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Numerical analysis is the study of algorithms that use numerical
approximation An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...
(as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
). It is the study of numerical methods that attempt at finding 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, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
s for simulating living cells in medicine and biology. Before modern computers,
numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
s often relied on hand
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
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. The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (
YBC 7289 YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest ...
), gives a sexagesimal numerical approximation of the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
, the length of the diagonal in a unit square. Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.


General introduction

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following: * Advanced numerical methods are essential in making numerical weather prediction feasible. * Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. * Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s numerically. * Hedge funds (private investment funds) use tools from all fields of numerical analysis to attempt to calculate the value of
stock In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...
s and
derivatives The derivative of a function is the rate of change of the function's output relative to its input value. Derivative may also refer to: In mathematics and economics * Brzozowski derivative in the theory of formal languages * Formal derivative, an ...
more precisely than other market participants. * Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research. * Insurance companies use numerical programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis.


History

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
,
Lagrange interpolation polynomial In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' a ...
,
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
, or Euler's method. The origins of modern numerical analysis are often linked to a 1947 paper by John von Neumann and
Herman Goldstine Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, the ...
, but others consider modern numerical analysis to go back to work by E. T. Whittaker in 1912. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done. The
Leslie Fox Prize for Numerical Analysis The Leslie Fox Prize for Numerical Analysis of the Institute of Mathematics and its Applications (IMA) is a biennial prize established in 1985 by the IMA in honour of mathematician Leslie Fox (1918-1992). The prize honours "young numerical analyst ...
was initiated in 1985 by the Institute of Mathematics and its Applications.


Direct and iterative methods

Consider the problem of solving :3''x''3 + 4 = 28 for the unknown quantity ''x''. For the iterative method, apply the bisection method to ''f''(''x'') = 3''x''3 − 24. The initial values are ''a'' = 0, ''b'' = 3, ''f''(''a'') = −24, ''f''(''b'') = 57. From this table it can be concluded that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.


Discretization and numerical integration

In a two-hour race, the speed of the car is measured at three instants and recorded in the following table. A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately . This would allow us to estimate the total distance traveled as + + = , which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity. Ill-conditioned problem: Take the function . Note that ''f''(1.1) = 10 and ''f''(1.001) = 1000: a change in ''x'' of less than 0.1 turns into a change in ''f''(''x'') of nearly 1000. Evaluating ''f''(''x'') near ''x'' = 1 is an ill-conditioned problem. Well-conditioned problem: By contrast, evaluating the same function near ''x'' = 10 is a well-conditioned problem. For instance, ''f''(10) = 1/9 ≈ 0.111 and ''f''(11) = 0.1: a modest change in ''x'' leads to a modest change in ''f''(''x''). Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in
infinite precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...
. Examples include
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
, the
QR factorization In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and an upper triangular matrix ''R''. QR decomp ...
method for solving systems of linear equations, and the simplex method of
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability). In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and
Jacobi iteration In numerical linear algebra, the Jacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The ...
. In computational matrix algebra, iterative methods are generally needed for large problems. Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the
conjugate gradient method In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterativ ...
. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.


Discretization

Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called '
discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
'. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a
continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
.


Generation and propagation of errors

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


Round-off

Round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).


Truncation and discretization error

Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of 3x^3+4=28, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01. Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type is even more inexact. A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.


