Wonani kulemba kwambula kuona kwa Kusanthula Vipendero

{{Short description|Kusanda ma algorithm agho ghakugwiliskira nchito malemba ghakuyana pafupi-pafupi zgolo lenecho}}
{{Use dmy dates|date=October 2020}}
[[Image:Ybc7289-bw.jpg|thumb|250px|right|Libwe la dongo la ku Babuloni [[YBC 7289]] (c. 1800–1600 BC) na mazgu ghakulongosola. Kuloska kwa zgolo la [[msisi ŵili wa 2]] ni ma lembo[[sexagesino]], iyo ni malembo gha [[mdoto]] pafupi fupi khumi na moza. 1 + 24/60 + 51/60<sup>2</sup> + 10/60<sup>3</sup> = 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>]]
'''Kusanthula vipendero''' ni chisambiro cha [[ndondomeko]] izo zikugwiliskika kusanga mazgoro agho [[ghakusendelera]] waka ku zgoro lenecho (mwakupambana na [[symbolic computation|kuseŵezga manambara mwakuchiti tisange manambara ghenecho nkhanira-nkhanira) pa masuzgo gha [[kusanthula kwa samuzi]] (Kupambana na [[samuzu ghapadera]]). Ntchigaŵa cha masambiro gha manambara agho ghakovwira kusanga nthowa zakumazgira masuzgo m'malo mwa kusanga mazgoro ghakwenelera ghankhanira nkhanira. Kusanthula wakugwiliskira nchito vinthu vyose vya sayansi, Ndipo mu vyaka vya m'ma 2000, likugwiliskirika nchito mu umoyo, sayansi, munkhwala, bizinesi na luso wuwo. Kukura kwa nkhongono ya kompyuta hŴakaŵa na mwaŵi wakugwiliskira ntchito nthowa zakusuzga chomene pa Kusanthula, kupeleka vyandulo vya masamu mu sayansi. Viyelezgero vinyake vya 'kusandula' ni ivi: [[viyanjano vyapadera vya waka]] nga umo viliri mu [[makina gha vyakucanya]] (kumanya umo mapulaneti, nyenyezi, na vipingausiku vikwendera), [[algebra ya mulingo ya vipendero]] mu kusanda mauthenga,<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> na [[chiyanjano cha upambano cha stokastic]] na [[maunyolo gha Markov]] pakuyelezgera maselo ghamoyo mu vyamankhwala na kusanda-umoyo.

Pambere makompyuta ghandaŵeko, [kusanthula]] Kanandi ŵakugwiliskira ntchito nthowa [[zakuŵikapo pakati]] na mawoko, Kufumiska mauthenga kufuma mu mathebulu ghakurughakuru. Kwambira m'ma 1900, makompyuta ghakuchita mawerengero gha mauteŵeti agho ghakukhumbikwa, kweni vigaŵa vinandi vya malemba agha vikulutilira kugwiliskirika nchito mu mapulogiramu gha mapulogiramu gha pa kompyuta.<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>

Fundo iyi yikamba kulembeka mu mabuku ghakwambilira gha masamu. Libwe la dongo 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]].

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

==General introduction==
The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:

* Advanced numerical methods are essential in making [[numerical weather prediction]] feasible.
* Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
* Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving [[partial differential equation]]s numerically.
* [[Hedge fund]]s (private investment funds) use tools from all fields of numerical analysis to attempt to calculate the value of [[stock]]s and [[Derivative (finance)|derivatives]] more precisely than other market participants.
* Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of [[operations research]].
* Insurance companies use numerical programs for [[Actuary|actuarial]] analysis.

The rest of this section outlines several important themes of numerical analysis.

