284:
891:, 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 'ill-conditioned', 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 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.
3924:
79:). It is the study of numerical methods that attempt to find 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:
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953:, 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.
870:
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,
291:
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
964:
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
307:
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
33:
1273:. 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
379:, 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
300:, 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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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
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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.
191:
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, many of which are infeasible to solve symbolically:
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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.
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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
110:
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.
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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
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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
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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
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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
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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
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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
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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 '
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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?
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Numerical analysis plays a crucial role in scientific computing, engineering simulations, financial modeling, and many other fields where mathematical modeling is essential.
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398:. 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.
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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.
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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.
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722:. 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
2105:
180:: Analyzing how the solution to a problem is affected by small changes in the input data, which helps in assessing the reliability of the numerical solution.
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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
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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
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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.
206:
Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving
152:: Understanding and minimizing the errors that arise in numerical calculations, such as round-off errors, truncation errors, and approximation errors.
1488:. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.
1322:
Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a
919:
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.
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Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type
332:
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
166:: Ensuring that small changes in the input or intermediate steps do not cause large changes in the output, which could lead to incorrect results.
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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.
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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
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deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the
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Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
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981:, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control
159:: Determining whether a numerical method will converge to the correct solution as more iterations or finer steps are taken.
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the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.
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if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is
173:: Developing algorithms that solve problems in a reasonable amount of time and with manageable computational resources.
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816:, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01.
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Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The
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algorithm is based on the singular value decomposition. The corresponding tool in statistics is called
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can be used to reduce optimization problems with constraints to unconstrained optimization problems.
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Jones, E., Oliphant, T., & Peterson, P. (2001). SciPy: Open source scientific tools for Python.
2017:
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Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB
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Scientific
Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations
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achieve very good numerical estimates of some functions. The canonical work in the field is the
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repository contains various collections of software routines for numerical problems, mostly in
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The numerical point of view goes back to the earliest mathematical writings. A tablet from the
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The field of optimization is further split in several subfields, depending on the form of the
387:. In computational matrix algebra, iterative methods are generally needed for large problems.
248:
The field of numerical analysis predates the invention of modern computers by many centuries.
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Numerical analysis is also concerned with computing (in an approximate way) the solution of
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are usually preferred for large systems. General iterative methods can be developed using a
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Bunks, C.; Chancelier, J.P.; Delebecque, F.; Goursat, M.; Nikoukhah, R.; Steer, S. (2012).
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The field of numerical analysis includes many sub-disciplines. Some of the major ones are:
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In the financial field, (private investment funds) and other financial institutions use
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1805:"The history and development of numerical analysis in Scotland: a personal perspective"
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1996:
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Bezanson, Jeff; Edelman, Alan; Karpinski, Stefan; Shah, Viral B. (1 January 2017).
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1368:. Commercial products implementing many different numerical algorithms include the
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Numerical solution of partial differential equations by the finite element method
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277:
2080:
Newton
Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms
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268:. The origins of modern numerical analysis are often linked to a 1947 paper by
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2490:
2349:
1398:
1334:. The theoretical justification of these methods often involves theorems from
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5166:
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2920:
2774:
18 May 2016 at the
Portuguese Web Archive Stefan Steinhaus, ScientificWeb.com
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978:
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to the exact solution only in the limit. A convergence test, often involving
218:
107:
64:
3976:
3087:
2874:
2106:"The Singular Value Decomposition and Its Applications in Image Compression"
1671:
1426:
1412:
5262:
4921:
4236:
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3600:
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966:
96:
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is used and the result is an approximation of the true solution (assuming
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and the derivative is known, then Newton's method is a popular choice.
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1513:
can be used to solve simple problems relating to numerical analysis.
276:, but others consider modern numerical analysis to go back to work by
4777:
4738:
4107:
2750:
2167:
1437:
1338:. This reduces the problem to the solution of an algebraic equation.
960:
Wind direction in blue, true trajectory in black, Euler method in red
934:
points, a line is computed that passes as close as possible to those
755:
exactly on a machine with finite memory (which is what all practical
60:
2083:. Computational Mathematics. Vol. 35 (2nd ed.). Springer.
1521:, including for matrices, which may be used in conjunction with its
663:
Well-conditioned problem: By contrast, evaluating the same function
4838:
4328:
3696:
3615:
3542:
1510:
1432:
There are several popular numerical computing applications such as
1258:
1105:
Much effort has been put in the development of methods for solving
217:
tools from numerical analysis to attempt to calculate the value of
131:
119:
37:
2643:
2223:
A first course in the numerical analysis of differential equations
3481:
2945:
2919:
2049:
1449:
1361:
1253:
Numerical integration, in some instances also known as numerical
236:
49:
2530:
2447:
Solving problems in scientific computing using Maple and Matlab®
1178:
1763:
Numerical analysis: Historical developments in the 20th century
1461:
1453:
1441:
1433:
1357:
992:
1464:(a C++ library). There are also programming languages such as
874:
40:(c. 1800–1600 BCE) with annotations. The approximation of the
1485:
1481:
1477:
1020:
391:
32:
2624:
2366:
Finite difference schemes and partial differential equations
2159:
2053:
Newton's method: An updated approach of
Kantorovich's theory
2018:"Methods of Conjugate Gradients for Solving Linear Systems"
293:
3235:
3203:, William H. Press (free, downloadable previous editions)
1697:
Introduction to numerical linear algebra and optimization
87:(predicting the motions of planets, stars and galaxies),
3247:
2416:
2050:
Ezquerro Fernández, J.A.; Hernández Verón, M.Á. (2017).
308:
longer and more complicated calculations could be done.
2025:
Journal of
Research of the National Bureau of Standards
1141:
for non-square matrices. Iterative methods such as the
1109:. Standard direct methods, i.e., methods that use some
729:
3106:. In Leader, I.; Gowers, T.; Barrow-Green, J. (eds.).
