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.
3931:
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
141:
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.
228:
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.
819:
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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2349:
1398:
1334:. The theoretical justification of these methods often involves theorems from
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5180:
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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:
3983:
3094:
2874:
2106:"The Singular Value Decomposition and Its Applications in Image Compression"
1671:
1426:
1412:
5276:
4935:
4243:
3612:
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3511:
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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2652:
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
4791:
4752:
4114:
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
4852:
4335:
3703:
3622:
3549:
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
3488:
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
3014:
Numerical
Analysis : Mathematics of Scientific Computing
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:
3242:
3210:, William H. Press (free, downloadable previous editions)
1697:
Introduction to numerical linear algebra and optimization
87:(predicting the motions of planets, stars and galaxies),
3254:
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:
3113:. 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.
3099:(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
3332:Introduction to Numerical Analysis for Engineering
3046:
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
5357:
3222:), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
3119:. Princeton University Press. pp. 604–614.
3079:. 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
3016:, 3rd Ed., AMS, ISBN 978-0-8218-4788-6 (2002).
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.
4375:Note: This template roughly follows the 2012
4351:
4013:
3999:
3385:
3025: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
3399:
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:
4358:
4344:
4006:
3992:
3392:
3378:
2990:
2362:
1992:Extrapolation methods: theory and practice
1021:Solving equations and systems of equations
3268:
3248:Digital Library of Mathematical Functions
3229:(Computational Science Education Project)
3102:
3041:
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:
3075:. 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:
5358:
5075:Knowledge representation and reasoning
4204:Atomic, molecular, and optical physics
3361:Numerical Analysis - Numerical Methods
3019:
2964:
2783:
2503:
2182:
1977:
1814:. World Scientific. pp. 161–177.
1802:
1655:
1499:also benefit from the availability of
5100:Philosophy of artificial intelligence
4339:
3987:
3373:
3365:California State University Fullerton
3336:Massachusetts Institute of Technology
3294:, Dennis Deturck and Herbert S. Wilf
3067:
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
4419:Energy consumption (Green computing)
4365:
3954:
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
27:Methods for numerical approximations
5105:Distributed artificial intelligence
4377:ACM Computing Classification System
3966:
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:
4610:Integrated development environment
3352:Introduction to Numerical Analysis
3342:Numerical Analysis for Engineering
3258:Handbook of Mathematical Functions
3116: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:
5387:
5085:Automated planning and scheduling
4615:Software configuration management
3215:First Steps in Numerical Analysis
3135:
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
5339:
5329:
5320:
5319:
3965:
3953:
3942:
3941:
3929:
3310:Numerical Methods for Physicists
3161:(searchable; pages are images).
3049: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
5330:
4733:Computational complexity theory
4325:Timeline of physics discoveries
3850:Computational complexity theory
3181:
3012:David Kincaid and Ward Cheney:
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
4517:Network performance evaluation
3319:Lectures in Numerical Analysis
3292: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:
4888:Multimedia information system
4873:Geographic information system
4863:Enterprise information system
4452:Computer systems organization