===History===
The field of numerical analysis predates the invention of modern computers by many centuries. [[Linear interpolation]] 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}} }}
</ref><ref>
{{harvnb|Brezinski|Wuytack|2001|p=2}}
</ref>
but others consider modern numerical analysis to go back to work by [[E. T. Whittaker]] in 1912.<ref name="watson" >
{{cite book |first=G.A. |last=Watson |chapter=The history and development of numerical analysis in Scotland: a personal perspective |chapter-url=https://core.ac.uk/download/pdf/206717434.pdf |title=The Birth of Numerical Analysis |publisher=World Scientific |date=2010 |isbn=9789814469456 |pages=161–177 }}
</ref>

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,<ref name="20c"/> since now longer and more complicated calculations could be done.

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

===Direct and iterative methods===
Consider the problem of solving

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

for the unknown quantity ''x''.

{| style="margin:auto; text-align:right"
 |+ Direct method
 |-
 | || 3''x''<sup>3</sup> + 4 = 28.
 |-
 | ''Subtract 4'' || 3''x''<sup>3</sup> = 24.
 |-
 | ''Divide by 3'' || ''x''<sup>3</sup> = &nbsp;8.
 |-
 | ''Take cube roots'' || ''x'' = &nbsp;2.
 |}

For the iterative method, apply the [[bisection method]] to ''f''(''x'') = 3''x''<sup>3</sup> &minus; 24. The initial values are ''a'' = 0, ''b'' = 3, ''f''(''a'') = &minus;24, ''f''(''b'') = 57.

{| style="margin:auto;" class="wikitable"
 |+ Iterative method
 |-
 ! ''a'' !! ''b'' !! mid !! ''f''(mid)
 |-
 | 0 || 3 || 1.5 || &minus;13.875
 |-
 | 1.5 || 3 || 2.25 || 10.17...
 |-
 | 1.5 || 2.25 || 1.875 || &minus;4.22...
 |-
 | 1.875 || 2.25 || 2.0625 || 2.32...
 |}

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.

====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===
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 (mathematics)|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 (set theory)|continuum]].

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

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

===Truncation and discretization error===
[[Truncation error]]s are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a [[discretization error]] because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of <math>3x^3+4=28</math>, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01.

Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type {{tmath|a+b+c+d+e}} is even more inexact.

A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.

===Numerical stability and well-posed problems===
[[Numerical stability]] 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. 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>0</sub> to <math>\sqrt{2}</math>, for instance ''x''<sub>0</sub> = 1.4, and then computing improved guesses ''x''<sub>1</sub>, ''x''<sub>2</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 method, called 'method X', is given by ''x''<sub>''k''+1</sub> = (''x''<sub>''k''</sub><sup>2</sup> − 2)<sup>2</sup> + ''x''<sub>''k''</sub>.{{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''<sub>0</sub> = 1.4 and ''x''<sub>0</sub> = 1.42.

{| class="wikitable"
|-
! Babylonian
! Babylonian
! Method X
! Method X
 |-
 | ''x''<sub>0</sub> = 1.4
 | ''x''<sub>0</sub> = 1.42
 | ''x''<sub>0</sub> = 1.4
 | ''x''<sub>0</sub> = 1.42
 |-
 | ''x''<sub>1</sub> = 1.4142857...
 | ''x''<sub>1</sub> = 1.41422535...
 | ''x''<sub>1</sub> = 1.4016
 | ''x''<sub>1</sub> = 1.42026896
 |-
 | ''x''<sub>2</sub> = 1.414213564...
 | ''x''<sub>2</sub> = 1.41421356242...
 | ''x''<sub>2</sub> = 1.4028614...
 | ''x''<sub>2</sub> = 1.42056...
 |-
 |
 |
 | ...
 | ...
 |-
 |
 |
 | ''x''<sub>1000000</sub> = 1.41421...
 | ''x''<sub>27</sub> = 7280.2284...
 |}

Observe that the Babylonian method converges quickly regardless of the initial guess, whereas Method X converges extremely slowly with initial guess ''x''<sub>0</sub> = 1.4 and diverges for initial guess ''x''<sub>0</sub> = 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...
* The example is a modification of one taken from Mathew; Numerical methods using MATLAB, 3rd ed.

