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Numerical analysis
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{{merge fromNumerical methoddate=January 2019}}{{more footnotesdate=November 2013}}{{Use dmy datesdate=July 2012}} the content below is remote from Wikipedia
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missing image!
 Ybc7289bw.jpg 
Babylonian clay tablet YBC 7289 (c. 1800â€“1600 BC) 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...Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics (predicting the motions of planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.Before the advent of modern computers, numerical methods often depended on hand interpolation formulas applied to data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas nevertheless continue to be used as part of the software algorithms.The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Computing the sides of a triangle in terms of square roots is a basic practical problem, for example in astronomy, carpentry, and construction.The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled CARPENTRY THEORY: Demonstrate knowledge of setting out a buildingNumerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to realworld measurements by translation into digits, it gives approximate solutions within specified error bounds. Ybc7289bw.jpg 
Babylonian clay tablet YBC 7289 (c. 1800â€“1600 BC) 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...Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
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 equations numerically.
 Hedge funds (private investment funds) use tools from all fields of numerical analysis to attempt 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. Historically, such algorithms were developed within the overlapping field of operations research.
 Insurance companies use numerical programs for actuarial 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, as is obvious from the names of important algorithms like Newton's method, 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 1000plus 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; marginleft: 1em;"


Direct vs iterative methodsConsider the problem of solving

3x3 + 4 = 28
for the unknown quantity x.{ style="margin:auto; textalign:right"
+ Direct method

  3x3 + 4 = 28.

 Subtract 4  3x3 = 24.

 Divide by 3  x3 = 8.

 Take cube roots  x = 2.
}
For the iterative method, apply the bisection method to f(x) = 3x3 − 24. The initial values are a = 0, b = 3, f(a) = −24, f(b) = 57.{ style="margin:auto;" class="wikitable"

  3x3 + 4 = 28.

 Subtract 4  3x3 = 24.

 Divide by 3  x3 = 8.

 Take cube roots  x = 2.
}
+ Iterative method

! a !! b !! mid !! f(mid)

 0  3  1.5  −13.875

 1.5  3  2.25  10.17...

 1.5  2.25  1.875  −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.
! a !! b !! mid !! f(mid)

 0  3  1.5  −13.875

 1.5  3  2.25  10.17...

 1.5  2.25  1.875  −4.22...

 1.875  2.25  2.0625  2.32...
}
Discretization and numerical integration
missing image!
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125px
In a twohour race, the speed of the car is measured at three instants and recorded in the following table.{ style="margin:auto;" class="wikitable"! Time! km/h Schumacher (Ferrari) in practice at USGP 2005.jpg 
125px
 1:40  
 180 
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.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.Roundoff
Roundoff errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).Truncation and discretization error
Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of 3x^3+4=28, after 10 or so iterations, it can be concluded that the root is roughly 1.99 (for example). Therefore there is a truncation error of 0.01.Once an error is generated, it will generally propagate through the calculation. For instance, already noted is that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type {{tmatha+b+c+d+e}} is even more inexact.The truncation error is created when a mathematical procedure is approximated. To integrate a function exactly it is required to find the sum of infinite trapezoids, but numerically only the sum of only finite trapezoids can be found, and hence the approximation of the mathematical procedure. Similarly, to differentiate a function, the differential element approaches zero but numerically only a finite value of the differential element can be chosen.Numerical stability and wellposed 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. This happens if the problem is 'wellconditioned', meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is 'illconditioned', then any small error in the data will grow to be a large error.Both the original problem and the algorithm used to solve that problem can be 'wellconditioned' or 'illconditioned', and any combination is possible.So an algorithm that solves a wellconditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a wellposed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a wellposed problem. Many algorithms solve this problem by starting with an initial approximation x0 to sqrt{2}, for instance x0 = 1.4, and then computing improved guesses x1, x2, etc. One such method is the famous Babylonian method, which is given by x'k+1 = xk/2 + 1/xk. Another method, called 'method X', is given by x'k+1 = (x'k2 âˆ’ 2)2 + x'k.This is a fixed point iteration for the equation x=(x^22)^2+x=f(x), whose solutions include sqrt{2}. The iterates always move to the right since f(x)geq x. Hence x_1=1.4sqrt{2} diverges. A few iterations of each scheme are calculated in table form below, with initial guesses x0 = 1.4 and x0 = 1.42.{ class="wikitable"