2185:"15. Monte carlo simulation and numerical integration"
2016:
Hestenes, Magnus R.; Stiefel, Eduard (December 1952).
1183:
Several important problems can be phrased in terms of
1175:
is another technique for solving nonlinear equations.
762:
2853:
2769:
Comparison of mathematical programs for data analysis
2751:
Speed comparison of various number crunching packages
2728:
Learning SciPy for numerical and scientific computing
1448:
as well as free and open-source alternatives such as
1066:
1031:
827:
780:
99:
for simulating living cells in medicine and biology.
3092:(examples of the importance of accurate arithmetic).
2250:
Numerical methods for partial differential equations
1285:), or, in modestly large dimensions, the method of
196:Advanced numerical methods are essential in making
3325:Introduction to Numerical Analysis for Engineering
3039:
2976:. Society for Industrial and Applied Mathematics.
2969:
1694:
1627:"Photograph, illustration, and description of the
1094:
1052:
857:
808:
682:= 10 is a well-conditioned problem. For instance,
52:figures. 1 + 24/60 + 51/60 + 10/60 = 1.41421296...
2592:Journal of Computational and Graphical Statistics
2417:Quarteroni, A.; Saleri, F.; Gervasio, P. (2014).
2304:The mathematical theory of finite element methods
27:Study of algorithms using numerical approximation
5343:
3215:), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
3112:. Princeton University Press. pp. 604–614.
3072:. Vol. 2. North-Holland. pp. 1214–39.
2931:(3rd ed.). Johns Hopkins University Press.
2627:"Julia: A Fresh Approach to Numerical Computing"
2533:Engineering and scientific computing with Scilab
1759:
1727:
902:
751:arise because it is impossible to represent all
2132:
2015:
1988:
1925:(2nd ed.). American Mathematical Society.
1922:Iterative methods for the solution of equations
1695:Ciarlet, P.G.; Miara, B.; Thomas, J.M. (1989).
969:for solving an ordinary differential equation.
401:As an example, consider the problem of solving
327:
235:Insurance companies use numerical programs for
2972:Accuracy and Stability of Numerical Algorithms
2585:"R: a language for data analysis and graphics"
2582:
2443:
225:more precisely than other market participants.
4368:Note: This template roughly follows the 2012
4344:
4006:
3992:
3378:
3018:Numerical Analysis and Scientific Computation
2887:
2555:
2470:
2393:Finite Volume Methods for Hyperbolic Problems
1834:
1179:Solving eigenvalue or singular value problems
317:Institute of Mathematics and its Applications
2946:Ralston Anthony; Rabinowitz Philips (2001).
2725:
2300:
2226:(2nd ed.). Cambridge University Press.
1949:Iterative methods for solving linear systems
1891:
993:Interpolation, extrapolation, and regression
949:Optimization: Suppose lemonade is sold at a
3392:
2703:SciPy and NumPy: an overview for developers
2420:Scientific computing with MATLAB and Octave
2334:(2nd ed.). Wellesley-Cambridge Press.
2191:. Vol. 1. Elsevier. pp. 731–800.
1868:Iterative methods for sparse linear systems
1631:tablet from the Yale Babylonian Collection"
1330:method, or (particularly in engineering) a
875:Numerical stability and well-posed problems
613:Ill-conditioned problem: Take the function
145:Key aspects of numerical analysis include:
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3985:
3385:
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2990:
2362:
1992:Extrapolation methods: theory and practice
1021:Solving equations and systems of equations
3261:
3241:Digital Library of Mathematical Functions
3222:(Computational Science Education Project)
3095:
3034:
2660:
2642:
2327:
2115:. College of the Redwoods. Archived from
2103:
2076:
1945:
1788:"An Introduction to Quantitative Finance"
1728:Trefethen, Lloyd; Bau III, David (1997).
1503:which can provide more accurate results.
1352:Comparison of numerical-analysis software
1299:Numerical ordinary differential equations
1292:
2849:
2700:
2331:An analysis of the finite element method
1798:
1796:
1303:Numerical partial differential equations
930:Regression: In linear regression, given
640:of less than 0.1 turns into a change in
282:
31:
3068:. Proc. IFIP Congress 71 in Ljubljana.
2818:
2444:Gander, W.; Hrebicek, J., eds. (2011).
2389:
2273:
2219:
1973:
1971:
1969:
1242:
313:Leslie Fox Prize for Numerical Analysis
14:
5344:
5061:Knowledge representation and reasoning
4197:Atomic, molecular, and optical physics
3354:Numerical Analysis - Numerical Methods
3012:
2964:
2783:
2503:
2182:
1977:
1814:. World Scientific. pp. 161–177.
1802:
1655:
1499:also benefit from the availability of
5086:Philosophy of artificial intelligence
4332:
3980:
3366:
3358:California State University Fullerton
3329:Massachusetts Institute of Technology
3287:, Dennis Deturck and Herbert S. Wilf
3060:
2559:Digital image processing using SCILAB
2160:
1918:
1898:(2nd ed.). Courier Corporation.
1793:
1755:
1753:
1751:
1408:Transactions on Mathematical Software
1391:published numerous algorithms in its
4412:Energy consumption (Green computing)
4358:
3947:
2950:(2nd ed.). Dover publications.
2948:A First Course in Numerical Analysis
2556:Thanki, R.M.; Kothari, A.M. (2019).
2246:
2133:Davis, P.J.; Rabinowitz, P. (2007).
1989:Brezinski, C.; Zaglia, M.R. (2013).
1966:
1864:
1017:-method is one way to achieve this.
730:Generation and propagation of errors
478:For the iterative method, apply the
5091:Distributed artificial intelligence
4370:ACM Computing Classification System
3959:
2189:Handbook of Computational Economics
1892:Hageman, L.A.; Young, D.M. (2012).
1760:Brezinski, C.; Wuytack, L. (2012).
1348:List of numerical-analysis software
1257:, asks for the value of a definite
763:Truncation and discretization error
660:= 1 is an ill-conditioned problem.