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

===Computing values of functions===
{| class="wikitable" style="float: right; width: 250px; clear: right; margin-left: 1em;"
 |
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 extrapolated that it will be 105 billion this year.

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

Regression: In linear regression, given ''n'' points, a line is computed that passes as close as possible to those ''n'' points.

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

Optimization: Suppose lemonade is sold at a [[lemonade stand]], at $1.00 per glass, that 197 glasses of lemonade can be sold per day, and that for each increase of $0.01, one less glass of lemonade will be sold per day. If $1.485 could be charged, profit would be maximized, but due to the constraint of having to charge a whole-cent amount, charging $1.48 or $1.49 per glass will both 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 100 fans are set up to blow air from one end of the room to the other and then a feather is dropped 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.
|}

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?

[[Extrapolation]] is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.<ref>{{cite book |last1=Brezinski |first1=C. |last2=Zaglia |first2=M.R. |title=Extrapolation methods: theory and practice |publisher=Elsevier |date=2013 |isbn=978-0-08-050622-7 |url={{GBurl|WGviBQAAQBAJ|pg=PR7}}}}</ref>

[[Regression analysis|Regression]] is also similar, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The [[numerical methods for linear least squares|least squares]]-method is one way to achieve this.

===Solving equations and systems of equations===
Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation <math>2x+5=3</math> is linear while <math>2x^2+5=3</math> is not.

Much effort has been put in the development of methods for solving [[systems of linear equations]]. Standard direct methods, i.e., methods that use some [[matrix decomposition]] are [[Gaussian elimination]], [[LU decomposition]], [[Cholesky decomposition]] for [[symmetric matrix|symmetric]] (or [[hermitian matrix|hermitian]]) and [[positive-definite matrix]], and [[QR decomposition]] for non-square matrices. Iterative methods such as the [[Jacobi method]], [[Gauss–Seidel method]], [[successive over-relaxation]] and [[conjugate gradient method]]<ref>{{cite journal |last1=Hestenes |first1=Magnus R. |last2=Stiefel |first2=Eduard |title=Methods of Conjugate Gradients for Solving Linear Systems |journal=Journal of Research of the National Bureau of Standards |volume=49 |issue=6 |pages=409– |date=December 1952 |doi=10.6028/jres.049.044 |url= https://nvlpubs.nist.gov/nistpubs/jres/049/jresv49n6p409_A1b.pdf}}</ref> are usually preferred for large systems. General iterative methods can be developed using a [[matrix splitting]].

[[Root-finding algorithm]]s are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is [[derivative|differentiable]] and the derivative is known, then Newton's method is a popular choice.<ref>{{cite book |last1=Ezquerro Fernández |first1=J.A. |last2=Hernández Verón |first2=M.Á. |title=Newton's method: An updated approach of Kantorovich's theory |publisher=Birkhäuser |date=2017 |isbn=978-3-319-55976-6  |url={{GBurl|A3orDwAAQBAJ|pg=PR11}}}}</ref><ref>{{cite book |first=Peter |last=Deuflhard |title=Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms |publisher=Springer |edition=2nd |series=Computational Mathematics |volume=35 |date=2006 |isbn=978-3-540-21099-3 |url={{GBurl|l20xK__HG_kC|pg=PP1}} }}</ref> [[Linearization]] is another technique for solving nonlinear 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 [[image compression|spectral image compression]] algorithm<ref>{{cite web |first1=C.J. |last1=Ogden |first2=T. |last2=Huff |title=The Singular Value Decomposition and Its Applications in Image Compression |date=1997 |work=Math 45 |publisher=College of the Redwoods |url=http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall97/Tammie/tammie.pdf|archive-url=https://web.archive.org/web/20060925193348/http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall97/Tammie/tammie.pdf |archive-date=25 September 2006 }}</ref> is based on the singular value decomposition. The corresponding tool in statistics is called [[principal component analysis]].

