Témbongkeun sumber pikeun Analisis numeris

{{tarjamahkeun|Inggris}}
[[Image:Ybc7289-bw.jpg|thumb|250px|right|Tablet lempung Babylonia  YBC 7289<br>(c. 1800–1600 BCE) [http://www.math.ubc.ca/~cass/Euclid/ybc/ybc.html] with annotations. (Image by Bill Casselman)]]
'''Analisis numeris''' nyaéta pangajaran [[algoritma]] keur pasualan-pasualan''matematika kontinyu'' (keur ngabédakeun jeung [[discrete mathematics|matematika diskrit]]).

One of the éarliest mathematical writings is the Babylonian tablet YBC 7289, which gives a [[sexagesimal]] numerical approximation of <math>\sqrt{2}</math>, the length of the diagonal in a unit square.<ref>The approximation of the [[square root of 2]] is four [[sexagesimal]] figures, which is about six [[decimal]] figures. 1 + 24/60 + 51/60<sup>2</sup> + 10/60<sup>3</sup> = 1.41421296... <br>[http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection] {{Webarchive|url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |date=2012-08-13 }}</ref>
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.<ref>The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled [http://www.nzqa.govt.nz/nqfdocs/units/pdf/13004.pdf CARPENTRY THEORY: Demonstrate knowledge of setting out a building]</ref> In a square wall section that is two méters by two méters, a diagonal béam has to be <math>\sqrt{8} \approx 2.83</math> méters long.<ref>By the [[Pythagorean theorem]], a square whose side is 2 meters has a diagonal of length <math>\sqrt{2^2+2^2}=\sqrt{8}</math> meters.</ref>

Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation to <math>\sqrt{2}</math>, modérn numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instéad, much of numerical analysis is concerned with obtaining approximate solutions while maintaining réasonable 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 equation]]s appéar in the movement of héavenly bodies (planets, stars and galaxies); [[Optimization (mathematics)|optimization]] occurs in portfolio management; [[numerical linear algebra]] is essential to quantitative psychology; [[stochastic differential equation]]s and [[Markov chain]]s are essential in simulating living cells for medicine and biology.

Before the advent of modérn computers numerical methods often depended on hand [[interpolation]] in large printed tables. 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.

==General introduction==
We will now outline several important themes of numerical analysis. The overall goal is the design and analysis of techniques to give approximate solutions to hard problems. To fix idéas, the réader might consider the following problems and methods:

* If a company wants to put a toothpaste commercial on television, it might produce five commercials and then choose the best one by testing éach one on a focus group. This would be an example of a [[Monte Carlo method|Monte Carlo optimization]] method.
* To send a rocket to the moon, rocket scientists will need a rocket simulator. This simulator will essentially be an integrator for an [[ordinary differential equation]].
* Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. These simulations are essentially solving [[partial differential equation]]s numerically.
* [[Hedge fund]]s (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.
* Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called [[operations research]].
* Insurance companies use numerical programs for [[Actuary|actuarial]] analysis.

===History===

The field of numerical analysis predates the invention of modérn computers by many centuries. [[Linear interpolation]] was alréady in use more than 2000 yéars ago. Many gréat mathematicians of the past were préoccupied by numerical analysis, as is obvious from the names of important algorithms like [[Newton's method]], [[Lagrange polynomial|Lagrange interpolation polynomial]], [[Gaussian elimination]], or [[Euler's method]].

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.

===Direct and iterative methods===

{| class="wikitable" style="float: right; width: 250px; margin-left: 1em;"
 |-
 |
'''Direct vs iterative methods'''

Consider the problem of solving

:3''x''<sup>3</sup>+4=28

for the unknown quantity ''x''.

{| align="center"
 |+ Direct Method
 |-
 | &nbsp; || 3''x''<sup>3</sup> + 4 = 28.
 |-
 | ''Subtract 4'' || 3''x''<sup>3</sup> = 24.
 |-
 | ''Divide by 3'' || ''x''<sup>3</sup> = 8.
 |-
 | ''Take cube roots'' || ''x'' = 2.
 |}

For the iterative method, apply the [[bisection method]] to ''f''(''x'') = 3''x''<sup>3</sup> + 4. The initial values are ''a'' = 0, ''b'' = 3, ''f''(''a'') = 4, ''f''(''b'') = 85.

{| align="center"
 |+ Iterative Method
 |-
 | ''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 ====

[[Image:Schumacher (Ferrari) in practice at USGP 2005.jpg|right|125px]]
In a two hour race, we have méasured the speed of the car at three instants and recorded them in the following table.

