Babylonian clay tablet YBC 7289
(c. 1800–1600 BCE) [1] with annotations. (Image by Bill Casselman)

Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the

One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of $\sqrt{2}$, the length of the diagonal in a unit square. Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. [1] Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction. [2] In a rectangular wall section that is 2. 40 meter by 3. 75 meter, a diagonal beam has to be 4. 45 meters long. [3]

Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation to $\sqrt{2}$, modern numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numerical linear algebra is essential to quantitative psychology; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function Numerical linear algebra is the study of Algorithms for performing Linear algebra computations most notably matrix operations on Computers A stochastic differential equation (SDE is a Differential equation in which one or more of the terms is a Stochastic process, thus resulting in a solution which is In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property.

Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of Nowadays (after mid 20th century) these tables have fallen into disuse, because computers can calculate the required functions. The interpolation algorithms nevertheless may be used as part of the software for solving differential equations and the like. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the

## 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. Numerical weather prediction uses current weather conditions as input into Mathematical models of the atmosphere to predict the weather.
• Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its
• 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 equations numerically. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i
• Hedge funds (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants. A hedge fund is a private Investment fund open to a limited range of investors which is permitted by regulators to undertake a wider range of activities than other investment
• Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called operations research. Operations Research (OR in North America South Africa and Australia and Operational Research in Europe is an interdisciplinary branch of applied Mathematics and
• Insurance companies use numerical programs for actuarial analysis. An actuary is a business professional who deals with the financial impact of risk and uncertainty

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 was already in use more than 2000 years ago. Linear interpolation is a method of Curve fitting using linear polynomials 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, or Euler's method. In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the In Numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation Polynomial for a given set of data points In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In Mathematics and Computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving

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. Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U 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. The history of computer hardware encompasses the hardware, its architecture, and its impact on software. 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.

### Direct and iterative methods

Direct vs iterative methods

Consider the problem of solving

3x3+4=28

for the unknown quantity x.

 3x3 + 4 = 28. Subtract 4 3x3 = 24. Divide by 3 x3 = 8. Take cube roots x = 2.

For the iterative method, apply the bisection method to f(x) = 3x3 + 4. In Mathematics, the bisection method is a Root-finding algorithm which repeatedly divides an interval in half and then selects the subinterval in which The initial values are a = 0, b = 3, f(a) = 4, f(b) = 85.

 a b mid f(mid) 0 3 1. 5 14. 125 1. 5 3 2. 25 38. 17. . . 1. 5 2. 25 1. 875 23. 77. . . 1. 875 2. 25 2. 0625 30. 32. . .

We conclude from this table 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.

#### Discretization and numerical integration

In a two hour race, we have measured the speed of the car at three instants and recorded them in the following table.

Time  0:20 1:00 1:40km/h  140  150  180

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 (2/3h x 140 km/h)=93. 3 km. This would allow us to estimate the total distance traveled as 93. 3 km + 100 km + 120 km = 313. 3 km, which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity. In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Ill posed problem: Take the function f(x) = 1/(x − 1). 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-posed problem: By contrast, the function $f(x)=\sqrt{x}$ is continuous and so evaluating it is well-posed.

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. The term computer numbering formats refers to the schemes implemented in Digital computer and Calculator hardware and software to represent numbers A common Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex method of linear programming. In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example In mathematical optimization theory, the simplex algorithm, created by the American Mathematician George Dantzig in 1947, is a popular In Mathematics, linear programming (LP is a technique for optimization of a Linear Objective function, subject to Linear equality In practice, finite precision is used and the result is an approximation of the true solution (assuming stability). In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. In the mathematical subfield of Numerical analysis, numerical stability is a desirable property of numerical Algorithms The precise definition of stability

In contrast to direct methods, iterative methods are not expected to terminate in a number of steps. In Computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations by finding successive Approximations Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or A convergence criterion is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even in infinite precision arithmetic these methods would not reach the solution in finitely many steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the In Mathematics, the bisection method is a Root-finding algorithm which repeatedly divides an interval in half and then selects the subinterval in which The Jacobi method is an algorithm in Linear algebra for determining the solutions of a System of linear equations with largest absolute values in each row and column 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. In mathematics the generalized minimal residual method (usually abbreviated GMRES) is an Iterative method for the numerical solution of a System of In Mathematics, the conjugate gradient method is an Algorithm for the Numerical solution of particular systems of linear equations, namely those 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. For example, the solution of a differential equation is a function. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the 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.

