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{{Short description|StudyMethods of algorithms usingfor numerical approximationapproximations}}

{{Use dmy dates|date=October 2020}}

[[Image:Ybc7289-bw.jpg|thumb|250px|right|Babylonian clay tablet [[YBC 7289]] (c. 1800–1600 BCBCE) with annotations. The approximation of the [[square root of 2]] is four [[sexagesimal]] figures, which is about six [[decimal]] figures. 1 + 24/60 + 51/60² + 10/60³ = 1.41421296...<ref>{{Cite web |url=http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |title=Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection |access-date=2 October 2006 |archive-date=13 August 2012 |archive-url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |url-status=dead }}</ref>]]

'''Numerical analysis''' is the study of [[algorithm]]s that use numerical [[approximation]] (as opposed to [[symbolic computation|symbolic manipulations]]) for the problems of [[mathematical analysis]] (as distinguished from [[discrete mathematics]]). It is the study of numerical methods that attempt atto findingfind approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: [[ordinary differential equation]]s as found in [[celestial mechanics]] (predicting the motions of planets, stars and galaxies), [[numerical linear algebra]] in data analysis,<ref>{{cite book |first=J.W. |last=Demmel |title=Applied numerical linear algebra |publisher=[[Society for Industrial and Applied Mathematics|SIAM]] |date=1997 |isbn=978-1-61197-144-6 |doi=10.1137/1.9781611971446 |url=https://epubs.siam.org/doi/epdf/10.1137/1.9781611971446.fm}}</ref><ref>{{cite book |last1=Ciarlet |first1=P.G. |last2=Miara |first2=B. |last3=Thomas |first3=J.M. |title=Introduction to numerical linear algebra and optimization |publisher=Cambridge University Press |date=1989 |isbn=9780521327886 |oclc=877155729 }}</ref><ref>{{cite book |last1=Trefethen |first1=Lloyd |last2=Bau III |first2=David |title=Numerical Linear Algebra |publisher=SIAM |date=1997 |isbn=978-0-89871-361-9 |url={{GBurl|4Mou5YpRD_kC|pg=PR7}}}}</ref> and [[stochastic differential equation]]s and [[Markov chain]]s for simulating living cells in medicine and biology.

Before modern computers, [[numerical method]]s often relied on hand [[interpolation]] 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.<ref name="20c">{{cite book |last1=Brezinski |first1=C. |last2=Wuytack |first2=L. |title=Numerical analysis: Historical developments in the 20th century |publisher=Elsevier |date=2012 |isbn=978-0-444-59858-5 |url={{GBurl|dt3Z1yu2VxwC|pg=PP6}}}}</ref>

The numerical point of view goes back to the earliest mathematical writings. A tablet from the [[Yale Babylonian Collection]] ([[YBC 7289]]), gives a [[sexagesimal]] numerical approximation of the [[square root of 2]], the length of the [[diagonal]] in a [[unit square]].

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Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.

==Applications==

==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 a wide variety of hard problems, the varietymany of which isare suggestedinfeasible byto thesolve followingsymbolically:

* Advanced numerical methods are essential in making [[numerical weather prediction]] feasible.

* Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.

* Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving [[partial differential equation]]s numerically.

* [[HedgeIn fund]]sthe financial field, (private investment funds) and other financial institutions use [[quantitative finance]] tools from numerical analysis to attempt to calculate the value of [[share capital|stock]]s and [[Derivative (finance)|derivatives]] more precisely than other market participants.<ref>

Stephen Blyth.

[https://wwwbooks.google.com/books/edition/An_Introduction_to_Quantitative_Finance/?id=SXbcAAAAQBAJ "An Introduction to Quantitative Finance"].

2013.

page VII.

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* Insurance companies use numerical programs for [[Actuary|actuarial]] analysis.