 Time  0:20 1:00 1:40&nbsp;km/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&nbsp;km/h)=93.3&nbsp;km. This would allow us to estimate the total distance traveled as 93.3&nbsp;km + 100&nbsp;km + 120&nbsp;km = 313.3&nbsp;km, which is an example of '''numerical integration''' (see below) using a [[Riemann sum]], because displacement is the [[integral]] of velocity.

'''Ill posed problem''': Take the function ''f''(''x'') = 1/(''x''&nbsp;&minus;&nbsp;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 néarly 1000. Evaluating ''f''(''x'') néar ''x'' = 1 is an ill-conditioned problem.

'''Well-posed problem''': By contrast, the function <math>f(x)=\sqrt{x}</math> 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 [[Computer numbering formats|infinite precision arithmetic]]. Examples include [[Gaussian elimination]], the [[QR]] factorization method for solving [[system of linear equations|systems
of linéar equations]], and the [[simplex method]] of [[linear programming]]. In practice, [[Floating point|finite precision]] is used and the result is an approximation of the true solution (assuming [[Numerically stable|stability]]).

In contrast to direct methods, [[iterative method]]s are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that [[convergence|converge]] to the exact solution only in the limit. 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 réach the solution in finitely many 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.

===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. 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 error]]s arise because it is impossible to represent all [[real number]]s exactly on a [[finite-state machine]] (which is what all practical [[digital computer]]s are).

On a pocket calculator, if one enters 0.0000000000001 (or the maximum number of zeros possible), then a +, and then 100000000000000 (again, the maximum number of zeros possible), one will obtain the number 100000000000000 again, and not 100000000000000.0000000000001. The calculator's answer is incorrect because of ''roundoff'' in the calculation.

====Truncation and discretization error====

[[Truncation]] errors are committed when an iterative method is terminated 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. For instance, in the iteration in the sidebar to compute the solution of <math>3x^3+4=28</math>, 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 alréady 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 léads 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. This is only possible if the problem is [[condition number|well-conditioned]], méaning that the solution changes by only a small amount if the problem data are changed by a small amount. 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 ''x''<sub>1</sub> to <math>\sqrt{2}</math>, for instance ''x''<sub>1</sub>=1.4, and then computing improved guesses ''x''<sub>2</sub>, ''x''<sub>3</sub>, etc... One such method is the famous [[Babylonian method]], which is given by ''x''<sub>''k''+1</sub> = ''x<sub>k</sub>''/2 + 1/''x<sub>k</sub>''. Another iteration, which we will call Method X, is given by ''x''<sub>''k'' + 1</sub> = (''x''<sub>''k''</sub><sup>2</sup>&minus;2)<sup>2</sup> + ''x''<sub>''k''</sub>.<ref>This is a [[fixed point iteration]] for the equation <math>x=(x^2-2)^2+x=f(x)</math>, whose solutions include <math>\sqrt{2}</math>. The iterates always move to the right since <math>f(x)\geq x</math>. Hence <math>x_1=1.4<\sqrt{2}</math> converges and <math>x_1=1.42>\sqrt{2}</math> diverges.</ref> We have calculated a few iterations of éach scheme in table form below, with initial guesses ''x''<sub>1</sub> = 1.4 and ''x''<sub>1</sub> = 1.42.

{| class="wikitable"
 |-
! Babylonian
! Babylonian
! Method X
! Method X
 |-
 | ''x''<sub>1</sub> = 1.4
 | ''x''<sub>1</sub> = 1.42
 | ''x''<sub>1</sub> = 1.4
 | ''x''<sub>1</sub> = 1.42
 |-
 | ''x''<sub>2</sub> = 1.4142857...
 | ''x''<sub>2</sub> = 1.41422535...
 | ''x''<sub>2</sub> = 1.4016
 | ''x''<sub>2</sub> = 1.42026896
 |-
 | ''x''<sub>3</sub> = 1.414213564...
 | ''x''<sub>3</sub> = 1.41421356242...
 | ''x''<sub>3</sub> = 1.4028614...
 | ''x''<sub>3</sub> = 1.42056...
 |-
 |
 |
 | ...
 | ...
 |-
 |
 |
 | ''x''<sub>1000000</sub> = 1.41421...
 | ''x''<sub>28</sub> = 7280.2284...
 |}

Observe that the Babylonian method converges fast regardless of the initial guess, wheréas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 1.42. Hence, the Babylonian method is said to be numerically stable, while Method X is said to be 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===

{| class="wikitable" style="float: right; width: 250px; clear: right; margin-left: 1em;"
 |
'''Interpolation''': We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linéar 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 yéar and was 100 billion dollars last yéar, we might extrapolate that it will be 105 billion dollars this yéar.