## The 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 finite-state machine (which is what all practical digital computers are). For the acrobatic movement roundoff see Roundoff. A round-off error, also called rounding error, is the difference between the In Mathematics, the real numbers may be described informally in several different ways A computer is a Machine that manipulates data according to a list of instructions.

#### Truncation and discretization error

Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution. In Mathematics, truncation is the term for limiting the number of digits right of the Decimal point, by discarding the least significant ones Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. In Numerical analysis, Computational physics, and Simulation, discretization error is Error resulting from the fact that a function For instance, in the iteration in the sidebar to compute the solution of 3x3 + 4 = 28, after 10 or so iterations, we conclude that the root is roughly 1. 99 (for example). We therefore have a truncation error of 0. 01.

Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.

#### Numerical stability and well posedness

This leads to the notion of numerical stability: an algorithm is numerically stable if an error, once it is generated, does not grow too much during the calculation. In the mathematical subfield of Numerical analysis, numerical stability is a desirable property of numerical Algorithms The precise definition of stability This is only possible 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. In Numerical analysis, the condition number associated with a problem is a measure of that problem's amenability to digital computation that is hownumerically well-conditioned Indeed, if a problem is ill-conditioned, then any error in the data will grow a lot.

However, an algorithm that solves a well-conditioned problem may or may not be numerically stable. 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 x1 to $\sqrt{2}$, for instance x1=1. 4, and then computing improved guesses x2, x3, etc. . . One such method is the famous Babylonian method, which is given by xk+1 = xk/2 + 1/xk. This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good Another iteration, which we will call Method X, is given by xk + 1 = (xk2−2)2 + xk. [4] We have calculated a few iterations of each scheme in table form below, with initial guesses x1 = 1. 4 and x1 = 1. 42.

BabylonianBabylonianMethod XMethod X
x1 = 1. 4x1 = 1. 42x1 = 1. 4x1 = 1. 42
x2 = 1. 4142857. . . x2 = 1. 41422535. . . x2 = 1. 4016x2 = 1. 42026896
x3 = 1. 414213564. . . x3 = 1. 41421356242. . . x3 = 1. 4028614. . . x3 = 1. 42056. . .
. . . . . .
x1000000 = 1. 41421. . . x28 = 7280. 2284. . .

Observe that the Babylonian method converges fast regardless of the initial guess, whereas Method X converges extremely slowly with initial guess 1. 4 and diverges for initial guess 1. 42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.

## Areas of study

The field of numerical analysis is divided in different disciplines according to the problem that is to be solved.

### Computing values of functions

 Interpolation: We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18. 5 degrees at 1:30pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year. Regression: In linear regression, given n points, we compute a line that passes as close as possible to those n points. Optimization: Say you sell lemonade at a lemonade stand, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of$0. 01, you will sell one less lemonade per day. If you could charge $1. 485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging$1. 49 per glass will yield the maximum income of \$220. 52 per day. Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation. In Mathematics and Computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving

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. In Numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an Algorithm for the efficient evaluation Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic. For the acrobatic movement roundoff see Roundoff. A round-off error, also called rounding error, is the difference between the In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number.

### Interpolation, extrapolation, and regression

Interpolation 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? A very simple method is to use linear interpolation, which assumes that the unknown function is linear between every pair of successive points. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of Linear interpolation is a method of Curve fitting using linear polynomials This can be generalized to polynomial interpolation, which is sometimes more accurate but suffers from Runge's phenomenon. In the mathematical subfield of Numerical analysis, polynomial interpolation is the Interpolation of a given Data set by a Polynomial In the mathematical field of Numerical analysis, Runge's phenomenon is a problem that occurs when using Polynomial interpolation with polynomials of Other interpolation methods use localized functions like splines or wavelets. In the mathematical field of Numerical analysis, a spline is a special function defined Piecewise by Polynomials In interpolating A wavelet is a mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution

Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points. In Mathematics, extrapolation is the process of constructing new data points outside a Discrete set of known data points

Regression is also similar, but it takes into account that the data is imprecise. In statistics regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a Dependent variable (response Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The least squares-method is one popular way to achieve this. The method of least squares is used to solve Overdetermined systems Least squares is often applied in statistical contexts particularly Regression analysis.