494:− 24. The initial values are
24:
4603:Integrated development environment
3345:Introduction to Numerical Analysis
3335:Numerical Analysis for Engineering
3251:Handbook of Mathematical Functions
3109:Princeton Companion of Mathematics
2996:Introduction to Numerical Analysis
2471:Barnes, B.; Fulford, G.R. (2011).
2464:
2437:
1844:. Vol. 10. World Scientific.
1748:
1422:Library of Mathematics Subroutines
1397:(code for these "AS" functions is
1224:and the constraint. For instance,
955:
940:
921:
714:'. For example, the solution of a
25:
5373:
5071:Automated planning and scheduling
4608:Software configuration management
3208:First Steps in Numerical Analysis
3128:
2583:Ihaka, R.; Gentleman, R. (1996).
1566:List of numerical analysis topics
1261:. Popular methods use one of the
945:How much for a glass of lemonade?
894:
705:
258:Lagrange interpolation polynomial
93:stochastic differential equations
5325:
5315:
5306:
5305:
3958:
3946:
3935:
3934:
3922:
3303:Numerical Methods for Physicists
3154:(searchable; pages are images).
3042:The Algebraic Eigenvalue Problem
2888:Marasco, A.; Romano, A. (2001).
2822:The MATHEMATICA® book, version 4
2790:(3rd ed.). Addison-Wesley.
2253:(3rd ed.). Academic Press.
2136:Methods of numerical integration
1659:Applied numerical linear algebra
5316:
4719:Computational complexity theory
4318:Timeline of physics discoveries
3843:Computational complexity theory
3174:
2881:
2843:
2812:
2777:
2762:
2744:
2719:
2694:
2685:
2618:
2576:
2549:
2524:
2497:
2410:
2383:
2356:
2328:Strang, G.; Fix, G.J. (2018) .
2321:
2301:Brenner, S.; Scott, R. (2013).
2294:
2267:
2240:
2213:
2176:
2153:
2126:
2097:
2070:
2043:
2009:
1982:
1939:
1912:
1885:
1841:The Birth of Numerical Analysis
1812:The Birth of Numerical Analysis
1517:, for example, has hundreds of
1313:ordinary differential equations
1202:
690:(11) = 0.1: a modest change in
608:
367:In contrast to direct methods,
322:
186:
126:numerical approximation of the
81:ordinary differential equations
4510:Network performance evaluation
3312:Lectures in Numerical Analysis
3285:Lectures on Numerical Analysis
2604:10.1080/10618600.1996.10474713
2396:. Cambridge University Press.
2104:Ogden, C.J.; Huff, T. (1997).
1858:
1838:; Cools, Ronald, eds. (2010).
1828:
1780:
1721:
1699:. Cambridge University Press.
1688:
1649:
1619:
1501:arbitrary-precision arithmetic
1317:partial differential equations
208:partial differential equations
13:
1:
4874:Multimedia information system
4859:Geographic information system
4849:Enterprise information system
4445:Computer systems organization
3164:Journal on Numerical Analysis
2998:(2nd ed.). McGraw-Hill.
2850:Shaw, W.T.; Tigg, J. (1993).
2197:10.1016/S1574-0021(96)01017-9
1607:
1189:singular value decompositions
903:Computing values of functions
648:) of nearly 1000. Evaluating
334:infinite precision arithmetic
315:was initiated in 1985 by the
5233:Computational social science
4821:Theoretical computer science
4641:Software development process
4417:Electronic design automation
4402:Very Large Scale Integration
3356:(archived), John H. Mathews
3170:, volumes 1–..., SIAM, 1964–
1612:
1591:Symbolic-numeric computation
1419:several times published its
1417:Naval Surface Warfare Center
1197:principal component analysis
743:
694:leads to a modest change in
636:(1.001) = 1000: a change in
328:Direct and iterative methods
198:numerical weather prediction
48:figures, which is about six
7:
5056:Natural language processing
4844:Information storage systems
4282:Quantum information science
3187:Encyclopedia of Mathematics
3133:
2726:Blanco-Silva, F.J. (2013).
2475:(2nd ed.). CRC Press.
1536:Category:Numerical analysts
1528:
1341:
1265:(like the midpoint rule or
1107:systems of linear equations
809:{\displaystyle 3x^{3}+4=28}
346:systems of linear equations
10:
5378:
4972:Human–computer interaction
4942:Intrusion detection system
4854:Social information systems
4839:Database management system
4113:Classical electromagnetism
3893:Films about mathematicians
3289:University of Pennsylvania
3101:"IV.21 Numerical analysis"
3066:A survey of error-analysis
2912:
2827:Cambridge University Press
2787:Programming in mathematica
2423:(4th ed.). Springer.
2307:(2nd ed.). Springer.
1571:Local linearization method
1345:
1296:
1246:
1206:
1193:spectral image compression
1151:successive over-relaxation
1095:{\displaystyle 2x^{2}+5=3}
733:
243:
116:Yale Babylonian Collection
5301:
5238:Computational engineering
5213:Computational mathematics
5190:
5137:
5099:
5046:
5008:
4970:
4912:
4829:
4775:
4737:
4689:
4626:
4559:
4523:
4480:
4444:
4377:
4366:
4290:
4227:
4155:
4071:
4043:
4015:
3916:
3866:
3823:
3733:
3695:
3662:
3614:
3586:
3533:
3480:
3462:Philosophy of mathematics
3437:
3402:
3226:U.S. Department of Energy
2819:Wolfram, Stephen (1999).
2507:Practical IDL programming
2390:LeVeque, Randall (2002).
2363:Strikwerda, J.C. (2004).
2077:Deuflhard, Peter (2006).