==History==

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. Many great mathematicians of the past were preoccupied by numerical analysis,<ref name="20c"/> as is obvious from the names of important algorithms like [[Newton's method]], [[Lagrange polynomial|Lagrange interpolation polynomial]], [[Gaussian elimination]], or [[Euler's method]]. The origins of modern numerical analysis are often linked to a 1947 paper by [[John von Neumann]] and [[Herman Goldstine]],<ref name="watson" /><ref>

{{cite book |editor1-link=Adhemar Bultheel |editor1-first=Adhemar |editor1-last=Bultheel |editor2-first=Ronald |editor2-last=Cools |title=The Birth of Numerical Analysis |volume=10 |publisher= World Scientific |date=2010 |isbn=978-981-283-625-0 |url={{GBurl|pKZpDQAAQBAJ|pg=PR17}} }}

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</ref>

[[File:Handbook of Mathematical Functions, by Abramowitz and Stegun, cover.jpg|right|thumb|128px|NIST publication]]

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.

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The [[Leslie Fox Prize for Numerical Analysis]] was initiated in 1985 by the [[Institute of Mathematics and its Applications]].

==Key concepts==

===Direct and iterative methods===

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in [[Arbitrary-precision arithmetic|infinite precision arithmetic]]. Examples include [[Gaussian elimination]], the [[QR decomposition|QR factorization]] method for solving [[system of linear equations|systems of linear equations]], and the [[simplex method]] of [[linear programming]]. In practice, [[floating-point arithmetic|finite precision]] is used and the result is an approximation of the true solution (assuming [[numerically stable|stability]]).

Consider the problem of solving

In contrast to direct methods, [[iterative method]]s are not expected to terminate in a finite number of steps, even if infinite precision were possible. Starting from an initial guess, iterative methods form successive approximations that [[Limit of a sequence|converge]] to the exact solution only in the limit. A convergence test, often involving [[Residual (numerical analysis)|the residual]], is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the [[bisection method]], and [[Jacobi iteration]]. In computational matrix algebra, iterative methods are generally needed for large problems.<ref>{{cite book |first=Y. |last=Saad |title=Iterative methods for sparse linear systems |publisher=SIAM |date=2003 |isbn=978-0-89871-534-7 |url={{GBurl|qtzmkzzqFmcC|pg=PR5}} }}</ref><ref>{{cite book |last1=Hageman |first1=L.A. |last2=Young |first2=D.M. |title=Applied iterative methods |publisher=Courier Corporation |edition=2nd |date=2012 |isbn=978-0-8284-0312-2 |url={{GBurl|se3YdgFgz4YC|pg=PR4}} }}</ref><ref>{{cite book |first=J.F. |last=Traub |title=Iterative methods for the solution of equations |publisher=American Mathematical Society |edition=2nd |date=1982 |isbn=978-0-8284-0312-2 |url={{GBurl|se3YdgFgz4YC|pg=PR4}}}} </ref><ref>{{cite book |first=A. |last=Greenbaum |title=Iterative methods for solving linear systems |publisher=SIAM |date=1997 |isbn=978-0-89871-396-1 |url={{GBurl|QpVpvE4gWZwC|pg=PP6}}}}</ref>

Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. [[GMRES]] and the [[conjugate gradient method]]. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

As an example, consider the problem of solving

:3''x''³ + 4 = 28

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From this table it can be concluded that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.

===Conditioning===

====Discretization and numerical integration====

[[Image:Schumacher (Ferrari) in practice at USGP 2005.jpg||thumb|right|upright=0.6]]

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

{| style="margin:auto;" class="wikitable"

! Time

| 0:20 || 1:00 || 1:40

|-

! 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 {{math|size=100%|1=({{val|2|/|3|u=h}}&nbsp;&times;&nbsp;{{val|140|u=km/h}})&nbsp;=&nbsp;{{val|93.3|u=km}}}}. This would allow us to estimate the total distance traveled as {{val|93.3|u=km}} + {{val|100|u=km}} + {{val|120|u=km}} = {{val|313.3|u=km}}, which is an example of '''numerical integration''' (see below) using a [[Riemann sum]], because displacement is the [[integral]] of velocity.

Ill-conditioned problem: Take the function {{math|size=100%|1=''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 nearly 1000. Evaluating ''f''(''x'') near ''x'' = 1 is an ill-conditioned problem.