Numerical stability and well-posed problems

Numerical stability In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorit ...
is a notion in numerical analysis. An algorithm is called 'numerically stable' if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is '
well-conditioned In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input ...
', 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. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation ''x''0 to \sqrt, for instance ''x''0 = 1.4, and then computing improved guesses ''x''1, ''x''2, etc. One such method is the famous
Babylonian method Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted \sqrt, \sqrt /math>, or S^) of a real number. Arithmetically, it means given S, a procedure for fin ...
, which is given by ''x''''k''+1 = ''xk''/2 + 1/''xk''. Another method, called 'method X', is given by ''x''''k''+1 = (''x''''k''2 − 2)2 + ''x''''k''. A few iterations of each scheme are calculated in table form below, with initial guesses ''x''0 = 1.4 and ''x''0 = 1.42. Observe that the Babylonian method converges quickly regardless of the initial guess, whereas Method X converges extremely slowly with initial guess ''x''0 = 1.4 and diverges for initial guess ''x''0 = 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable. :Numerical stability is affected by the number of the significant digits the machine keeps. If a machine is used that keeps only the four most significant decimal digits, a good example on loss of significance can be given by the two equivalent functions : f(x)=x\left(\sqrt-\sqrt\right) and g(x)=\frac. :Comparing the results of :: f(500)=500 \left(\sqrt-\sqrt \right)=500 \left(22.38-22.36 \right)=500(0.02)=10 :and : \beging(500)&=\frac\\ &=\frac\\ &=\frac=11.17 \end : by comparing the two results above, it is clear that loss of significance (caused here by
catastrophic cancellation In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_ ...
from subtracting approximations to the nearby numbers \sqrt and \sqrt, despite the subtraction being computed exactly) has a huge effect on the results, even though both functions are equivalent, as shown below :: \begin f(x)&=x \left(\sqrt-\sqrt \right)\\ &=x \left(\sqrt-\sqrt \right)\frac\\ &=x\frac\\ &=x\frac \\ &=x\frac \\ &=\frac \\ &=g(x) \end : The desired value, computed using infinite precision, is 11.174755... * The example is a modification of one taken from Mathew; Numerical methods using MATLAB, 3rd ed.


Areas of study

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


Computing values of functions

One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control
round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s arising from the use of
floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
.


Interpolation, extrapolation, and regression

Interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? Extrapolation is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.
Regression Regression or regressions may refer to: Science * Marine regression, coastal advance due to falling sea level, the opposite of marine transgression * Regression (medicine), a characteristic of diseases to express lighter symptoms or less extent ( ...
is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The
least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
-method is one way to achieve this.


Solving equations and systems of equations

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation 2x+5=3 is linear while 2x^2+5=3 is not. Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and
conjugate gradient method In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterativ ...
are usually preferred for large systems. General iterative methods can be developed using a matrix splitting. Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.


Solving eigenvalue or singular value problems

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics is called
principal component analysis Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
.


Optimization

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints. The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance,
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method. The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.


Evaluating integrals

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of
sparse grid Sparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician Sergey A. Smolyak, a student of Lazar Lyusternik, and are based on a sparse tensor ...
s.


Differential equations

Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations. Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.


Software

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The
Netlib Netlib is a repository of software for scientific computing maintained by AT&T, Bell Laboratories, the University of Tennessee and Oak Ridge National Laboratory. Netlib comprises many separate programs and libraries. Most of the code is written in ...
repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the
GNU Scientific Library The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C; wrappers are available for other programming languages. The GSL is part of the GNU Project and is d ...
. Over the years the Royal Statistical Society published numerous algorithms in its ''Applied Statistics'' (code for these "AS" functions i
here
;
ACM ACM or A.C.M. may refer to: Aviation * AGM-129 ACM, 1990–2012 USAF cruise missile * Air chief marshal * Air combat manoeuvring or dogfighting * Air cycle machine * Arica Airport (Colombia) (IATA: ACM), in Arica, Amazonas, Colombia Computing * ...
similarly, in its ''
Transactions on Mathematical Software ''ACM Transactions on Mathematical Software'' (''TOMS'') is a quarterly scientific journal that aims to disseminate the latest findings of note in the field of numeric, symbolic, algebraic, and geometric computing applications. It is one of the old ...
'' ("TOMS" code i
here
. The Naval Surface Warfare Center several times published it
''Library of Mathematics Subroutines''
(cod
here
. There are several popular numerical computing applications such as MATLAB,
TK Solver TK Solver (originally TK!Solver) is a mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc. History Invented by Milos Konopasek in the late 19 ...
, S-PLUS, and
IDL IDL may refer to: Computing * Interface description language, any computer language used to describe a software component's interface ** IDL specification language, the original IDL created by Lamb, Wulf and Nestor at Queen's University, Canada ...
as well as free and open source alternatives such as
FreeMat FreeMat is a free open-source numerical computing environment and programming language, similar to MATLAB and GNU Octave. In addition to supporting many MATLAB functions and some IDL functionality, it features a codeless interface to external C ...
,
Scilab Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simulat ...
, GNU Octave (similar to Matlab), and
IT++ IT++ is a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics. It is being developed by researchers in these areas and is widely used by researchers, both in the commu ...
(a C++ library). There are also programming languages such as R (similar to S-PLUS), Julia, and Python with libraries such as NumPy, SciPy and SymPy. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude. Many computer algebra systems such as
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
also benefit from the availability of arbitrary-precision arithmetic which can provide more accurate results. Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.
Excel ExCeL London (an abbreviation for Exhibition Centre London) is an exhibition centre, international convention centre and former hospital in the Custom House area of Newham, East London. It is situated on a site on the northern quay of the ...
, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".


