Numerical analysis is the study of

^{3} + 4 = 28
for the unknown quantity ''x''.
For the iterative method, apply the ^{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.

_{0} to $\backslash sqrt$, for instance ''x''_{0} = 1.4, and then computing improved guesses ''x''_{1}, ''x''_{2}, etc. One such method is the famous Babylonian method, which is given by ''x''_{''k''+1} = ''x_{k}''/2 + 1/''x_{k}''. Another method, called 'method X', is given by ''x''_{''k''+1} = (''x''_{''k''}^{2} − 2)^{2} + ''x''_{''k''}. A few iterations of each scheme are calculated in table form below, with initial guesses ''x''_{0} = 1.4 and ''x''_{0} = 1.42.
Observe that the Babylonian method converges quickly regardless of the initial guess, whereas Method X converges extremely slowly with initial guess ''x''_{0} = 1.4 and diverges for initial guess ''x''_{0} = 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.
:Numerical stability is affected by the number of the significant digits the machine keeps. If a machine is used that keeps only the four most significant decimal digits, a good example on loss of significance can be given by the two equivalent functions
:$f(x)=x\backslash left(\backslash sqrt-\backslash sqrt\backslash right)$ and $g(x)=\backslash frac.$
:Comparing the results of
:: $f(500)=500\; \backslash left(\backslash sqrt-\backslash sqrt\; \backslash right)=500\; \backslash left(22.38-22.36\; \backslash right)=500(0.02)=10$
:and
:$\backslash beging(500)\&=\backslash frac\backslash \backslash \; \&=\backslash frac\backslash \backslash \; \&=\backslash frac=11.17\; \backslash end$
: by comparing the two results above, it is clear that loss of significance (caused here by

here

; ACM similarly, in its '' Transactions on Mathematical Software'' ("TOMS" code i

here

. The

''Library of Mathematics Subroutines''

(cod

here

. There are several popular numerical computing applications such as

Springer

1959–

volumes 1–66, 1959–1994

(searchable; pages are images). *'' Journal on Numerical Analysis'

(SINUM)

volumes 1–..., SIAM, 1964–

William H. Press (free, downloadable previous editions)

( archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner

''CSEP'' (Computational Science Education Project)

Numerical Methods

ch 3. in the ''

Numerical Interpolation, Differentiation and Integration

ch 25. in the ''Handbook of Mathematical Functions'' (

Numerical Methods

(), Stuart Dalziel

Lectures on Numerical Analysis

Dennis Deturck and Herbert S. Wilf

Numerical methods

John D. Fenton

Numerical Methods for Physicists

Anthony O’Hare

Lectures in Numerical Analysis

( archived), R. Radok Mahidol University

Introduction to Numerical Analysis for Engineering

Henrik Schmidt

''Numerical Analysis for Engineering''

D. W. Harder

Introduction to Numerical Analysis

Doron Levy

Numerical Analysis - Numerical Methods

(archived), John H. Mathews California State University Fullerton {{DEFAULTSORT:Numerical Analysis Mathematical physics Computational science

algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

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, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied in th ...

(as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding 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, 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 equations 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 equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...

s and Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

s for simulating living cells in medicine and biology.
Before modern computers, numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathe ...

s often relied on hand interpolation
In the 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 often has ...

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 or sexagenary, is a numeral system with sixty as its 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 form ...

numerical approximation of the square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...

, 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 δ ...

in a unit square.
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.
General introduction

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following: * 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 solvingpartial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...

s numerically.
* Hedge fund
A hedge fund is a pooled investment fund that trades in relatively liquid assets and is able to make extensive use of more complex trading, portfolio-construction, and risk management techniques in an attempt to improve performance, such as s ...

s (private investment funds) use tools from all fields of numerical analysis to attempt to calculate the value of stock
In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...

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 ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve dec ...

.
* Insurance companies use numerical programs for actuarial analysis.
The rest of this section outlines several important themes of numerical analysis.
History

The field of numerical analysis predates the invention of modern computers by many centuries.Linear interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known poi ...

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
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a rea ...

, Lagrange interpolation polynomial, Gaussian elimination, or Euler's 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. It is the most basic explicit ...

