the length of the
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
in a
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...
. Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.


Applications

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to a wide variety of hard problems, many of which are infeasible to solve symbolically: * Advanced numerical methods are essential in making numerical weather prediction feasible. * Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. * Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s numerically. * In the financial field, (private investment funds) and other financial institutions use quantitative finance tools from numerical analysis to attempt to calculate the value of
stock Stocks (also capital stock, or sometimes interchangeably, shares) consist of all the Share (finance), shares by which ownership of a corporation or company is divided. A single share of the stock means fractional ownership of the corporatio ...
s and derivatives more precisely than other market participants. * Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of
operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ...
. * Insurance companies use numerical programs for actuarial analysis.


History

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial,
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...
, or Euler's method. The origins of modern numerical analysis are often linked to a 1947 paper by
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
and Herman Goldstine, but others consider modern numerical analysis to go back to work by E. T. Whittaker in 1912. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done. The Leslie Fox Prize for Numerical Analysis was initiated in 1985 by the Institute of Mathematics and its Applications.


Key concepts


Direct and iterative methods

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...
, the QR factorization method for solving systems of linear equations, and the simplex method of
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomia ...
. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability). In contrast to direct methods,
iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...
s are not expected to terminate in a finite number of steps, even if infinite precision were possible. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems. Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method. As an example, consider the problem of solving :3''x''3 + 4 = 28 for the unknown quantity ''x''. For the iterative method, apply the bisection method to ''f''(''x'') = 3''x''3 − 24. The initial values are ''a'' = 0, ''b'' = 3, ''f''(''a'') = −24, ''f''(''b'') = 57. From this table it can be concluded that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.


Conditioning

Ill-conditioned problem: Take the function . Note that ''f''(1.1) = 10 and ''f''(1.001) = 1000: a change in ''x'' of less than 0.1 turns into a change in ''f''(''x'') of nearly 1000. Evaluating ''f''(''x'') near ''x'' = 1 is an ill-conditioned problem. Well-conditioned problem: By contrast, evaluating the same function near ''x'' = 10 is a well-conditioned problem. For instance, ''f''(10) = 1/9 ≈ 0.111 and ''f''(11) = 0.1: a modest change in ''x'' leads to a modest change in ''f''(''x'').


Discretization

Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called '
discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
'. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.


Generation and propagation of errors

The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.


Round-off

Round-off errors arise because it is impossible to represent all
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s exactly on a machine with finite memory (which is what all practical digital computers are).


Truncation and discretization error

Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of 3x^3+4=28, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01. Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type is even more inexact. A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.


Numerical stability and well-posed problems

An algorithm is called '' numerically stable'' if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is '' well-conditioned'', meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error. Both the original problem and the algorithm used to solve that problem can be well-conditioned or ill-conditioned, and any combination is possible. So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem.


Areas of study

The field of numerical analysis includes many sub-disciplines. Some of the major ones are:


Computing values of functions

One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of
floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
.


Interpolation, extrapolation, and regression

Interpolation In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? Extrapolation is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found. Regression is also similar, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The least squares-method is one way to achieve this.


Solving equations and systems of equations

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation 2x+5=3 is linear while 2x^2+5=3 is not. Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...
, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting. Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.


Solving eigenvalue or singular value problems

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.


Optimization

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints. The field of optimization is further split in several subfields, depending on the form of the
objective function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
and the constraint. For instance,
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomia ...
deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method. The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.


Evaluating integrals

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or
Gaussian quadrature In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for . Th ...
. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use
Monte Carlo Monte Carlo ( ; ; or colloquially ; , ; ) is an official administrative area of Monaco, specifically the Ward (country subdivision), ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is located. Informally, the name also refers to ...
or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.


Differential equations

Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations. Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a
finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...
method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. This reduces the problem to the solution of an algebraic equation.


Software

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the
GNU Scientific Library The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
. Over the years the Royal Statistical Society published numerous algorithms in its ''Applied Statistics'' (code for these "AS" functions i
here
; ACM similarly, in its '' Transactions on Mathematical Software'' ("TOMS" code i
here
. The Naval Surface Warfare Center several times published it
''Library of Mathematics Subroutines''
(cod
here
. There are several popular numerical computing applications such as MATLAB, TK Solver, S-PLUS, and IDL (programming language), IDL as well as free and open-source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (programming language), R (similar to S-PLUS), Julia (programming language), Julia, and Python (programming language), Python with libraries such as NumPy, SciPy and SymPy. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude. Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary-precision arithmetic which can provide more accurate results. Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis. Microsoft_Excel#, Excel, for example, has hundreds of Microsoft Excel#Functions, available functions, including for matrices, which may be used in conjunction with its Microsoft Excel#Add-ins, built in "solver".


See also

*:Numerical analysts *Analysis of algorithms *Approximation theory *Computational science *Computational physics *Gordon Bell Prize *Interval arithmetic *List of numerical analysis topics *Local linearization method *Numerical differentiation *Numerical Recipes *Probabilistic numerics *Symbolic-numeric computation *Validated numerics


Notes


References


Citations


Sources

* * * * * David Kincaid and Ward Cheney: ''Numerical Analysis : Mathematics of Scientific Computing'', 3rd Ed., AMS, ISBN 978-0-8218-4788-6 (2002). * * * (examples of the importance of accurate arithmetic). *


External links


Journals

*''Numerische Mathematik'', volumes 1–...
Springer
1959–
volumes 1–66, 1959–1994
(searchable; pages are images). *''Journal on Numerical Analysis'
(SINUM)
volumes 1–..., SIAM, 1964–


Online texts

*

William H. Press (free, downloadable previous editions)

(Internet Archive, archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
''CSEP'' (Computational Science Education Project)
U.S. Department of Energy (Internet Archive, archived 2017-08-01)
Numerical Methods
ch 3. in the ''Digital Library of Mathematical Functions''
Numerical Interpolation, Differentiation and Integration
ch 25. in the ''Handbook of Mathematical Functions'' ( Abramowitz and Stegun)
Tobin A. Driscoll and Richard J. Braun: ''Fundamentals of Numerical Computation'' (free online version)


Online course material



(), Stuart Dalziel University of Cambridge
Lectures on Numerical Analysis
Dennis Deturck and Herbert S. Wilf University of Pennsylvania
Numerical methods
John D. Fenton University of Karlsruhe
Numerical Methods for Physicists
Anthony O’Hare Oxford University
Lectures in Numerical Analysis
(Internet Archive, archived), R. Radok Mahidol University
Introduction to Numerical Analysis for Engineering
Henrik Schmidt Massachusetts Institute of Technology
''Numerical Analysis for Engineering''
D. W. Harder University of Waterloo
Introduction to Numerical Analysis
Doron Levy University of Maryland
Numerical Analysis - Numerical Methods
(archived), John H. Mathews California State University Fullerton {{DEFAULTSORT:Numerical Analysis Numerical analysis, Mathematical physics Computational science