Well-conditioned problem: By contrast, evaluating the same function {{math|size=100%|1=''f''(''x'') = 1/(''x''&nbsp;&minus;&nbsp;1)}} 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 [[Arbitrary-precision arithmetic|infinite precision arithmetic]]. Examples include [[Gaussian elimination]], the [[QR decomposition|QR factorization]] method for solving [[system of linear equations|systems of linear equations]], and the [[simplex method]] of [[linear programming]]. In practice, [[floating-point arithmetic|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 finite number of steps. Starting from an initial guess, iterative methods form successive approximations that [[Limit of a sequence|converge]] to the exact solution only in the limit. A convergence test, often involving [[Residual (numerical analysis)|the residual]], is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the [[bisection method]], and [[Jacobi iteration]]. In computational matrix algebra, iterative methods are generally needed for large problems.<ref>{{cite book |first=Y. |last=Saad |title=Iterative methods for sparse linear systems |publisher=SIAM |date=2003 |isbn=978-0-89871-534-7 |url={{GBurl|qtzmkzzqFmcC|pg=PR5}} }}</ref><ref>{{cite book |last1=Hageman |first1=L.A. |last2=Young |first2=D.M. |title=Applied iterative methods |publisher=Courier Corporation |edition=2nd |date=2012 |isbn=978-0-8284-0312-2 |url={{GBurl|se3YdgFgz4YC|pg=PR4}} }}</ref><ref>{{cite book |first=J.F. |last=Traub |title=Iterative methods for the solution of equations |publisher=American Mathematical Society |edition=2nd |date=1982 |isbn=978-0-8284-0312-2 |url={{GBurl|se3YdgFgz4YC|pg=PR4}}}} </ref><ref>{{cite book |first=A. |last=Greenbaum |title=Iterative methods for solving linear systems |publisher=SIAM |date=1997 |isbn=978-0-89871-396-1 |url={{GBurl|QpVpvE4gWZwC|pg=PP6}}}}</ref>

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===

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==Generation and propagation of errors==

{{further|Error propagation}}

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.

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===Numerical stability and well-posed problems===

[[Numerical stability]] 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.<ref name="stab">{{harvnb|Higham|2002}}</ref> This happens if the problem is ''[[condition number|well-conditioned]]'', meaning that the solution changes by only a small amount if the problem data are changed by a small amount.<ref name="stab"/> To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error.<ref name="stab"/>

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.

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''₀ to <math>\sqrt{2}</math>, 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''.{{NoteTag|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.}} A few iterations of each scheme are calculated in table form below, with initial guesses ''x''₀ = 1.4 and ''x''₀ = 1.42.

{| class="wikitable"

|-

! Babylonian

! Babylonian

! Method X

! Method X

|-

| ''x''₀ = 1.4

| ''x''₀ = 1.42

| ''x''₀ = 1.4

| ''x''₀ = 1.42

|-

| ''x''₁ = 1.4142857...

| ''x''₁ = 1.41422535...

| ''x''₁ = 1.4016

| ''x''₁ = 1.42026896

|-

| ''x''₂ = 1.414213564...

| ''x''₂ = 1.41421356242...

| ''x''₂ = 1.4028614...

| ''x''₂ = 1.42056...

|-

|

|

| ...

| ...

|-

|

|

| ''x''_1000000 = 1.41421...

| ''x''₂₇ = 7280.2284...

|}

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

:<math>

f(x)=x\left(\sqrt{x+1}-\sqrt{x}\right)

</math> and <math>

g(x)=\frac{x}{\sqrt{x+1}+\sqrt{x}}.

</math>

:Comparing the results of

:: <math> f(500)=500 \left(\sqrt{501}-\sqrt{500} \right)=500 \left(22.38-22.36 \right)=500(0.02)=10</math>

:and

:<math>

\begin{alignat}{3}g(500)&=\frac{500}{\sqrt{501}+\sqrt{500}}\\

&=\frac{500}{22.38+22.36}\\

&=\frac{500}{44.74}=11.17

\end{alignat}

</math>

: by comparing the two results above, it is clear that [[loss of significance]] (caused here by [[catastrophic cancellation]] from subtracting approximations to the nearby numbers <math>\sqrt{501}</math> and <math>\sqrt{500}</math>, despite the subtraction being computed exactly) has a huge effect on the results, even though both functions are equivalent, as shown below

:: <math> \begin{alignat}{4}

f(x)&=x \left(\sqrt{x+1}-\sqrt{x} \right)\\

&=x \left(\sqrt{x+1}-\sqrt{x} \right)\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}\\