[[Image:Linear-regression.png|right|100px|A line through 20 points?]]

'''Regression''': In linéar regression, given ''n'' points, we compute a line that passes as close as possible to those ''n'' points.

[[Image:LemonadeJuly2006.JPG|right|100px|How much for a glass of lemonade?]]

'''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 éach incréase 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.

[[Image:Wind-particle.png|right|Wind direction in blue, true trajectory in black, Euler method in red.]]

'''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 féather into the wind, what happens? The féather will follow the air currents, which may be very complex. One approximation is to méasure the speed at which the air is blowing néar the féather every second, and advance the simulated féather as if it were moving in a straight line at that same speed for one second, before méasuring the wind speed again. This is called the [[Euler method]] for solving an ordinary differential equation.
|}

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 error]]s arising from the use of [[floating point]] arithmetic.

===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 linéar between every pair of successive points. This can be generalized to [[polynomial interpolation]], which is sometimes more accurate but suffers from [[Runge's phenomenon]]. Other interpolation methods use localized functions like [[spline (mathematics)|spline]]s or [[wavelet]]s.

[[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.

[[Regression analysis|Regression]] is also similar, but it takes into account that the data is imprecise. Given some points, and a méasurement 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.

===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 linéar or not. For instance, the equation <math>2x+5=3</math> is linéar while <math>2x^2+5=3</math> is not.

Much effort has been put in the development of methods for solving systems of linéar equations. Standard direct methods i.e. methods that use some [[matrix decomposition]] are [[Gaussian elimination]], [[LU decomposition]], [[Cholesky decomposition]] for [[symmetric matrix|symmetric]] (or [[hermitian matrix|hermitian]]) and [[positive-definite matrix]], and [[QR decomposition]] for non-square matrices. [[Iterative method]]s such as the [[Jacobi method]], [[Gauss-Seidel method]], [[successive over-relaxation]] and [[conjugate gradient method]] are usually preferred for large systems.

[[Root-finding algorithm]]s are used to solve nonlinéar equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is [[derivative|differentiable]] and the derivative is known, then [[Newton's method]] is a popular choice. [[Linearization]] is another technique for solving nonlinéar equations.

===Solving eigenvalue or singular value problems===

Several important problems can be phrased in terms of [[eigenvalue decomposition]]s or [[singular value decomposition]]s. For instance, the [[spectral image compression]] algorithm <ref>[http://online.redwoods.cc.ca.us/instruct/darnold/maw/single.htm The Singular Value Decomposition and Its Applications in Image Compression] {{Webarchive|url=https://web.archive.org/web/20061004041704/http://online.redwoods.cc.ca.us/instruct/darnold/maw/single.htm |date=2006-10-04 }}</ref> 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 éach web page according to which subject it belongs to.

===Optimization===
{{utama|Optimization (mathematics)}}

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some [[constraint]]s.

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]] déals with the case that both the objective function and the constraints are linéar. A famous method in linéar 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===
{{utama|Numerical integration}}

Numerical integration, in some instances also known as numerical [[quadrature (mathematics)|quadrature]], asks for the value of a definite [[integral]]. 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 method|Monte Carlo]] or [[quasi-Monte Carlo method]]s (see [[Monte Carlo integration]]), or, in modestly large dimensions, the method of [[sparse grid]]s.

===Differential equations===
:''Main articles: [[Numerical ordinary differential equations]], [[Numerical partial differential equations]].''

Numerical analysis is also concerned with computing (in an approximate way) the solution of [[differential equation]]s, both ordinary differential equations and [[partial differential equation]]s.

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 théoretical justification of these methods often involves théorems from [[functional analysis]]. This reduces the problem to the solution of an algebraic equation.

==Software==
{{utama|List of numerical analysis software}}

Since the late twentieth century, most algorithms are implemented and run on a computer. The [[Netlib]] repository contains various collections of software routines for numerical problems, mostly in [[Fortran]] and [[C (programming language)|C]]. Commercial products implementing many different numerical algorithms include the [[IMSL Numerical Libraries|IMSL]] and [[Numerical Algorithms Group|NAG]] libraries; a free alternative is the [[GNU Scientific Library]].
 