===Optimization===
{{Main|Mathematical optimization}}

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 method.

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

===Evaluating integrals===
{{Main|Numerical integration}}

Numerical integration, in some instances also known as numerical [[quadrature (mathematics)|quadrature]], asks for the value of a definite [[integral]].<ref>{{cite book |last1=Davis |first1=P.J. |last2=Rabinowitz |first2=P. |title=Methods of numerical integration |publisher=Courier Corporation |date=2007 |isbn=978-0-486-45339-2 |url={{GBurl|gGCKdqka0HAC|pg=PR5}}}}</ref> Popular methods use one of the [[Newton–Cotes formulas]] (like the midpoint rule or [[Simpson's rule]]) or [[Gaussian quadrature]].<ref>{{MathWorld|author=Weisstein, Eric W. |title=Gaussian Quadrature |id=GaussianQuadrature}}</ref> 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]]<ref>{{cite book |first=John |last=Geweke |chapter=15. Monte carlo simulation and numerical integration |chapter-url=https://www.sciencedirect.com/science/article/pii/S1574002196010179) |doi=10.1016/S1574-0021(96)01017-9|title=Handbook of Computational Economics |publisher=Elsevier |volume=1 |date=1996 |isbn=9780444898579 |pages=731–800  }}</ref>), or, in modestly large dimensions, the method of [[sparse grid]]s.

===Differential equations===
{{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}}

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]].

Over the years the [[Royal Statistical Society]] published numerous algorithms in its [[Journal of the Royal Statistical Society, Series C (Applied Statistics)|''Applied Statistics'']] (code for these "AS" functions is [https://jblevins.org/mirror/amiller/#apstat here]); 
[[Association for Computing Machinery|ACM]] similarly, in its ''[[Transactions on Mathematical Software]]'' ("TOMS" code is [https://jblevins.org/mirror/amiller/#toms here]).
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 |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|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>[https://web.archive.org/web/20061005024002/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://arquivo.pt/wayback/20160518062220/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>

Also, any [[spreadsheet]] [[software]] can be used to solve simple problems relating to numerical analysis. 
[[Microsoft_Excel#|Excel]], for example, has hundreds of [[Microsoft Excel#Functions|available functions]], including for matrices, which may be used in conjunction with its [[Microsoft Excel#Add-ins|built in "solver"]].

==See also==
{{div col |colwidth=20em}}
*[[:Category:Numerical analysts]]
*[[Analysis of algorithms]]
*[[Computational science]]
*[[Computational physics]]
*[[Gordon Bell Prize]]
*[[Interval arithmetic]]
*[[List of numerical analysis topics]]
*[[Local linearization method]]
*[[Numerical differentiation]]
*[[Numerical Recipes]]
*[[Probabilistic numerics]]
*[[Symbolic-numeric computation]]
*[[Validated numerics]]
{{div col end}}

==Notes==
{{NoteFoot}}

==References==
===Citations===
{{Reflist}}

===Sources===
{{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 |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 }}
* {{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 }}
* {{cite conference |last = Kahan |first = W. |author-link= William Kahan |title = A survey of error-analysis  |journal = Info. Processing 71 |conference = Proc. IFIP Congress 71 in Ljubljana |volume = 2 |pages = 1214–39 |publisher = North-Holland |year = 1972 |isbn=978-0-7204-2063-0 |oclc=25116949 }} (examples of the importance of accurate arithmetic).
*{{cite book |author-link=Lloyd N. Trefethen |last=Trefethen |first=Lloyd N. |chapter=IV.21 Numerical analysis |chapter-url=http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |editor1-first=I. |editor1-last=Leader  |editor2-first=T. |editor2-last=Gowers |editor3-first=J. |editor3-last=Barrow-Green |title=Princeton Companion of Mathematics |publisher=Princeton University Press |date=2008 |isbn=978-0-691-11880-2  |pages=604–614 |url={{GBurl|GLumDwAAQBAJ|pg=PR5}}}}
{{refend}}