1895:Applied iterative methods
1576:Numerical differentiation
1460:(similar to Matlab), and
1389:Royal Statistical Society
1279:quasi-Monte Carlo methods
1209:Mathematical optimization
1185:eigenvalue decompositions
1155:conjugate gradient method
987:floating-point arithmetic
858:{\displaystyle a+b+c+d+e}
413:for the unknown quantity
396:conjugate gradient method
102:Before modern computers,
5248:Computational healthcare
5243:Differentiable computing
5162:Graphics processing unit
4588:Domain-specific language
4457:Computational complexity
4219:Condensed matter physics
3898:Recreational mathematics
1731:Numerical Linear Algebra
1602:
1493:computer algebra systems
1135:positive-definite matrix
985:arising from the use of
926:A line through 20 points
89:numerical linear algebra
5223:Computational chemistry
5157:Photograph manipulation
5048:Artificial intelligence
4864:Decision support system
3783:Mathematical statistics
3773:Mathematical psychology
3743:Engineering mathematics
3677:Algebraic number theory
3298:University of Karlsruhe
3280:University of Cambridge
3156:(in English and German)
3152:volumes 1–66, 1959–1994
2280:. Courier Corporation.
2139:. Courier Corporation.
1672:10.1137/1.9781611971446
1476:with libraries such as
1283:Monte Carlo integration
1165:Root-finding algorithms
879:An algorithm is called
686:(10) = 1/9 ≈ 0.111 and
75:(as distinguished from
36:Babylonian clay tablet
5288:Educational technology
5119:Reinforcement learning
4869:Process control system
4767:Computational geometry
4757:Algorithmic efficiency
4752:Analysis of algorithms
4407:Systems on Chip (SoCs)
4303:Nobel Prize in Physics
4165:Relativistic mechanics
3929:Mathematics portal
3778:Mathematical sociology
3758:Mathematical economics
3753:Mathematical chemistry
3682:Analytic number theory
3563:Differential equations
3349:University of Maryland
3340:University of Waterloo
3262:Online course material
2756:5 October 2006 at the
2369:(2nd ed.). SIAM.
1946:Greenbaum, A. (1997).
1586:Probabilistic numerics
1541:Analysis of algorithms
1382:GNU Scientific Library
1309:differential equations
1293:Differential equations
1123:Cholesky decomposition
1096:
1054:
1053:{\displaystyle 2x+5=3}
961:
946:
927:
917:gross domestic product
915:Extrapolation: If the
867:is even more inexact.
859:
810:
296:publication edited by
288:
91:in data analysis, and
71:) for the problems of
69:symbolic manipulations
53:
5362:Computational science
5258:Electronic publishing
5228:Computational biology
5218:Computational physics
5114:Unsupervised learning
5028:Distributed computing
4904:Information retrieval
4811:Mathematical analysis
4801:Mathematical software
4691:Theory of computation
4656:Software construction
4646:Requirements analysis
4524:Software organization
4452:Computer architecture
4422:Hardware acceleration
4387:Printed circuit board
4308:Philosophy of physics
3908:Mathematics education
3838:Theory of computation
3558:Hypercomplex analysis
3255:Abramowitz and Stegun
3141:Numerische Mathematik
2784:Maeder, R.E. (1997).
2701:Bressert, E. (2012).
2504:Gumley, L.E. (2001).
2183:Geweke, John (1996).
2163:"Gaussian Quadrature"
2122:on 25 September 2006.
1803:Watson, G.A. (2010).
1656:Demmel, J.W. (1997).
1551:Computational physics
1546:Computational science
1468:(similar to S-PLUS),
1324:finite element method
1263:Newton–Cotes formulas
1249:Numerical integration
1097:
1055:
959:
944:
925:
860:
811:
734:Further information:
716:differential equation
676: − 1)
626: − 1)
305:mechanical calculator
298:Abramowitz and Stegun
286:
106:often relied on hand
73:mathematical analysis
35:
5357:Mathematical physics
5018:Concurrent computing
4990:Ubiquitous computing
4962:Application security
4957:Information security
4786:Discrete mathematics
4762:Randomized algorithm
4714:Computability theory
4699:Model of computation
4671:Software maintenance
4666:Software engineering
4628:Software development
4578:Programming language
4573:Programming paradigm
4490:Network architecture
4267:Mathematical physics
3888:Informal mathematics
3768:Mathematical physics
3763:Mathematical finance
3748:Mathematical biology
3687:Diophantine geometry
3274:28 July 2009 at the
3182:"Numerical analysis"
2274:Johnson, C. (2012).
2220:Iserles, A. (2009).
2037:10.6028/jres.049.044
1919:Traub, J.F. (1982).
1332:finite volume method
1243:Evaluating integrals
1237:Lagrange multipliers
1191:. For instance, the
1115:Gaussian elimination
1111:matrix decomposition
1064:
1029:
825:
778:
772:discretization error
338:Gaussian elimination
262:Gaussian elimination
250:Linear interpolation
215:quantitative finance
130:, the length of the
77:discrete mathematics
5293:Document management
5283:Operations research
5208:Enterprise software
5124:Multi-task learning
5109:Supervised learning
4831:Information systems
4661:Software deployment
4618:Software repository
4472:Real-time computing
4242:Atmospheric physics
4081:Classical mechanics
4009:branches of physics
3903:Mathematics and art
3813:Operations research
3568:Functional analysis
3230:archived 2017-08-01
3097:Trefethen, Lloyd N.
3070:Info. Processing 71
3046:. Clarendon Press.
2966:Higham, Nicholas J.
2929:Matrix Computations
2925:Charles F. Van Loan
2247:Ames, W.F. (2014).
2161:Weisstein, Eric W.
1561:Interval arithmetic
1519:available functions
1387:Over the years the
1380:alternative is the
1336:functional analysis
1271:Gaussian quadrature
1147:Gauss–Seidel method
523:
422:
344:method for solving
336:. Examples include
230:operations research
85:celestial mechanics
63:that use numerical
5352:Numerical analysis
5076:Search methodology
5023:Parallel computing
4980:Interaction design
4889:Computing platform
4816:Numerical analysis
4806:Information theory
4598:Software framework
4561:Software notations
4500:Network components
4397:Integrated circuit
4298:History of physics
3848:Numerical analysis
3457:Mathematical logic
3452:Information theory
3320:Mahidol University
3278:), Stuart Dalziel
3020:. Addison Wesley.