[[MATLAB]] is a popular commercial programming language for numerical scientific calculations, but there are commercial alternatives such as [[S-PLUS]] and [[IDL programming language|IDL]], or semi-free alternative [[Scilab]] as well as free and open source alternatives such as [[FreeMat]], [[GNU Octave]] (similar to Matlab), [[IT++]] (a C++ library), [[R (programming language)|R]] (similar to S-PLUS) and certain variants of [[Python (programming language)|Python]]. Performance varies widely: while vector and matrix operations are usually fast, scalar loops vary in speed by more than an order of magnitude.<ref>[http://www.sciviews.org/benchmark/ Speed comparison of various number crunching packages] {{Webarchive|url=https://web.archive.org/web/20061005024002/http://www.sciviews.org/benchmark/ |date=2006-10-05 }}</ref>

Many [[computer algebra system]]s such as [[Mathematica]] or [[Maple computer algebra system|Maple]] ([[free software]] systems include [[Software for Algebra and Geometry Experimentation|SAGE]], [[Maxima (software)|Maxima]], [[Axiom computer algebra system|Axiom]], and [[Yacas]]), can also be used for numerical computations. However, their strength typically lies in symbolic computations.  Also, any [[spreadsheet]] software can be used to solve simple problems relating to numerical analysis.

==Tempo oge==

*[[Scientific computing]]
*[[List of numerical analysis topics]]
*[[Gram-Schmidt process]]
*[[Halting problem]]
*[[Numerical differentiation]]

==Catetan==
<references/>

==Rujukan==
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This template can be used for additional references.
*{{cite book |last= |first= |authorlink= |coauthors= |title= |year= |publisher= |location= |id= }}
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*{{cite book |last=Gilat |first=Amos |authorlink= |coauthors= |title=MATLAB: An Introduction with Applications | edition=2nd edition |year=2004 |publisher=John Wiley & Sons |location= |id=ISBN 0-471-69420-7 }}
*{{cite book |last=Hildebrand |first=F. B. | title=Introduction to Numerical Analysis | edition=2nd edition |year=1974 |publisher=McGraw-Hill |location= |id= ISBN 0-070-28761-9}}
*{{cite book |last=Leader |first=Jeffery J. | title=Numerical Analysis and Scientific Computation |year=2004 |publisher=Addison Wesley |location= |id= ISBN 0-201-73499-0 }}
* Trefethen, Lloyd N. (2006). [http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/NAessay.pdf "Numerical analysis"], 20 pages. To appéar in: Timothy Gowers and June Barrow-Green (editors), ''Princeton Companion of Mathematics'', Princeton University Press.

==Tumbu kaluar==
{{Wikibooks}}
* [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN362160546 Numerische Mathematik], volumes 1-66, Springer, 1959-1994 (séarchable; pages are images). {{en icon}} {{de icon}}
*[http://www.mathcom.com/corpdir/techinfo.mdir/index.html Scientific computing FAQ] {{Webarchive|url=https://web.archive.org/web/20080225141305/http://www.mathcom.com/corpdir/techinfo.mdir/index.html |date=2008-02-25 }}
*[http://dmoz.org/Science/Math/Numerical_Analysis/ Numerical analysis DMOZ category]
*[http://norma.mas.ecp.fr/wikimas/ScientificComputingSoftware Lists of free software for scientific computing and numerical analysis] {{Webarchive|url=https://web.archive.org/web/20081223214554/http://norma.mas.ecp.fr/wikimas/ScientificComputingSoftware |date=2008-12-23 }} {{en icon}} {{fr icon}}
*[http://www.indiana.edu/~statmath/bysubject/numerics.html Numerical Computing Resources on the Internet] {{Webarchive|url=https://web.archive.org/web/20080317102746/http://www.indiana.edu/~statmath/bysubject/numerics.html |date=2008-03-17 }} - a list maintained by Indiana University Stat/Math Center
*[http://www.nr.com/ Numerical Recipes Homepage - with free, complete downloadable books] {{Webarchive|url=https://web.archive.org/web/20080516044136/http://www.nr.com/ |date=2008-05-16 }}
*[http://www.apropos-logic.com/nc/ Java Number Cruncher features free, downloadable code samples that graphically illustrate common numerical algorithms]
*[http://math.fullerton.edu/mathews/numerical.html Numerical Analysis Project by John H. Mathews]
*[http://www.fceia.unr.edu.ar/~fisicomp/apuntes/biblios/altnr.htm Alternatives to Numerical Recipes] {{Webarchive|url=https://web.archive.org/web/20090210071240/http://www.fceia.unr.edu.ar/~fisicomp/apuntes/biblios/altnr.htm |date=2009-02-10 }}

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