See also

* :Numerical analysts *
Analysis of algorithms In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that re ...
*
Computational science Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...
* Computational physics * Gordon Bell Prize * Interval arithmetic * List of numerical analysis topics *
Local linearization method In numerical analysis, the local linearization (LL) method is a general strategy for designing Numerical integration, numerical integrators for differential equations based on a local (piecewise) linearization of the given equation on consecutive ti ...
* Numerical differentiation *
Numerical Recipes ''Numerical Recipes'' is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since 1 ...
* Probabilistic numerics *
Symbolic-numeric computation In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic Symbolic may refer to: * Symbol, something that represents an idea, a process, or a physical entity Mathematics, logic, and computing ...
* Validated numerics


Notes


References


Citations


Sources

* * * * * * (examples of the importance of accurate arithmetic). *


External links


Journals

*'' Numerische Mathematik'', volumes 1–...
Springer
1959–
volumes 1–66, 1959–1994
(searchable; pages are images). *''
Journal on Numerical Analysis A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organization *Diary, a record of what happened over the course of a day or other period *Daybook, also known as a general journal, a ...
'
(SINUM)
volumes 1–..., SIAM, 1964–


Online texts

*

William H. Press (free, downloadable previous editions)

( archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
''CSEP'' (Computational Science Education Project)
U.S. Department of Energy The United States Department of Energy (DOE) is an executive department of the U.S. federal government that oversees U.S. national energy policy and manages the research and development of nuclear power and nuclear weapons in the United States. ...
( archived 2017-08-01)
Numerical Methods
ch 3. in the '' Digital Library of Mathematical Functions''
Numerical Interpolation, Differentiation and Integration
ch 25. in the ''Handbook of Mathematical Functions'' ( Abramowitz and Stegun)


Online course material


Numerical Methods
(), Stuart Dalziel University of Cambridge
Lectures on Numerical Analysis
Dennis Deturck and Herbert S. Wilf University of Pennsylvania
Numerical methods
John D. Fenton
University of Karlsruhe The Karlsruhe Institute of Technology (KIT; german: Karlsruher Institut für Technologie) is a public research university in Karlsruhe, Germany. The institute is a national research center of the Helmholtz Association. KIT was created in 2009 w ...

Numerical Methods for Physicists
Anthony O’Hare Oxford University
Lectures in Numerical Analysis
( archived), R. Radok
Mahidol University Mahidol University (Mahidol), an autonomous research institution in Thailand, had its origin in the establishment of Siriraj Hospital in 1888. Mahidol had an acceptance rate for Medicine of 0.4% as of the 2016 academic year. Becoming the Univers ...

Introduction to Numerical Analysis for Engineering
Henrik Schmidt Massachusetts Institute of Technology
''Numerical Analysis for Engineering''
D. W. Harder University of Waterloo
Introduction to Numerical Analysis
Doron Levy University of Maryland
Numerical Analysis - Numerical Methods
(archived), John H. Mathews California State University Fullerton {{DEFAULTSORT:Numerical Analysis Mathematical physics Computational science