&=x\frac{(\sqrt{x+1})^2-(\sqrt{x})^2}{\sqrt{x+1}+\sqrt{x}}\\

&=x\frac{x+1-x}{\sqrt{x+1}+\sqrt{x}} \\

&=x\frac{1}{\sqrt{x+1}+\sqrt{x}} \\

&=\frac {x}{\sqrt{x+1}+\sqrt{x}} \\

&=g(x)

\end{alignat}</math>

: The desired value, computed using infinite precision, is 11.174755...{{NoteTag|The example is a modification of one taken from {{harvtxt|Mathews|Fink|1999}}.<ref>{{cite book|page=28|contribution=Example 1.17|title=Numerical Methods Using MATLAB|edition=3rd|first1=John H.|last1=Mathews|first2=Kurtis D.|last2=Fink|publisher=Prentice Hall|year=1999}}</ref>}}

==Areas of study==

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Interpolation: Observing that the temperature varies 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 last year, it might be extrapolated that it will be 105 billion this year.

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

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Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some [[Constraint (mathematics)|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]] deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the [[simplex algorithm|simplex method]].

The method of [[Lagrange multipliers]] can be used to reduce optimization problems with constraints to unconstrained optimization problems.

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{{Main|Numerical ordinary differential equations|Numerical partial 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]].<ref>{{cite book |first=A. |last=Iserles |title=A first course in the numerical analysis of differential equations |publisher=Cambridge University Press |edition=2nd |date=2009 |isbn=978-0-521-73490-5 |url={{GBurl|M0tkw4oUucoC|pg=PR5}} }}</ref>

Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace.<ref>{{cite book |first=W.F. |last=Ames |title=Numerical methods for partial differential equations |publisher=Academic Press |edition=3rd |date=2014 |isbn=978-0-08-057130-0 |url={{GBurl|KmjiBQAAQBAJ|pg=PP7}} }}</ref> This can be done by a [[finite element method]],<ref>{{cite book |first=C. |last=Johnson |title=Numerical solution of partial differential equations by the finite element method |publisher=Courier Corporation |date=2012 |isbn=978-0-486-46900-3 |url={{GBurl|0IFCAwAAQBAJ|p=2}} }}</ref><ref>{{cite book |last1=Brenner |first1=S. |last2=Scott |first2=R. |title=The mathematical theory of finite element methods |publisher=Springer |edition=2nd |date=2013 |isbn=978-1-4757-3658-8 |url={{GBurl|ServBwAAQBAJ|pg=PR11}}}}</ref><ref>{{cite book |last1=Strang |first1=G. |last2=Fix |first2=G.J. |orig-year=1973 |title=An analysis of the finite element method |publisher=Wellesley-Cambridge Press |date=2018 |isbn=9780980232783 |url=https://archive.org/details/analysisoffinite0000stra |oclc=1145780513 |edition=2nd}}</ref> a [[finite difference]] method,<ref>{{cite book |last=Strikwerda |first=J.C. |title=Finite difference schemes and partial differential equations |publisher=SIAM |edition=2nd |date=2004 |isbn=978-0-89871-793-8 |url={{GBurl|mbdt5XT25AsC|pg=PP5}}}}</ref> or (particularly in engineering) a [[finite volume method]].<ref>{{cite book |first=Randall |last=LeVeque |title=Finite Volume Methods for Hyperbolic Problems |publisher=Cambridge University Press |date=2002 |isbn=978-1-139-43418-8 |url={{GBurl|mfAfAwAAQBAJ|pg=PT6}}}}</ref> The theoretical justification of these methods often involves theorems from [[functional analysis]]. This reduces the problem to the solution of an algebraic equation.

==Software==

{{Main|List of numerical-analysis software|Comparison of numerical-analysis software|List of programming languages by type#Numerical analysis|l3=Numerical analysis programming languages|List of numerical libraries}}

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The [[Netlib]] repository contains various collections of software routines for numerical problems, mostly in [[Fortran]] and [[C (programming language)|C]]. Commercial products implementing many different numerical algorithms include the [[IMSL Numerical Libraries|IMSL]] and [[Numerical Algorithms Group|NAG]] libraries; a [[free software|free-software]] alternative is the [[GNU Scientific Library]].