! Babylonian! Babylonian! Method X! Method X

 x0 = 1.4
 x0 = 1.42
 x0 = 1.4
 x0 = 1.42

 x1 = 1.4142857...
 x1 = 1.41422535...
 x1 = 1.4016
 x1 = 1.42026896

 x2 = 1.414213564...
 x2 = 1.41421356242...
 x2 = 1.4028614...
 x2 = 1.42056...



 ...
 ...



 x1000000 = 1.41421...
 x27 = 7280.2284...
}
Observe that the Babylonian method converges quickly regardless of the initial guess, whereas Method X converges extremely slowly with initial guess x0 = 1.4 and diverges for initial guess x0 = 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.
 x0 = 1.4
 x0 = 1.42
 x0 = 1.4
 x0 = 1.42

 x1 = 1.4142857...
 x1 = 1.41422535...
 x1 = 1.4016
 x1 = 1.42026896

 x2 = 1.414213564...
 x2 = 1.41421356242...
 x2 = 1.4028614...
 x2 = 1.42056...



 ...
 ...



 x1000000 = 1.41421...
 x27 = 7280.2284...
}
Numerical stability is affected by the number of the significant digits the machine keeps on, 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 these two equivalent functions
f(x)=xleft(sqrt{x+1}sqrt{x}right)text{ and } g(x)=frac{x}{sqrt{x+1}+sqrt{x}}.
Comparing the results of
begin{alignat}{3}g(500)&=frac{500}{sqrt{501}+sqrt{500}}
f(500)=500 left(sqrt{501}sqrt{500} right)=500 left(22.3822.36 right)=500(0.02)=10
and
&=frac{500}{22.38+22.36}
&=frac{500}{44.74}=11.17
end{alignat}
&=frac{500}{44.74}=11.17
by comparing the two results above, it is clear that loss of significance (caused here by 'catastrophic cancelation') has a huge effect on the results, even though both functions are equivalent, as shown below
f(x)&=x left(sqrt{x+1}sqrt{x} right)
begin{alignat}{4}
&=x left(sqrt{x+1}sqrt{x} right)frac{sqrt{x+1}+sqrt{x}}{sqrt{x+1}+sqrt{x}}
&=xfrac{(sqrt{x+1})^2(sqrt{x})^2}{sqrt{x+1}+sqrt{x}}
&=xfrac{x+1x}{sqrt{x+1}+sqrt{x}}
&=xfrac{1}{sqrt{x+1}+sqrt{x}}
&=frac {x}{sqrt{x+1}+sqrt{x}}
&=g(x)
end{alignat}
&=xfrac{(sqrt{x+1})^2(sqrt{x})^2}{sqrt{x+1}+sqrt{x}}
&=xfrac{x+1x}{sqrt{x+1}+sqrt{x}}
&=xfrac{1}{sqrt{x+1}+sqrt{x}}
&=frac {x}{sqrt{x+1}+sqrt{x}}
&=g(x)
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 subdisciplines. Some of the major ones are:Computing values of functions{ class"wikitable" style"float: right; width: 250px; clear: right; marginleft: 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 missing image! Linearregression.svg">right100pxA line through 20 pointsRegression: In linear regression, given n points, a line is computed that passes as close as possible to those n points.LemonadeJuly2006.JPG 
missing image!
 Windparticle.png 
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 isdropped 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 roundoff errors arising from the use of floating point arithmetic. Windparticle.png 
Wind direction in blue, true trajectory in black, Euler method in red.
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.Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The least squaresmethod is one way to achieve this.Solving equations and systems of equations
Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation 2x+5=3 is linear while 2x^2+5=3 is not.Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positivedefinite matrix, and QR decomposition for nonsquare matrices. Iterative methods such as the Jacobi method, Gaussâ€“Seidel method, successive overrelaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.Rootfinding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.Solving eigenvalue or singular value problems
Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithmThe Singular Value Decomposition and Its Applications in Image Compression {{webarchive url=https://web.archive.org/web/20061004041704weblink date=4 October 2006 }} is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.Optimization
Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.Evaluating integrals
Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newtonâ€“Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasiMonte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.Differential equations
Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.Partial differential equations are solved by first discretizing the equation, bringing it into a finitedimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.Software
Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a freesoftware alternative is the GNU Scientific Library.There are several popular numerical computing applications such as MATLAB, TK Solver, SPLUS, and IDL as well as free and open source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to SPLUS) and Python with libraries such as NumPy, SciPy 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.Speed comparison of various number crunching packages {{webarchive url=https://web.archive.org/web/20061005024002weblink date=5 October 2006 }}Comparison of mathematical programs for data analysis Stefan Steinhaus, ScientificWeb.comMany computer algebra systems such as Mathematica also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results.Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.See also
 Analysis of algorithms
 Computational science
 Interval arithmetic
 List of numerical analysis topics
 Numerical differentiation
 Numerical Recipes
 Symbolicnumeric computation
Notes
References
 BOOK, Gene H. Golub, Golub, Gene H. and Charles F. Van Loan, Matrix Computations, third, Johns Hopkins University Press, 080185413X, 1986,
 BOOK, Nicholas J., Higham, Nicholas Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, 0898713552, 1996,
 BOOK, Hildebrand, F. B., Francis B. Hildebrand, Introduction to Numerical Analysis, 2nd, 1974, McGrawHill, 0070287619,
 BOOK, Leader, Jeffery J., Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2004, Addison Wesley, 0201734990,
 BOOK, Wilkinson, J.H., James H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press), 1965,
 JOURNAL, Kahan, W., William Kahan, "A survey of erroranalysis," in Info. Processing 71 (Proc. IFIP Congress 71 in Ljubljana), vol. 2, pp. 1214â€“39, NorthHolland Publishing, Amsterdam, 1972, (examples of the importance of accurate arithmetic).
 Trefethen, Lloyd N. (2006). "Numerical analysis", 20 pages. In: Timothy Gowers and June BarrowGreen (editors), Princeton Companion of Mathematics, Princeton University Press.
External links
{{Sister project links wikt=no  b=Numerical Methods  n=no  q=Numerical analysis  s=no  v=no  voy=no  species=no  d=no}}Journals
 gdz.sub.unigoettingen, Numerische Mathematik, volumes 166, Springer, 19591994 (searchable; pages are images). {{en icon}} {{de icon}}
 springerlink.com, Numerische Mathematik, volumes 1112, Springer, 1959â€“2009
 SIAM Journal on Numerical Analysis, volumes 147, SIAM, 1964â€“2009
Online texts
 {{springertitle=Numerical analysisid=p/n120130}}
 Numerical Recipes, William H. Press (free, downloadable previous editions)
 weblink" title="web.archive.org/web/20120225082123weblink">First Steps in Numerical Analysis (archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
 CSEP (Computational Science Education Project), U.S. Department of Energy
Online course material
 Numerical Methods, Stuart Dalziel University of Cambridge
 Lectures on Numerical Analysis, Dennis Deturck and Herbert S. Wilf University of Pennsylvania
 Numerical methods, John D. Fenton University of Karlsruhe
 Numerical Methods for Physicists, Anthony Oâ€™Hare Oxford University
 weblink" title="web.archive.org/web/20120225082123weblink">Lectures in Numerical Analysis (archived), R. Radok Mahidol University
 Introduction to Numerical Analysis for Engineering, Henrik Schmidt Massachusetts Institute of Technology
 Numerical Analysis for Engineering, D. W. Harder University of Waterloo
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