Numerical analysis is the study of
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s 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
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
(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 equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s as found in
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
(predicting the motions of planets, stars and galaxies),
numerical linear algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. ...
in data analysis, and
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock pr ...
s 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
Comprising some 45,000 items, the Yale Babylonian Collection is an independent branch of the Yale University Library housed on the Yale University campus in Sterling Memorial Library at New Haven, Connecticut, United States. In 2017, the collec ...
(
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
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
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 geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
in a
unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinate ...
.
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
Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in th ...
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 fund
A hedge fund is a pooled investment fund that trades in relatively liquid assets and is able to make extensive use of more complex trading, portfolio-construction, and risk management techniques in an attempt to improve performance, such as sho ...
s (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
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
.
* 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
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known poin ...
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
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit met ...
. The origins of modern numerical analysis are often linked to a 1947 paper by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
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
Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
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
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
, 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
A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or (historically) a simulation such as an analog computer or a slide rule. Most mechanical calculators wer ...
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 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
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and the ...
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
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
, because displacement is the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
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
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
, and the
simplex method
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are ...
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
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
**Asymptotic stability
**Linear stability
**Lyapunov stability
**Orbital stability
**Structural stabilit ...
).
In contrast to direct methods,
iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived fr ...
s are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that
converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
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
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and the ...
, 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 In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace wit ...
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
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
. 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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s exactly on a machine with finite memory (which is what all practical
digital computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These pro ...
s are).
Truncation and discretization error
Truncation error
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process.
Examples Infinite series
A summation series for e^x is given by an infinite series such as
e^x=1+ x+ \frac + \frac ...
s are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a
discretization error
In numerical analysis, computational physics, and simulation, discretization error is the error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a ...
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
, 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
, 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 = ''x
k''/2 + 1/''x
k''. 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
:
and
:Comparing the results of
::
:and
:
: by comparing the two results above, it is clear that
loss of significance
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_ ...
(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
and
, despite the subtraction being computed exactly) has a huge effect on the results, even though both functions are equivalent, as shown below
::
: 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
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Hor ...
, 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
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between know ...
is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.
Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The
least 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
is linear while
is not.
Much effort has been put in the development of methods for solving
systems of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
. Standard direct methods, i.e., methods that use some
matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of ...
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
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a pe ...
,
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
for
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
(or
hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
) and
positive-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
, and
QR decomposition
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 decompo ...
for non-square matrices. Iterative methods such as the
Jacobi method
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 ...
,
Gauss–Seidel method
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl ...
,
successive over-relaxation In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging ...
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
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depen ...
.
Root-finding algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...
s are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
and the derivative is known, then Newton's method is a popular choice.
Linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, lineariz ...
is another technique for solving nonlinear equations.
Solving eigenvalue or singular value problems
Several important problems can be phrased in terms of
eigenvalue decomposition
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matr ...
s or
singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...
s. 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
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
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
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
. Popular methods use one of the
Newton–Cotes formulas
In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called ''quadrature'') based on evaluating the integrand at ...
(like the midpoint rule or
Simpson's rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads
\int_a^b f(x) \, ...
) or
Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...
. 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
Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
or
quasi-Monte Carlo method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). This is in contrast to the regu ...
s (see
Monte Carlo integration
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand a ...
), 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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, a
finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
method, or (particularly in engineering) a
finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
. The theoretical justification of these methods often involves theorems from
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. 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
Free software or libre software is computer software distributed under terms that allow users to run the software for any purpose as well as to study, change, and distribute it and any adapted versions. Free software is a matter of liberty, no ...
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
The Royal Statistical Society (RSS) is an established statistical society. It has three main roles: a British learned society for statistics, a professional body for statisticians and a charity which promotes statistics for the public good.
...
published numerous algorithms in its Journal of the Royal Statistical Society, Series C (Applied Statistics), ''Applied Statistics'' (code for these "AS" functions i
here;
Association for Computing Machinery, ACM similarly, in its ''Transactions on Mathematical Software'' ("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, S-PLUS, and IDL (programming language), IDL as well as free and open source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (programming language), R (similar to S-PLUS), Julia (programming language), Julia, and Python (programming language), 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 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.
Microsoft_Excel#, Excel, for example, has hundreds of Microsoft Excel#Functions, available functions, including for matrices, which may be used in conjunction with its Microsoft Excel#Add-ins, built in "solver".
See also
*:Numerical analysts
*Analysis of algorithms
*Computational science
*Computational physics
*Gordon Bell Prize
*Interval arithmetic
*List of numerical analysis topics
*Local linearization method
*Numerical differentiation
*Numerical Recipes
*Probabilistic numerics
*Symbolic-numeric computation
*Validated numerics
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'
(SINUM) volumes 1–..., SIAM, 1964–
Online texts
*
William H. Press (free, downloadable previous editions)
(Internet Archive, archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
''CSEP'' (Computational Science Education Project) U.S. Department of Energy (Internet Archive, 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
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
)
Online course material
Numerical Methods(), Stuart Dalziel University of Cambridge
Lectures on Numerical Analysis Dennis Deturck and Herbert S. Wilf University of Pennsylvania
Numerical methods John D. Fenton University of Karlsruhe
Numerical Methods for Physicists Anthony O’Hare Oxford University
Lectures in Numerical Analysis(Internet Archive, archived), R. Radok Mahidol University
Introduction to Numerical Analysis for Engineering Henrik Schmidt Massachusetts Institute of Technology
''Numerical Analysis for Engineering'' D. W. Harder University of Waterloo
Introduction to Numerical Analysis Doron Levy University of Maryland
Numerical Analysis - Numerical Methods(archived), John H. Mathews California State University Fullerton
{{DEFAULTSORT:Numerical Analysis
Numerical analysis,
Mathematical physics
Computational science