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The [[Naval Surface Warfare Center]] several times published its [https://apps.dtic.mil/sti/pdfs/ADA476840.pdf ''Library of Mathematics Subroutines''] (code [https://jblevins.org/mirror/amiller/#nswc here]).

There are several popular numerical computing applications such as [[MATLAB]],<ref>{{cite book |last1=Quarteroni |first1=A. |last2=Saleri |first2=F. |last3=Gervasio |first3=P. |title=Scientific computing with MATLAB and Octave |publisher=Springer |edition=4th |date=2014 |isbn=978-3-642-45367-0 |url={{GBurl|_0m9BAAAQBAJ|pg=PR11}}}}</ref><ref name="gh">{{cite book |editor1-last=Gander |editor1-first=W. |editor2-last=Hrebicek |editor2-first=J. |title=Solving problems in scientific computing using Maple and Matlab® |publisher=Springer |date=2011 |isbn=978-3-642-18873-2 |url={{GBurl|di2qCAAAQBAJ|pg=PR14}}}}</ref><ref name="bf">{{cite book |last1=Barnes |first1=B. |last2=Fulford |first2=G.R. |title=Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB |publisher=CRC Press |edition=2nd |date=2011 |isbn=978-1-4200-8350-7 |oclc=1058138488 }}</ref> [[TK Solver]], [[S-PLUS]], and [[IDL (programming language)|IDL]]<ref>{{cite book |first=L.E. |last=Gumley |title=Practical IDL programming |publisher=Elsevier |date=2001 |isbn=978-0-08-051444-4 |url={{GBurl|1d-tNpm_x4gC|pg=PR9}}}}</ref> as well as free and open -source alternatives such as [[FreeMat]], [[Scilab]],<ref>{{cite book |last1=Bunks |first1=C. |last2=Chancelier |first2=J.P. |last3=Delebecque |first3=F. |last4=Goursat |first4=M. |last5=Nikoukhah |first5=R. |last6=Steer |first6=S. |title=Engineering and scientific computing with Scilab |publisher=Springer |date=2012 |isbn=978-1-4612-7204-5 }}</ref><ref>{{cite book |last1=Thanki |first1=R.M. |last2=Kothari |first2=A.M. |title=Digital image processing using SCILAB |publisher=Springer |date=2019 |isbn=978-3-319-89533-8 |url={{GBurl|VydaDwAAQBAJ|pg=PR9}}}}</ref> [[GNU Octave]] (similar to Matlab), and [[IT++]] (a C++ library). There are also programming languages such as [[R (programming language)|R]]<ref>{{cite journal |last1=Ihaka |first1=R. |last2=Gentleman |first2=R. |title=R: a language for data analysis and graphics |journal=Journal of Computational and Graphical Statistics |volume=5 |issue=3 |pages=299–314 |date=1996 |doi=10.1080/10618600.1996.10474713 |s2cid=60206680 |url=https://www.stat.auckland.ac.nz/~ihaka/downloads/R-paper.pdf}}</ref> (similar to S-PLUS), [[Julia (programming language)|Julia]],<ref>{{Cite journal|last1=Bezanson|first1=Jeff|last2=Edelman|first2=Alan|last3=Karpinski|first3=Stefan|last4=Shah|first4=Viral B.|date=2017-01-01|title=Julia: A Fresh Approach to Numerical Computing|url=https://epubs.siam.org/doi/abs/10.1137/141000671|journal=SIAM Review|volume=59|issue=1|pages=65–98|doi=10.1137/141000671|arxiv=1411.1607 |issn=0036-1445|hdl=1721.1/110125|s2cid=13026838 |hdl-access=free}}</ref> and [[Python (programming language)|Python]] with libraries such as [[NumPy]], [[SciPy]]<ref>Jones, E., Oliphant, T., & Peterson, P. (2001). SciPy: Open source scientific tools for Python.</ref><ref>{{cite book |first=E. |last=Bressert |title=SciPy and NumPy: an overview for developers |publisher=O'Reilly |date=2012 |isbn=9781306810395 }}</ref><ref>{{cite book |first=F.J. |last=Blanco-Silva |title=Learning SciPy for numerical and scientific computing |publisher=Packt |date=2013 |isbn=9781782161639 }}</ref> 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.<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=5 October 2006 }}</ref><ref>[http://www.scientificweb.com/ncrunch/ncrunch5.pdf Comparison of mathematical programs for data analysis] {{Webarchive|url=http://arquivo.pt/wayback/20160518062220/http://www.scientificweb.com/ncrunch/ncrunch5.pdf |date=18 May 2016 }} Stefan Steinhaus, ScientificWeb.com</ref>