3014:Leader, Jeffery J.
2859:. Addison-Wesley.
1596:Validated numerics
1405:similarly, in its
1394:Applied Statistics
1226:linear programming
1222:objective function
1092:
1050:
962:
947:
928:
882:numerically stable
855:
806:
521:
420:
354:linear programming
289:
57:Numerical analysis
54:
18:Numerical calculus
5339:
5338:
5268:Electronic voting
5198:Quantum Computing
5191:Applied computing
5177:Image compression
4947:Hardware security
4937:Security services
4894:Digital marketing
4681:Open-source model
4593:Modeling language
4505:Network scheduler
4326:
4325:
4313:Physics education
4262:Materials science
4229:Interdisciplinary
4187:Quantum mechanics
3974:
3973:
3573:Harmonic analysis
3327:, Henrik Schmidt
3307:Oxford University
3305:, Anthony O’Hare
3296:, John D. Fenton
3294:Numerical methods
3268:Numerical Methods
3236:Numerical Methods
3200:Numerical Recipes
3144:, volumes 1–...,
3119:978-0-691-11880-2
3079:978-0-7204-2063-0
3053:978-0-19-853418-1
2992:Hildebrand, F. B.
2901:978-0-8176-4205-1
2866:978-0-201-54217-2
2653:10.1137/141000671
2569:978-3-319-89533-8
2542:978-1-4612-7204-5
2517:978-0-08-051444-4
2482:978-1-4200-8350-7
2457:978-3-642-18873-2
2430:978-3-642-45367-0
2403:978-1-139-43418-8
2376:978-0-89871-793-8
2314:978-1-4757-3658-8
2287:978-0-486-46900-3
2260:978-0-08-057130-0
2233:978-0-521-73490-5
2146:978-0-486-45339-2
2090:978-3-540-21099-3
2063:978-3-319-55976-6
2002:978-0-08-050622-7
1959:978-0-89871-396-1
1932:978-0-8284-0312-2
1905:978-0-8284-0312-2
1878:978-0-89871-534-7
1865:Saad, Y. (2003).
1851:978-981-283-625-0
1836:Bultheel, Adhemar
1790:. 2013. page VII.
1773:978-0-444-59858-5
1741:978-0-89871-361-9
1681:978-1-61197-144-6
1637:on 13 August 2012
1581:Numerical Recipes
1556:Gordon Bell Prize
1523:built in "solver"
1328:finite difference
974:
973:
768:Truncation errors
757:digital computers
736:Error propagation
603:
602:
522:Iterative method
476:
475:
369:iterative methods
104:numerical methods
16:(Redirected from
5369:
5329:
5328:
5319:
5318:
5309:
5308:
5129:Cross-validation
5101:Machine learning
4985:Social computing
4952:Network security
4747:Algorithm design
4676:Programming team
4636:Control variable
4613:Software library
4551:Software quality
4546:Operating system
4495:Network protocol
4360:Computer science
4353:
4346:
4339:
4330:
4329:
4252:Chemical physics
4192:Particle physics
4118:Classical optics
4001:
3994:
3987:
3978:
3977:
3962:
3961:
3950:
3949:
3938:
3937:
3927:
3926:
3858:Computer algebra
3833:Computer science
3553:Complex analysis
3387:
3380:
3373:
3364:
3363:
3249:, ch 25. in the
3195:
3157:
3123:
3105:
3091:
3057:
3045:
3031:
3009:
2987:
2975:
2961:
2942:
2906:
2905:
2885:
2879:
2878:
2858:
2847:
2841:
2840:
2816:
2810:
2809:
2781:
2775:
2766:
2760:
2748:
2742:
2741:
2723:
2717:
2716:
2698:
2692:
2689:
2683:
2682:
2664:
2646:
2622:
2616:
2615:
2589:
2580:
2574:
2573:
2553:
2547:
2546:
2528:
2522:
2521:
2501:
2495:
2494:
2468:
2462:
2461:
2441:
2435:
2434:
2414:
2408:
2407:
2387:
2381:
2380:
2360:
2354:
2353:
2325:
2319:
2318:
2298:
2292:
2291:
2271:
2265:
2264:
2244:
2238:
2237:
2217:
2211:
2210:
2180:
2174:
2173:
2172:
2157:
2151:
2150:
2130:
2124:
2123:
2121:
2110:
2101:
2095:
2094:
2074:
2068:
2067:
2047:
2041:
2040:
2022:
2013:
2007:
2006:
1986:
1980:
1975:
1964:
1963:
1943:
1937:
1936:
1916:
1910:
1909:
1889:
1883:
1882:
1862:
1856:
1855:
1832:
1826:
1825:
1809:
1800:
1791:
1784:
1778:
1777:
1757:
1746:
1745:
1725:
1719:
1718:
1692:
1686:
1685:
1653:
1647:
1646:
1644:
1642:
1633:. Archived from
1623:
1411:("TOMS" code is
1159:matrix splitting
1139:QR decomposition
1119:LU decomposition
1101:
1099:
1098:
1093:
1079:
1078:
1060:is linear while
1059:
1057:
1056:
1051:
983:round-off errors
907:
906:
888:well-conditioned
866:
864:
862:
861:
856:
815:
813:
812:
807:
793:
792:
749:Round-off errors
677:
627:
524:
520:
480:bisection method
423:
419:
385:Jacobi iteration
381:bisection method
358:finite precision
342:QR factorization
287:NIST publication
274:Herman Goldstine
270:John von Neumann
128:square root of 2
59:is the study of
42:square root of 2
21:
5377:
5376:
5372:
5371:
5370:
5368:
5367:
5366:
5342:
5341:
5340:
5335:
5326:
5297:
5278:Word processing
5186:
5172:Virtual reality
5133:
5095:
5066:Computer vision
5042:
5038:Multiprocessing
5004:
4966:
4932:Security hacker
4908:
4884:Digital library
4825:
4776:Mathematics of
4771:
4733:
4709:Automata theory
4704:Formal language
4685:
4651:Software design
4622:
4555:
4541:Virtual machine
4519:
4515:Network service
4476:
4467:Embedded system
4440:
4373:
4362:
4357:
4327:
4322:
4286:
4272:Medical physics
4223:
4182:Nuclear physics
4151:
4145:Non-equilibrium
4067:
4039:
4011:
4005:
3975:
3970:
3921:
3912:
3862:
3819:
3798:Systems science
3729:
3725:Homotopy theory
3691:
3658:
3610:
3582:
3529:
3476:
3447:Category theory
3433:
3398:
3391:
3338:, D. W. Harder
3276:Wayback Machine
3264:
3238:, ch 3. in the
3180:
3177:
3155:
3136:
3131:
3126:
3120:
3103:
3080:
3054:
3036:Wilkinson, J.H.