. The origins of modern numerical analysis are often linked to a 1947 paper by John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cov ...

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
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...

publication edited by Abramowitz and Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and ...

, 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.
Direct and iterative methods

Consider the problem of solving :3''x''bisection method
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and th ...

to ''f''(''x'') = 3''x''Discretization and numerical integration

In a two-hour race, the speed of the car is measured at three instants and recorded in the following table. A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately . This would allow us to estimate the total distance traveled as + + = , which is an example of numerical integration (see below) using a Riemann sum, because displacement is theintegral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with dif ...

of velocity.
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'').
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, 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 are represented by linear relationships. Linear programming is ...

. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
**Structural stab ...

).
In contrast to direct methods, iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pr ...

s are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT
Converge ICT Solutions Inc., commonly referred to as Con ...

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
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and th ...

, 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 In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace wit ...

and the conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an itera ...

. 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.
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 numeric ...

'. For example, the solution of a differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

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
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal numbe ...

.
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 numbers exactly on a machine with finite memory (which is what all practicaldigital computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These progr ...

s 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

Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algo ...

is a notion in numerical analysis. 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. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation ''x''catastrophic cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
For example, if there are two studs, one L_ ...

from subtracting approximations to the nearby numbers $\backslash sqrt$ and $\backslash sqrt$, despite the subtraction being computed exactly) has a huge effect on the results, even though both functions are equivalent, as shown below
:: $\backslash begin\; f(x)\&=x\; \backslash left(\backslash sqrt-\backslash sqrt\; \backslash right)\backslash \backslash \; \&=x\; \backslash left(\backslash sqrt-\backslash sqrt\; \backslash right)\backslash frac\backslash \backslash \; \&=x\backslash frac\backslash \backslash \; \&=x\backslash frac\; \backslash \backslash \; \&=x\backslash frac\; \backslash \backslash \; \&=\backslash frac\; \backslash \backslash \; \&=g(x)\; \backslash end$
: The desired value, computed using infinite precision, is 11.174755...
* The example is a modification of one taken from Mathew; Numerical methods using MATLAB, 3rd ed.
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.Interpolation, extrapolation, and regression

Interpolation
In the 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 often has ...

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
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between know ...

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
Regression or regressions may refer to:
Science
* Marine regression, coastal advance due to falling sea level, the opposite of marine transgression
* Regression (medicine), a characteristic of diseases to express lighter symptoms or less extent ...

is also similar, but it takes into account that the data is 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
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...

-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 somematrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class o ...

are Gaussian elimination, LU decomposition
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a ...

, Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effic ...

for symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

(or hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature ...

) and positive-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...

, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method
In numerical linear algebra, the Jacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. Th ...

, Gauss–Seidel method, successive over-relaxation In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging ...

and conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an itera ...

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
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...

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 orsingular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...

s. For instance, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

.
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 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 are represented by linear relationships. Linear programming is ...

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 definiteintegral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with dif ...

. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. 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 or quasi-Monte Carlo method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). This is in contrast to the regu ...

s (see Monte Carlo integration
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at ...

), 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, afinite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for th ...

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 (e.g. inner product, norm, topology, etc.) and the linear functions defined on ...

. 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. TheNetlib Netlib is a repository of software for scientific computing maintained by AT&T, Bell Laboratories, the University of Tennessee and Oak Ridge National Laboratory. Netlib comprises many separate programs and libraries
A library is a collecti ...

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
Free software or libre software is computer software distributed under terms that allow users to run the software for any purpose as well as to study, change, and distribute it and any adapted versions. Free software is a matter of liberty, no ...

alternative is the GNU Scientific Library.
Over the years the Royal Statistical Society
The Royal Statistical Society (RSS) is an established statistical society. It has three main roles: a British learned society for statistics, a professional body for statisticians and a charity which promotes statistics for the public good.
...

published numerous algorithms in its ''Applied Statistics'' (code for these "AS" functions ihere

; ACM similarly, in its '' Transactions on Mathematical Software'' ("TOMS" code i

here

. The

Naval Surface Warfare Center
*
A Naval Surface Warfare Center (NSWC) is part of the Naval Sea Systems Command (NAVSEA) operated by the United States Navy
The United States Navy (USN) is the maritime service branch of the United States Armed Forces and one of the ...

several times published it''Library of Mathematics Subroutines''

(cod

here

. There are several popular numerical computing applications such as

MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementati ...