3171: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
5247:Computational social science
4835:Theoretical computer science
4648:Software development process
4424:Electronic design automation
4409:Very Large Scale Integration
3363:(archived), John H. Mathews
3177:, 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:
5070:Natural language processing
4858:Information storage systems
4289:Quantum information science
3194:Encyclopedia of Mathematics
3140:
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:
5392:
4986:Human–computer interaction
4956:Intrusion detection system
4868:Social information systems
4853:Database management system
4120:Classical electromagnetism
3900:Films about mathematicians
3296:University of Pennsylvania
3108:"IV.21 Numerical analysis"
3073: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
5315:
5252:Computational engineering
5227:Computational mathematics
5204:
5151:
5113:
5060:
5022:
4984:
4926:
4843:
4789:
4751:
4696:
4633:
4566:
4530:
4487:
4451:
4384:
4373:
4297:
4234:
4162:
4078:
4050:
4022:
3923:
3873:
3830:
3740:
3702:
3669:
3621:
3593:
3540:
3487:
3469:Philosophy of mathematics
3444:
3409:
3233: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,
5262:Computational healthcare
5257:Differentiable computing
5176:Graphics processing unit
4595:Domain-specific language
4464:Computational complexity
4226:Condensed matter physics
3905: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
5237:Computational chemistry
5171:Photograph manipulation
5062:Artificial intelligence
4878:Decision support system
3790:Mathematical statistics
3780:Mathematical psychology
3750:Engineering mathematics
3684:Algebraic number theory
3305:University of Karlsruhe
3287:University of Cambridge
3163:(in English and German)
3159: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
5302:Educational technology
5133:Reinforcement learning
4883:Process control system
4781:Computational geometry
4771:Algorithmic efficiency
4766:Analysis of algorithms
4414:Systems on Chip (SoCs)
4310:Nobel Prize in Physics
4172:Relativistic mechanics
3936:Mathematics portal
3785:Mathematical sociology
3765:Mathematical economics
3760:Mathematical chemistry
3689:Analytic number theory
3570:Differential equations
3356:University of Maryland
3347:University of Waterloo
3269: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:
5376:Computational science
5272:Electronic publishing
5242:Computational biology
5232:Computational physics
5128:Unsupervised learning
5042:Distributed computing
4918:Information retrieval
4825:Mathematical analysis
4815:Mathematical software
4698:Theory of computation
4663:Software construction
4653:Requirements analysis
4531:Software organization
4459:Computer architecture
4429:Hardware acceleration
4394:Printed circuit board
4315:Philosophy of physics
3915:Mathematics education
3845:Theory of computation
3565:Hypercomplex analysis
3262:Abramowitz and Stegun
3148: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:
5371:Mathematical physics
5032:Concurrent computing
5004:Ubiquitous computing
4976:Application security
4971:Information security
4800:Discrete mathematics
4776:Randomized algorithm
4728:Computability theory
4706:Model of computation
4678:Software maintenance
4673:Software engineering
4635:Software development
4585:Programming language
4580:Programming paradigm
4497:Network architecture
4274:Mathematical physics
3895:Informal mathematics
3775:Mathematical physics
3770:Mathematical finance
3755:Mathematical biology
3694:Diophantine geometry
3281:28 July 2009 at the
3189:"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
5307:Document management
5297:Operations research
5222:Enterprise software
5138:Multi-task learning
5123:Supervised learning
4845:Information systems
4668:Software deployment
4625:Software repository
4479:Real-time computing
4249:Atmospheric physics
4088:Classical mechanics
4016:branches of physics
3910:Mathematics and art
3820:Operations research
3575:Functional analysis
3237:archived 2017-08-01
3104:Trefethen, Lloyd N.
3077:Info. Processing 71
3053:. 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
5366:Numerical analysis
5090:Search methodology
5037:Parallel computing
4994:Interaction design
4903:Computing platform
4830:Numerical analysis
4820:Information theory
4605:Software framework
4568:Software notations
4507:Network components
4404:Integrated circuit
4305:History of physics
3855:Numerical analysis
3464:Mathematical logic
3459:Information theory
3327:Mahidol University
3285:), Stuart Dalziel
3027:. Addison Wesley.