3028:
3006:
2984:
2958:
2939:
2915:
2910:
2909:
2902:
2886:
2882:
2867:
2856:
2848:
2844:
2837:
2817:
2813:
2798:
2782:
2778:
2767:
2763:
2758:Wayback Machine
2749:
2745:
2738:
2724:
2720:
2713:
2699:
2695:
2690:
2686:
2623:
2619:
2587:
2581:
2577:
2570:
2554:
2550:
2543:
2529:
2525:
2518:
2502:
2498:
2483:
2469:
2465:
2458:
2442:
2438:
2431:
2415:
2411:
2404:
2388:
2384:
2377:
2361:
2357:
2342:
2326:
2322:
2315:
2299:
2295:
2288:
2272:
2268:
2261:
2245:
2241:
2234:
2218:
2214:
2207:
2181:
2177:
2158:
2154:
2147:
2131:
2127:
2119:
2108:
2102:
2098:
2091:
2075:
2071:
2064:
2048:
2044:
2020:
2014:
2010:
2003:
1987:
1983:
1976:
1967:
1960:
1944:
1940:
1933:
1917:
1913:
1906:
1890:
1886:
1879:
1863:
1859:
1852:
1833:
1829:
1822:
1807:
1801:
1794:
1786:Stephen Blyth.
1785:
1781:
1774:
1758:
1749:
1742:
1726:
1722:
1707:
1693:
1689:
1682:
1654:
1650:
1640:
1638:
1625:
1624:
1620:
1615:
1610:
1605:
1600:
1531:
1354:
1346:Main articles:
1344:
1305:
1297:Main articles:
1295:
1251:
1245:
1211:
1205:
1181:
1074:
1070:
1065:
1062:
1061:
1030:
1027:
1026:
1023:
995:
905:
897:
877:
826:
823:
822:
820:
788:
784:
779:
776:
775:
765:
746:
738:
732:
708:
664:
632:(1.1) = 10 and
614:
611:
585:−4.22...
510:) = −24,
465:Take cube roots
356:. In practice,
330:
325:
278:E. T. Whittaker
254:Newton's method
246:
189:
67:(as opposed to
28:
23:
22:
15:
12:
11:
5:
5375:
5365:
5364:
5359:
5354:
5337:
5336:
5334:
5333:
5323:
5313:
5302:
5299:
5298:
5296:
5295:
5290:
5285:
5280:
5275:
5270:
5265:
5260:
5255:
5250:
5245:
5240:
5235:
5230:
5225:
5220:
5215:
5210:
5205:
5200:
5194:
5192:
5188:
5187:
5185:
5184:
5182:Solid modeling
5179:
5174:
5169:
5164:
5159:
5154:
5149:
5143:
5141:
5135:
5134:
5132:
5131:
5126:
5121:
5116:
5111:
5105:
5103:
5097:
5096:
5094:
5093:
5088:
5083:
5081:Control method
5078:
5073:
5068:
5063:
5058:
5052:
5050:
5044:
5043:
5041:
5040:
5035:
5033:Multithreading
5030:
5025:
5020:
5014:
5012:
5006:
5005:
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4997:
4992:
4987:
4982:
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4974:
4968:
4967:
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4954:
4949:
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4939:
4934:
4929:
4927:Formal methods
4924:
4918:
4916:
4910:
4909:
4907:
4906:
4901:
4899:World Wide Web
4896:
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4886:
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4150:
4149:
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4135:Thermodynamics
4132:
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4115:
4110:
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4098:
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3876:Mathematicians
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3868:Related topics
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3827:
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3803:Control theory
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3790:
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3770:
3765:
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3578:Measure theory
3575:
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3260:
3259:
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3216:
3204:
3196:
3176:
3173:
3172:
3171:
3160:
3159:
3158:
3135:
3132:
3130:
3129:External links
3127:
3125:
3124:
3118:
3093:
3078:
3058:
3052:
3032:
3026:
3010:
3004:
2988:
2982:
2962:
2957:978-0486414546
2956:
2943:
2937:
2921:Golub, Gene H.
2916:
2914:
2911:
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2907:
2900:
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2684:
2617:
2598:(3): 299–314.
2575:
2568:
2548:
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2523:
2516:
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2125:
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2069:
2062:
2056:. Birkhäuser.