### 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 2x2 + 5 = 3 is not.

Much effort has been put in the development of methods for solving systems of linear equations. In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example Standard direct methods, i. e. , methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. In the mathematical discipline of Linear algebra, a matrix decomposition is a Factorization of a matrix into some Canonical form. In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In Linear algebra, the LU decomposition is a Matrix decomposition which writes a matrix as the product of a lower and upper Triangular matrix In Mathematics, the Cholesky decomposition is named after André-Louis Cholesky, who found that a symmetric Positive-definite matrix can be In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that In Linear algebra, a positive-definite matrix is a (Hermitian matrix which in many ways is analogous to a Positive Real number. In Linear algebra, the QR decomposition (also called the QR factorization) of a matrix is a decomposition of the matrix into an orthogonal Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems. In Computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations by finding successive Approximations The Jacobi method is an algorithm in Linear algebra for determining the solutions of a System of linear equations with largest absolute values in each row and column The Gauss–Seidel method is a technique used to solve a Linear system of equations. Successive over-relaxation ( SOR) is a Numerical method used to speed up convergence of the Gauss–Seidel method for solving a Linear system of equations In Mathematics, the conjugate gradient method is an Algorithm for the Numerical solution of particular systems of linear equations, namely those

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). A root-finding algorithm is a numerical method or Algorithm, for finding a value x such that f ( x) = 0 for a given function If the function is differentiable and the derivative is known, then Newton's method is a popular choice. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the Linearization is another technique for solving nonlinear equations. In Mathematics and its applications linearization refers to finding the Linear approximation to a function at a given point

### Solving eigenvalue or singular value problems

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators In Linear algebra, the singular value decomposition ( SVD) is an important factorization of a rectangular real or complex matrix For instance, the spectral image compression algorithm [5] is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis. One application is to automatically find the 100 top subjects of discussion on the web, and to then classify each web page according to which subject it belongs to.

### Optimization

Optimization problems ask for the point at which a given function is maximized (or minimized). In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function 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 deals with the case that both the objective function and the constraints are linear. In Mathematics, linear programming (LP is a technique for optimization of a Linear Objective function, subject to Linear equality A famous method in linear programming is the simplex method. In mathematical optimization theory, the simplex algorithm, created by the American Mathematician George Dantzig in 1947, is a popular

The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems. In mathematical optimization problems the method of Lagrange multipliers, named after Joseph Louis Lagrange, is a method for finding the extrema of

### Evaluating integrals

Main article: Numerical integration

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. In Numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes rules, are a group of formulas for Numerical integration (also called In Numerical analysis, Simpson's rule is a method for Numerical integration, the numerical approximation of Definite integrals Specifically it is the following In Numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a Weighted sum of function 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 methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids. Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results In Numerical analysis, a quasi-Monte Carlo method is a method for the computation of an Integral (or some other problem that is based on Low-discrepancy sequences In Mathematics, Monte Carlo integration is Numerical quadrature using pseudorandom numbers That is Monte Carlo integration methods are Algorithms Sparse grids are a numerical technique to represent integrate or interpolate high Dimensional functions

### Differential equations

Main articles: Numerical ordinary differential equations, Numerical partial differential equations. Numerical ordinary differential equations is the part of Numerical analysis which studies the numerical solution of ordinary differential equations (ODEs Numerical partial differential equations is the branch of Numerical analysis that studies the numerical solution of Partial differential equations (PDEs

Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

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 method, or (particularly in engineering) a finite volume method. The finite element method (FEM (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of Partial differential A finite difference is a mathematical expression of the form f ( x + b) &minus f ( x + a) The finite volume method is a method for representing and evaluating Partial differential equations as algebraic equations The theoretical justification of these methods often involves theorems from functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article This reduces the problem to the solution of an algebraic equation.