3021: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 Analysis
5353:
5352:
5282:Electronic voting
5212:Quantum Computing
5205:Applied computing
5191:Image compression
4961:Hardware security
4951:Security services
4908:Digital marketing
4688:Open-source model
4600:Modeling language
4512:Network scheduler
4333:
4332:
4320:Physics education
4269:Materials science
4236:Interdisciplinary
4194:Quantum mechanics
3981:
3980:
3580:Harmonic analysis
3334:, Henrik Schmidt
3314:Oxford University
3312:, Anthony O’Hare
3303:, John D. Fenton
3301:Numerical methods
3275:Numerical Methods
3243:Numerical Methods
3207:Numerical Recipes
3151:, volumes 1–...,
3126:978-0-691-11880-2
3086:978-0-7204-2063-0
3060:978-0-19-853418-1
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
5383:
5343:
5342:
5333:
5332:
5323:
5322:
5143:Cross-validation
5115:Machine learning
4999:Social computing
4966:Network security
4761:Algorithm design
4683:Programming team
4643:Control variable
4620:Software library
4558:Software quality
4553:Operating system
4502:Network protocol
4367:Computer science
4360:
4353:
4346:
4337:
4336:
4259:Chemical physics
4199:Particle physics
4125:Classical optics
4008:
4001:
3994:
3985:
3984:
3969:
3968:
3957:
3956:
3945:
3944:
3934:
3933:
3865:Computer algebra
3840:Computer science
3560:Complex analysis
3394:
3387:
3380:
3371:
3370:
3256:, ch 25. in the
3202:
3164:
3130:
3112:
3098:
3064:
3052:
3038:
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:
5391:
5390:
5386:
5385:
5384:
5382:
5381:
5380:
5356:
5355:
5354:
5349:
5340:
5311:
5292:Word processing
5200:
5186:Virtual reality
5147:
5109:
5080:Computer vision
5056:
5052:Multiprocessing
5018:
4980:
4946:Security hacker
4922:
4898:Digital library
4839:
4790:Mathematics of
4785:
4747:
4723:Automata theory
4718:Formal language
4692:
4658:Software design
4629:
4562:
4548:Virtual machine
4526:
4522:Network service
4483:
4474:Embedded system
4447:
4380:
4369:
4364:
4334:
4329:
4293:
4279:Medical physics
4230:
4189:Nuclear physics
4158:
4152:Non-equilibrium
4074:
4046:
4018:
4012:
3982:
3977:
3928:
3919:
3869:
3826:
3805:Systems science
3736:
3732:Homotopy theory
3698:
3665:
3617:
3589:
3536:
3483:
3454:Category theory
3440:
3405:
3398:
3345:, D. W. Harder
3283:Wayback Machine
3271:
3245:, ch 3. in the
3187:
3184:
3162:
3143:
3138:
3133:
3127:
3110:
3087:
3061:
3043:Wilkinson, J.H.
3035:
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:
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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:
5389:
5379:
5378:
5373:
5368:
5351:
5350:
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5337:
5327:
5316:
5313:
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5310:
5309:
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5299:
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5264:
5259:
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5234:
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5214:
5208:
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5196:Solid modeling
5193:
5188:
5183:
5178:
5173:
5168:
5163:
5157:
5155:
5149:
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5135:
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5119:
5117:
5111:
5110:
5108:
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5102:
5097:
5095:Control method
5092:
5087:
5082:
5077:
5072:
5066:
5064:
5058:
5057:
5055:
5054:
5049:
5047:Multithreading
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4941:Formal methods
4938:
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4920:
4915:
4913:World Wide Web
4910:
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4142:Thermodynamics
4139:
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3883:Mathematicians
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3875:Related topics
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3810:Control theory
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3585:Measure theory
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3167:
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3165:
3142:
3139:
3137:
3136:External links
3134:
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3100:
3085:
3065:
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3017:
3010:
3004:
2988:
2982:
2962:
2957:978-0486414546
2956:
2943:
2937:
2921:Golub, Gene H.
2916:
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2598:(3): 299–314.
2575:
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2056:. Birkhäuser.