==External links==
{{Sister project links| wikt=no|c=Category:Numerical analysis | b=Numerical Methods | n=no | q=Numerical analysis | s=no | v=no | voy=no | species=no | d=no}}

===Journals===
*''[[Numerische Mathematik]]'', volumes 1–..., [https://www.springer.com/mathematics/numerical+and+computational+mathematics/journal/211 Springer], 1959–
**[http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN362160546 volumes 1–66, 1959–1994] (searchable; pages are images). {{in lang|en|de}}
*''[[Journal on Numerical Analysis]]'' [https://epubs.siam.org/journal/sjnaam (SINUM)], volumes 1–..., SIAM, 1964–

===Online texts===
*{{springer|title=Numerical analysis|id=p/n120130}}
*[https://web.archive.org/web/20150905141405/http://www.nr.com/oldverswitcher.html ''Numerical Recipes''], William H. Press (free, downloadable previous editions)
*[https://web.archive.org/web/20120225082123/http://kr.cs.ait.ac.th/~radok/math/mat7/stepsa.htm ''First Steps in Numerical Analysis''] ([[Internet Archive|archived]]), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
*[https://web.archive.org/web/20170801213333/http://www.phy.ornl.gov/csep/CSEP/TEXTOC.html ''CSEP'' (Computational Science Education Project)], [[U.S. Department of Energy]] ([[Internet Archive|archived 2017-08-01]]) 
*[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]])

===Online course material===
*[http://www.damtp.cam.ac.uk/user/fdl/people/sd103/lectures/nummeth98/index.htm#L_1_Title_Page Numerical Methods] ({{Webarchive|url=https://web.archive.org/web/20090728181209/http://www.damtp.cam.ac.uk/user/fdl/people/sd103/lectures/nummeth98/index.htm#L_1_Title_Page |date=28 July 2009 }}), Stuart Dalziel [[University of Cambridge]]
*[http://www.math.upenn.edu/~wilf/DeturckWilf.pdf Lectures on Numerical Analysis], Dennis Deturck and Herbert S. Wilf [[University of Pennsylvania]]
*[http://johndfenton.com/Lectures/Numerical-Methods/Numerical-Methods.pdf Numerical methods], John D. Fenton [[University of Karlsruhe]]
*[http://www-teaching.physics.ox.ac.uk/computing/NumericalMethods/NMfP.pdf Numerical Methods for Physicists], Anthony O’Hare [[Oxford University]]
*[https://web.archive.org/web/20120225082123/http://kr.cs.ait.ac.th/~radok/math/mat7/stepsa.htm Lectures in Numerical Analysis] ([[Internet Archive|archived]]), R. Radok [[Mahidol University]]
*[http://ocw.mit.edu/courses/mechanical-engineering/2-993j-introduction-to-numerical-analysis-for-engineering-13-002j-spring-2005/ Introduction to Numerical Analysis for Engineering], Henrik Schmidt [[Massachusetts Institute of Technology]]
*[http://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/ ''Numerical Analysis for Engineering''], D. W. Harder [[University of Waterloo]]
*[https://www.math.umd.edu/~diom/courses/AMSC466/Levy-notes.pdf Introduction to Numerical Analysis], Doron Levy [[University of Maryland]]
*[https://web.archive.org/web/20070310212643/http://math.fullerton.edu/mathews/n2003/NumericalUndergradMod.html Numerical Analysis - Numerical Methods] (archived), John H. Mathews [[California State University Fullerton]]

{{Areas of mathematics}}
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{{DEFAULTSORT:Numerical Analysis}}
[[Category:Numerical analysis| ]]
[[Category:Mathematical physics]]
[[Category:Computational science]]