2042:
2008:
2001:
1981:
1965:
1958:
1938:
1931:
1911:
1904:
1884:
1877:
1857:
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1553:
1548:
1543:
1538:
1532:
1530:
1527:
1343:
1340:
1294:
1291:
1267:Simpson's rule
1247:Main article:
1244:
1241:
1235:The method of
1230:simplex method
1207:Main article:
1204:
1201:
1180:
1177:
1169:differentiable
1091:
1088:
1085:
1082:
1077:
1073:
1069:
1049:
1046:
1043:
1040:
1037:
1034:
1022:
1019:
994:
991:
972:
971:
951:lemonade stand
904:
901:
896:
895:Areas of study
893:
876:
873:
854:
851:
848:
845:
842:
839:
836:
833:
830:
805:
802:
799:
796:
791:
787:
783:
764:
761:
745:
742:
731:
728:
712:discretization
707:
706:Discretization
704:
610:
607:
601:
600:
597:
594:
591:
587:
586:
583:
580:
577:
573:
572:
569:
566:
563:
559:
558:
557:−13.875
555:
552:
549:
545:
544:
538:
535:
530:
474:
473:
467:
461:
460:
454:
448:
447:
440:
434:
433:
426:
421:Direct method
411:
410:
350:simplex method
329:
326:
324:
321:
266:Euler's method
245:
242:
241:
240:
233:
226:
211:
204:
201:
188:
185:
150:Error Analysis
26:
9:
6:
4:
3:
2:
5374:
5363:
5360:
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5350:
5349:
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5294:
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5281:
5279:
5276:
5274:
5271:
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5266:
5264:
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5209:
5206:
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5199:
5196:
5195:
5193:
5189:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5167:Mixed reality
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5144:
5142:
5140:
5136:
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5026:
5024:
5021:
5019:
5016:
5015:
5013:
5011:
5007:
5001:
5000:Accessibility
4998:
4996:
4995:Visualization
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4975:
4973:
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4963:
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4902:
4900:
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4882:
4880:
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4774:
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4755:
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4710:
4707:
4705:
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4692:
4688:
4682:
4679:
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4659:
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4493:
4491:
4488:
4487:
4485:
4483:
4479:
4473:
4470:
4468:
4465:
4463:
4462:Dependability
4460:
4458:
4455:
4453:
4450:
4449:
4447:
4443:
4437:
4433:
4430:
4428:
4425:
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4403:
4400:
4398:
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4393:
4390:
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4384:
4382:
4380:
4376:
4371:
4365:
4361:
4354:
4349:
4347:
4342:
4340:
4335:
4334:
4331:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4295:
4293:
4289:
4283:
4280:
4278:
4277:Ocean physics
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
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4238:
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4234:
4232:
4230:
4226:
4220:
4217:
4213:
4212:Modern optics
4210:
4208:
4205:
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4199:
4198:
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4129:
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4092:
4089:
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4084:
4083:
4082:
4079:
4078:
4076:
4074:
4070:
4062:
4061:Computational
4059:
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4057:
4054:
4052:
4049:
4048:
4046:
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4030:
4029:
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4018:
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3997:
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3988:
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3979:
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3851:
3849:
3846:
3844:
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3836:
3834:
3831:
3830:
3828:
3826:
3825:Computational
3822:
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3804:
3801:
3800:
3799:
3796:
3794:
3791:
3789:
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3708:
3706:
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3700:
3698:
3694:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3669:
3667:
3665:
3664:Number theory
3661:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3621:
3619:
3617:
3613:
3607:
3604:
3602:
3599:
3597:
3596:Combinatorics
3594:
3593:
3591:
3589:
3585:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3548:Real analysis
3546:
3544:
3541:
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3538:
3536:
3532:
3526:
3523:
3521:
3518:
3516:
3513:
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3436:
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3427:
3425:
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108:interpolation
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65:approximation
62:
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5263:Cyberwarfare
4922:Cryptography
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4237:Astrophysics
4051:Experimental
3963:
3951:
3939:
3920:
3853:Optimization
3847:
3715:Differential
3639:Differential
3606:Order theory
3601:Graph theory
3505:Group theory
3334:
3318:), R. Radok
3250:
3239:
3219:
3207:
3199:
3185:
3175:Online texts
3162:
3139:
3108:
3069:
3065:
3041:
3017:
2995:
2971:
2947:
2928:
2894:. Springer.
2890:
2883:
2852:
2845:
2821:
2814:
2786:
2779:
2764:
2746:
2727:
2721:
2705:. O'Reilly.
2702:
2696:
2687:
2637:(1): 65–98.
2634:
2630:
2620:
2595:
2591:
2578:
2562:. Springer.
2558:
2551:
2535:. Springer.
2532:
2526:
2510:. Elsevier.
2506:
2499:
2472:
2466:
2450:. Springer.
2446:
2439:
2419:
2412:
2392:
2385:
2365:
2358:
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2323:
2303:
2296:
2276:
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2178:
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2128:
2117:the original
2112:
2099:
2079:
2072:
2052:
2045:
2028:
2024:
2011:
1995:. Elsevier.
1991:
1984:
1948:
1941:
1921:
1914:
1894:
1887:
1867:
1860:
1840:
1830:
1811:
1782:
1766:. Elsevier.
1762:
1730:
1723:
1696:
1690:
1658:
1651:
1639:. Retrieved
1635:the original
1628:
1621:
1505:
1490:
1431:
1421:
1406:
1393:
1386:
1355:
1321:
1306:
1287:sparse grids
1252:
1234:
1219:
1212:
1203:Optimization
1182:
1163:
1104:
1024:
1008:
1002:
996:
975:
967:Euler method
963:
948:
935:
931:
929:
914:
910:
898:
886:
880:
878:
869:
818:
766:
753:real numbers
747:
739:
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699:
695:
691:
687:
683:
679:
673:
669:
665:
662:
657:
653:
649:
645:
641:
637:
633:
629:
628:. Note that
623:
619:
615:
612:
609:Conditioning
604:
540:
532:
527:
515:
511:
507:
503:
499:
495:
491:
487:
483:
477:
469:
464:
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451:
443:
437:
429:
414:
412:
406:
400:
389:
377:the residual
366:
331:
323:Key concepts
310:
302:
290:
247:
210:numerically.