2042:
2008:
2001:
1981:
1965:
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1911:
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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:
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851:
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842:
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802:
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783:
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761:
745:
742:
731:
728:
712:discretization
707:
706:Discretization
704:
610:
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597:
594:
591:
587:
586:
583:
580:
577:
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572:
569:
566:
563:
559:
558:
557:−13.875
555:
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535:
530:
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421:Direct method
411:
410:
350:simplex method
329:
326:
324:
321:
266:Euler's method
245:
242:
241:
240:
233:
226:
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204:
201:
188:
185:
150:Error Analysis
26:
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6:
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5209:
5207:
5203:
5197:
5194:
5192:
5189:
5187:
5184:
5182:
5181:Mixed reality
5179:
5177:
5174:
5172:
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5167:
5164:
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5015:
5014:Accessibility
5012:
5010:
5009:Visualization
5007:
5005:
5002:
5000:
4997:
4995:
4992:
4991:
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4480:
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4472:
4470:
4469:Dependability
4467:
4465:
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4457:
4456:
4454:
4450:
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4437:
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4313:
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4308:
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4296:
4290:
4287:
4285:
4284:Ocean physics
4282:
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4270:
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4219:Modern optics
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4069:
4068:Computational
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3832:Computational
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3808:
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3695:
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3690:
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3682:
3680:
3677:
3676:
3674:
3672:
3671:Number theory
3668:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
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3634:
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3606:
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3603:Combinatorics
3601:
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3598:
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3592:
3586:
3583:
3581:
3578:
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3555:Real analysis
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108:interpolation
105:
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97:Markov chains
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65:approximation
62:
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47:
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39:
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5277:Cyberwarfare
4936:Cryptography
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4244:Astrophysics
4058:Experimental
3970:
3958:
3946:
3927:
3860:Optimization
3854:
3722:Differential
3646:Differential
3613:Order theory
3608:Graph theory
3512:Group theory
3341:
3325:), R. Radok
3257:
3246:
3226:
3214:
3206:
3192:
3182:Online texts
3169:
3146:
3115:
3076:
3072:
3048:
3024:
3013:
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:
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2365:
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2303:
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2276:
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2178:
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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:
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753:real numbers
747:
739:
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699:
695:
691:
687:
683:
679:
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669:
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662:
657:
653:
649:
645:
641:
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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:
29:
5287:Video games
5267:Digital art
5024:Concurrency
4893:Data mining
4805:Probability
4538:Interpreter
4147:Statistical
4063:Theoretical
4040:Engineering
3972:WikiProject
3815:Game theory
3795:Probability
3532:Homological
3522:Multilinear
3502:Commutative
3479:Type theory
3446:Foundations
3402: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
5360:Categories
5345:Glossaries
5217:E-commerce
4810:Statistics
4753:Algorithms
4711:Stochastic
4543:Middleware
4399:Peripheral
4264:Geophysics
4254:Biophysics
4098:Analytical
4051:Approaches
3800:Statistics
3679:Arithmetic
3641:Arithmetic
3507:Elementary
3474: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
5166:Rendering
5161:Animation
4792:computing
4743:Semantics
4434:Processor
4214:Molecular
4115:Acoustics
4108:Continuum
4103:Celestial
4093:Newtonian
4080:Classical
4023:Divisions
3727:Geometric
3717:Algebraic
3656:Euclidean
3631:Algebraic
3527:Universal
3199:EMS Press
3069:Kahan, W.
3045:(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
5325:Category
5153:Graphics
4928:Security
4590:Compiler
4489:Networks
4386:Hardware
3948:Category
3704:Topology
3651:Discrete
3636:Analytic
3623:Geometry
3595:Discrete
3550:Calculus
3542:Analysis
3497:Abstract
3436:Glossary
3419:Timeline
3323:archived
3279:Archived
3220:archived
3155:, 1959–
3153:Springer
3141:Journals
3106:(2008).
3095:25116949
3071:(1972).
3023:(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
5335:Outline
4298:Related
4182:General
4177:Special
4035:Applied
3960:Commons
3742:Applied
3712:General
3489:Algebra
3414:History
3201:, 2001
3175:(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
4209:Atomic
4164:Modern
4014:Major
3661:Finite
3517:Linear
3424:Future
3400:Major
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3083:
3057:
3031:
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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
4738:Logic
4572:tools
3888:lists
3431:Lists
3404:areas
3111:(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
502:= 3,
498:= 0,
490:) = 3
392:GMRES
264:, or
134:in a
4570:and
4443:Form
4439:Size
4135:Wave
4030:Pure
3227:CSEP
3121:ISBN
3091:OCLC
3081:ISBN
3055:ISBN
3029: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
2282:ISBN
2255:ISBN
2228:ISBN
2201:ISBN
2141:ISBN
2085:ISBN
2058:ISBN
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
1125:for
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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
4130:Ray
2657:hdl
2649:doi
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1429:).
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