Many [[computer algebra system]]s such as [[Mathematica]] also benefit from the availability of [[arbitrary-precision arithmetic]] which can provide more accurate results.<ref>{{cite book |first=R.E. |last=Maeder |title=Programming in mathematica |publisher=Addison-Wesley |edition=3rd |date=1997 |isbn=9780201854497 |oclc=1311056676 |url=https://archive.org/details/programminginmat0000maed_l2m6}}</ref><ref>{{cite book |first=Stephen |last=Wolfram |date=1999 |title=The MATHEMATICA® book, version 4 |publisher=[[Cambridge University Press]] |url={{GBurl|Xny77v_QPkEC|pg=PR19}} |isbn=9781579550042 }}</ref><ref>{{cite book |last1=Shaw |first1=W.T. |last2=Tigg |first2=J. |title=Applied Mathematica: getting started, getting it done |publisher=Addison-Wesley |date=1993 |isbn=978-0-201-54217-2 |oclc=28149048 |url=http://www.gbv.de/dms/bowker/toc/9780201542172.pdf}}</ref><ref>{{cite book |last1=Marasco |first1=A. |last2=Romano |first2=A. |title=Scientific Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations |publisher=Springer |date=2001 |isbn=978-0-8176-4205-1 |url={{GBurl|iFRqemnmMqUC|pg=PR7}}}}</ref>

Line 270 ⟶ 195:

*[[:Category:Numerical analysts]]

*[[Analysis of algorithms]]

*[[Approximation theory]]

*[[Computational science]]

*[[Computational physics]]

Line 293 ⟶ 219:

{{refbegin}}

* {{cite book |author1 = Golub, Gene H. |author-link = Gene H. Golub |author2 = Charles F. Van Loan |author2-link = Charles F. Van Loan |title=Matrix Computations |edition=3rd |publisher=Johns Hopkins University Press |isbn=0-8018-5413-X |year = 1986 }}

* {{cite book |author1 = Ralston Anthony |author2 = Rabinowitz Philips | title=A First Course in Numerical Analysis | edition=2nd |publisher=Dover publications | isbn=978-0486414546 |year=2001 }}

*{{cite book |first=Nicholas J. |last=Higham |author-link=Nicholas Higham |title = Accuracy and Stability of Numerical Algorithms |url=https://archive.org/details/accuracystabilit0000high |url-access=registration |publisher=Society for Industrial and Applied Mathematics |isbn=0-89871-355-2 |orig-year=1996 |year=2002}}

* {{cite book |last=Hildebrand |first=F. B. |author-link=Francis B. Hildebrand | title=Introduction to Numerical Analysis |edition=2nd |year=1974 |publisher=McGraw-Hill |isbn= 0-07-028761-9 }}

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

* {{cite book |last=Leader |first=Jeffery J. |author-link=Jeffery J. Leader |title=Numerical Analysis and Scientific Computation |year=2004 |publisher=Addison Wesley |isbn= 0-201-73499-0 }}

* {{cite book|last= Wilkinson |first =J.H. |author-link=James H. Wilkinson |title=The Algebraic Eigenvalue Problem |url= https://archive.org/details/algebraiceigenva0000wilk |url-access= registration |publisher = Clarendon Press |orig-year=1965 |year=1988 |isbn=978-0-19-853418-1 }}

Line 316 ⟶ 244:

*[https://dlmf.nist.gov/3 Numerical Methods], ch 3. in the ''[[Digital Library of Mathematical Functions]]''

*[https://personal.math.ubc.ca/~cbm/aands/page_875.htm Numerical Interpolation, Differentiation and Integration], ch 25. in the ''Handbook of Mathematical Functions'' ([[Abramowitz and Stegun]])

*[https://fncbook.com/ Tobin A. Driscoll and Richard J. Braun: ''Fundamentals of Numerical Computation'' (free online version)]

===Online course material===