## Software

Since the late twentieth century, most algorithms are implemented and run on a computer. Listed here are a number of computer programs used for performing numerical calculations acslX is a software application for modeling and evaluating the performance The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Netlib is a repository of software for Scientific computing maintained by AT&T, Bell Laboratories, the University of Tennessee and Oak Ridge Fortran (previously FORTRAN) is a general-purpose, procedural, imperative Programming language that is especially suited to tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library. IMSL (International Mathematics and Statistics Library is a collection of software libraries of Numerical analysis functionality that are implemented in widely used The Numerical Algorithms Group ( NAG) is a non-profit Software company, whose head office is in Oxford, UK. In Computing, the GNU Scientific Library (or GSL) is a Software library written in the C programming language for numerical calculations

MATLAB is a popular commercial programming language for numerical scientific calculations, but there are commercial alternatives such as S-PLUS, LabVIEW, and IDL, or semi-free alternatives Scilab as well as free and open source alternatives such as FreeMat, GNU Octave (similar to Matlab), IT++ (a C++ library), R (similar to S-PLUS) and certain variants of Python. MATLAB is a numerical computing environment and Programming language. S-PLUS is a commercial advanced statistics package sold by Insightful Corporation LabVIEW (short for Lab oratory V irtual I nstrumentation E ngineering W orkbench is a platform and development environment for a IDL, short for interactive data language, is a Programming language that is a popular data analysis language among scientists Scilab is a numerical computational package developed since 1990 by researchers from the INRIA and the École nationale des ponts et chaussées (ENPC FreeMat is a free numerical computing environment and Programming language, similar to MATLAB and GNU Octave. Octave is a free Computer program for performing numerical computations which is mostly compatible with MATLAB. IT++ is a C++ library composed of classes and functions for Linear algebra (matrices and vectors Signal processing and Telecommunication systems The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude. [6]

Many computer algebra systems such as Mathematica can also be used for numerical computations. A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics. Mathematica is a computer program used widely in scientific engineering and mathematical fields However, their strength typically lies in symbolic computations. Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis. A spreadsheet is a Computer application that simulates a paper worksheet

## Notes

1. ^ The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. Computational science (or scientific computing) is the field of study concerned with constructing Mathematical models and numerical solution techniques and using computers This is a list of Numerical analysis topics, by Wikipedia page In Mathematics, particularly Linear algebra and Numerical analysis, the Gram–Schmidt process is a method for orthogonalizing a set of In computability theory, the halting problem is a Decision problem which can be stated as follows given a description of a program and a finite input Numerical differentiation is a technique of Numerical analysis to produce an estimate of the Derivative of a Mathematical function or function Subroutine The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 1 + 24/60 + 51/602 + 10/603 = 1. 41421296. . .
Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
2. ^ The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled CARPENTRY THEORY: Demonstrate knowledge of setting out a building
3. ^ By the Pythagorean theorem, a rectangle whose sides are 2. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry 40 meter and 3. 75 meter has a diagonal of length $\sqrt{2.40^2+3.75^2}\approx 4.45$ meters.
4. ^ This is a fixed point iteration for the equation x = (x2 − 2)2 + x = f(x), whose solutions include $\sqrt{2}$. In Numerical analysis, fixed point iteration is a method of computing fixed points of Iterated functions More specifically given a function f The iterates always move to the right since $f(x)\geq x$. Hence $x_1=1.4<\sqrt{2}$ converges and $x_1=1.42>\sqrt{2}$ diverges.
5. ^ The Singular Value Decomposition and Its Applications in Image Compression
6. ^ Speed comparison of various number crunching packages

## References

• Gilat, Amos (2004). MATLAB: An Introduction with Applications, 2nd edition, John Wiley & Sons. ISBN 0-471-69420-7.
• Hildebrand, F. B. (1974). Introduction to Numerical Analysis, 2nd edition, McGraw-Hill. ISBN 0-070-28761-9.
• Leader, Jeffery J. (2004). Numerical Analysis and Scientific Computation. Addison Wesley. ISBN 0-201-73499-0.
• Trefethen, Lloyd N. (2006). "Numerical analysis", 20 pages. To appear in: Timothy Gowers and June Barrow-Green (editors), Princeton Companion of Mathematics, Princeton University Press.