190:
187:Applications
182:
178:Conditioning
177:
175:
170:
168:
163:
161:
156:
154:
149:
147:
144:
140:
113:
101:
83:as found in
56:
55:
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5273:Video games
5253:Digital art
5010:Concurrency
4879:Data mining
4791:Probability
4531:Interpreter
4140:Statistical
4056:Theoretical
4033:Engineering
3965:WikiProject
3808:Game theory
3788:Probability
3525:Homological
3515:Multilinear
3495:Commutative
3472:Type theory
3439:Foundations
3395:mathematics
2631:SIAM Review
2031:(6): 409–.
1978:Higham 2002
1508:spreadsheet
1497:Mathematica
1275:Monte Carlo
1215:constraints
472:= 2.
459:= 8.
452:Divide by 3
223:derivatives
157:Convergence
136:unit square
124:sexagesimal
122:), gives a
46:sexagesimal
5346:Categories
5331:Glossaries
5203:E-commerce
4796:Statistics
4739:Algorithms
4536:Middleware
4392:Peripheral
4257:Geophysics
4247:Biophysics
4091:Analytical
4044:Approaches
3793:Statistics
3672:Arithmetic
3634:Arithmetic
3500:Elementary
3467:Set theory
2806:1311056676
2491:1058138488
2350:1145780513
1608:References
1506:Also, any
1458:GNU Octave
1255:quadrature
1010:Regression
438:Subtract 4
432:+ 4 = 28.
348:, and the
171:Efficiency
61:algorithms
5152:Rendering
5147:Animation
4778:computing
4729:Semantics
4427:Processor
4207:Molecular
4108:Acoustics
4101:Continuum
4096:Celestial
4086:Newtonian
4073:Classical
4016:Divisions
3720:Geometric
3710:Algebraic
3649:Euclidean
3624:Algebraic
3520:Universal
3192:EMS Press
3062:Kahan, W.
3038:(1988) .
2968:(2002) .
2730:. Packt.
2671:0036-1445
2644:1411.1607
2168:MathWorld
1715:877155729
1641:2 October
1613:Citations
1438:TK Solver
1131:hermitian
1127:symmetric
744:Round-off
724:continuum
571:10.17...
362:stability
280:in 1912.
239:analysis.
237:actuarial
200:feasible.
164:Stability
5311:Category
5139:Graphics
4914:Security
4583:Compiler
4482:Networks
4379:Hardware
3941:Category
3697:Topology
3644:Discrete
3629:Analytic
3616:Geometry
3588:Discrete
3543:Calculus
3535:Analysis
3490:Abstract
3429:Glossary
3412:Timeline
3316:archived
3272:Archived
3213:archived
3148:, 1959–
3146:Springer
3134:Journals
3099:(2008).
3088:25116949
3064:(1972).
3016:(2004).
2994:(1974).
2927:(1986).
2875:28149048
2772:Archived
2754:Archived
2679:13026838
2612:60206680
1952:. SIAM.
1871:. SIAM.
1734:. SIAM.
1529:See also
1511:software
1495:such as
1342:Software
1259:integral
1102:is not.
938:points.
720:function
599:2.32...
518:) = 57.
409:+ 4 = 28
394:and the
373:converge
132:diagonal
120:YBC 7289
44:is four
38:YBC 7289
5321:Outline
4291:Related
4175:General
4170:Special
4028:Applied
3953:Commons
3735:Applied
3705:General
3482:Algebra
3407:History
3194:, 2001
3168:(SINUM)
2913:Sources
2113:Math 45
1629:root(2)
1450:FreeMat
1415:). The
1362:Fortran
1311:, both
865:
821:
672:) = 1/(
656:) near
622:) = 1/(
244:History
50:decimal
4202:Atomic
4157:Modern
4007:Major
3654:Finite
3510:Linear
3417:Future
3393:Major
3116:
3086:
3076:
3050:
3024:
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1678:
1474:Python
1472:, and
1454:Scilab
1444:, and
1442:S-PLUS
1434:MATLAB
1425:(code
1358:Netlib
1137:, and
1133:) and
759:are).
596:2.0625
543:(mid)
446:= 24.
383:, and
340:, the
219:stocks
4724:Logic
4565:tools
3881:lists
3424:Lists
3397:areas
3104:(PDF)
2857:(PDF)
2675:S2CID
2639:arXiv
2608:S2CID
2588:(PDF)
2120:(PDF)
2109:(PDF)
2021:(PDF)
1808:(PDF)
1603:Notes
1515:Excel
1491:Many
1486:SymPy
1482:SciPy
1478:NumPy
1470:Julia
1281:(see
1269:) or
718:is a
678:near
590:1.875
582:1.875
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498:= 0,
490:) = 3
392:GMRES
264:, or
134:in a
4563:and
4436:Form
4432:Size
4128:Wave
4023:Pure
3220:CSEP
3114:ISBN
3084:OCLC
3074:ISBN
3048:ISBN
3022:ISBN
3000:ISBN
2978:ISBN
2952:ISBN
2933:ISBN
2896:ISBN
2871:OCLC
2861:ISBN
2831:ISBN
2802:OCLC
2792:ISBN
2732:ISBN
2707:ISBN
2667:ISSN
2564:ISBN
2537:ISBN
2512:ISBN
2487:OCLC
2477:ISBN
2452:ISBN
2425:ISBN
2398:ISBN
2371:ISBN
2346:OCLC
2336:ISBN
2309:ISBN
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2255:ISBN
2228:ISBN
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2141:ISBN
2085:ISBN
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1997:ISBN
1954:ISBN
1927:ISBN
1900:ISBN
1873:ISBN
1846:ISBN
1816:ISBN
1768:ISBN
1736:ISBN
1711:OCLC
1701:ISBN
1676:ISBN
1664:SIAM
1643:2006
1484:and
1462:IT++
1427:here
1413:here
1401:);
1399:here
1372:and
1370:IMSL
1364:and
1350:and
1326:, a
1315:and
1301:and
1153:and
1129:(or
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593:2.25
579:2.25
568:2.25
311:The
303:The
294:NIST
272:and
221:and
95:and
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2657:hdl
2649:doi
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1429:).
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