, TK Solver
TK Solver (originally TK!Solver) is a mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc.
History
Invented by Milos Konopasek in the late ...

, S-PLUS
S-PLUS is a commercial implementation of the S programming language sold by TIBCO Software Inc.
It features object-oriented programming capabilities and advanced analytical algorithms.
Due to the increasing popularity of the open source S succ ...

, and IDL as well as free and open source alternatives such as FreeMat
FreeMat is a free open-source numerical computing environment and programming language, similar to MATLAB and GNU Octave. In addition to supporting many MATLAB functions and some IDL functionality, it features a codeless interface to external C, ...

, Scilab
Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simu ...

, GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a lan ...

(similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to S-PLUS), Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e. ...

, and Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (p ...

with libraries such as NumPy, SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, si ...

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 system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...

s such as Mathematica also benefit from the availability of arbitrary-precision arithmetic
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are l ...

which can provide more accurate results.
Also, any spreadsheet
A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in ce ...

software can be used to solve simple problems relating to numerical analysis.
Excel
ExCeL London (an abbreviation for Exhibition Centre London) is an exhibition centre, international convention centre and former hospital in the Custom House area of Newham, East London. It is situated on a site on the northern quay of the ...

, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".
See also

* :Numerical analysts * Analysis of algorithms * Computational science * Computational physics * Gordon Bell Prize *Interval arithmetic
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...

* List of numerical analysis topics
* Local linearization method
* Numerical differentiation
*Numerical Recipes
''Numerical Recipes'' is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since ...

* Probabilistic numerics
* Symbolic-numeric computation
* Validated numerics
Notes

References

Citations

Sources

* * * * * * (examples of the importance of accurate arithmetic). *External links

Journals

*''Numerische Mathematik
''Numerische Mathematik'' is a peer-reviewed mathematics journal on numerical analysis. It was established in 1959 and is published by Springer Science+Business Media. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2 ...

'', 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)

( archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner

''CSEP'' (Computational Science Education Project)

U.S. Department of Energy
The United States Department of Energy (DOE) is an executive department of the U.S. federal government that oversees U.S. national energy policy and manages the research and development of nuclear power and nuclear weapons in the United States ...

( archived 2017-08-01) Numerical Methods

ch 3. in the ''

Digital Library of Mathematical Functions
The Digital Library of Mathematical Functions (DLMF) is an online project at the National Institute of Standards and Technology (NIST) to develop a database of mathematical reference data for special functions and their applications. It is inten ...

''Numerical Interpolation, Differentiation and Integration

ch 25. in the ''Handbook of Mathematical Functions'' (

Abramowitz and Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and ...

)
Online course material

Numerical Methods

(), Stuart Dalziel

University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...

Lectures on Numerical Analysis

Dennis Deturck and Herbert S. Wilf

University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universiti ...

Numerical methods

John D. Fenton

University of Karlsruhe
The Karlsruhe Institute of Technology (KIT; german: Karlsruher Institut für Technologie) is a public university, public research university in Karlsruhe, Germany. The institute is a national research center of the Helmholtz Association.
KIT wa ...

Numerical Methods for Physicists

Anthony O’Hare

Oxford University
Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...

Lectures in Numerical Analysis

( archived), R. Radok Mahidol University

Introduction to Numerical Analysis for Engineering

Henrik Schmidt

Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...

''Numerical Analysis for Engineering''

D. W. Harder

University of Waterloo
The University of Waterloo (UWaterloo, UW, or Waterloo) is a public research university with a main campus in Waterloo, Ontario, Canada. The main campus is on of land adjacent to "Uptown" Waterloo and Waterloo Park. The university also operates ...

Introduction to Numerical Analysis

Doron Levy

University of Maryland
The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public land-grant research university in College Park, Maryland. Founded in 1856, UMD is the flagship institution of the University System of ...

Numerical Analysis - Numerical Methods

(archived), John H. Mathews California State University Fullerton {{DEFAULTSORT:Numerical Analysis Mathematical physics Computational science