Numerical analysis is the study of
algorithmIn mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s that use numerical
approximationAn approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
(as opposed to general
symbolic manipulationsSymbolic computation or algebraic computation, relates to the use of machines, such as computers, to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols...
) for the problems of
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
(as distinguished from
discrete mathematicsDiscrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
).
One of the earliest mathematical writings is the Babylonian tablet BC 7289, which gives a sexagesimal numerical approximation of
, the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction.
Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of
, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations.
Ordinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s appear in the
movement of heavenly bodies (planets, stars and galaxies)Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
; optimization occurs in portfolio management;
numerical linear algebraNumerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, Telecommunication, computational...
is important for data analysis;
stochastic differential equationA stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
s and
Markov chainA Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
s are essential in simulating living cells for medicine and biology.
Before the advent of modern computers numerical methods often depended on hand
interpolationIn the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
in large printed tables. Since the mid 20th century, computers calculate the required functions instead. The interpolation algorithms nevertheless may be used as part of the software for solving differential equations.
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 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 the 1950s that numerical weather predictions produced realistic...
feasible.
- Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s.
- 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, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s numerically.
- Hedge fund
A hedge fund is a private pool of capital actively managed by an investment adviser. Hedge funds are only open for investment to a limited number of accredited or qualified investors who meet criteria set by regulators. These investors can be institutions, such as pension funds, university...
s (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants.
- Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...
.
- Insurance companies use numerical programs for actuarial
An actuary is a business professional who deals with the financial impact of risk and uncertainty. Actuaries provide expert assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms ....
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 interpolationLinear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation, which can also be used as a verb .-Linear interpolation between two known points:...
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 methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
,
Lagrange interpolation polynomialIn numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...
,
Gaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
, or Euler's method.
To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by
Abramowitz and StegunAbramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards...
, 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 calculatorA mechanical calculator is a device used to perform the basic operations of arithmetic. Mechanical calculators are comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator....
was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.
Direct and iterative methods
|-
|
Direct vs iterative methods
Consider the problem of solving
- 3x3 + 4 = 28
for the unknown quantity
x.
|+ Direct method
|-
| >
x3 + 4 = 28.
>-
| Subtract 4 |
x3 = 24.
>-
| Divide by 3 |
x3 = 8.
>-
| Take cube roots |
For the iterative method, apply the
bisection methodThe bisection method in mathematics is a root-finding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow...
to
f(
x) = 3
x3 − 24. The initial values are
a = 0,
b = 3,
f(
a) = −24,
f(
b) = 57.
|+ Iterative method
|-
!
a !!
b !! mid !!
f(mid)
|-
| 0 >
3 |
1.5 |
|
>-
| 1.5
3 |
2.25 |
|
>-
| 1.5
2.25 |
1.875 |
|
>-
| 1.875
2.25 |
2.0625 |
We conclude from this table 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, we have measured the speed of the car at three instants and recorded them in the following table.
| Time |
0:20 |
1:00 |
1:40 |
| km/h |
140 |
150 |
180 |
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 (2/3h × 140 km/h) = 93.3 km. This would allow us to estimate the total distance traveled as 93.3 km + 100 km + 120 km = 313.3 km, which is an example of
numerical integration (see below) using a
Riemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
, because displacement is the
integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of velocity.
Ill posed problem: Take the function
f(
x) = 1/(
x − 1). 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-posed problem: By contrast, the function
is continuous and so evaluating it is well-posed, at least for
x being not close to zero.>
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 arithmeticA computer number format is the internal representation of numeric values in digital computer and calculator hardware and software.-Bits:The concept of a bit can be understood as a value of either 1 or 0, on or off, yes or no, true or false, or encoded by a switch or toggle of some kind...
. Examples include
Gaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
, the
QRIn numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR transformation was developed in the late 1950s by John G.F. Francis and by Vera N. Kublanovskaya , working independently...
factorization method for solving
systems
of linear equations, and the simplex method of
linear programmingLinear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
. In practice,
finite precisionIn computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
is used and the result is an approximation of the true solution (assuming stability).
In contrast to direct methods,
iterative methodIn computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
s are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that
convergeThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
to the exact solution only in the limit. A convergence test 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 methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
, the
bisection methodThe bisection method in mathematics is a root-finding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow...
, and Jacobi iteration. 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 methodIn mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
. 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
discretizationIn mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...
. For example, the solution of a
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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.
The 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 errorA round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...
s arise because it is impossible to represent all
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s exactly on a machine with finite memory (which is what all practical digital computers are).
Truncation and discretization error
TruncationIn mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...
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 errorIn numerical analysis, computational physics, and simulation, discretization error is 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 lattice...
because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of
, after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.
Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.
What does it mean when we say that the truncation error is created when we approximate a mathematical procedure? We know that to integrate a function exactly requires one to find the sum of infinite trapezoids. But numerically one can find the sum of only finite trapezoids, and hence the approximation of the mathematical procedure. Similarly, to differentiate a function, the differential element approaches to zero but numerically we can only choose a finite value of the differential element.
Numerical stability and well-posed problems
Numerical stabilityIn the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
is an important 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-conditionedIn the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...
, 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 and/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
x1 to
, for instance
x1=1.4, and then computing improved guesses
x2,
x3, etc.. One such method is the famous Babylonian method, which is given by
xk+1 =
xk/2 + 1/
xk. Another iteration, which we will call Method X, is given by
xk + 1 = (
xk2−2)
2 +
xk. We have calculated a few iterations of each scheme in table form below, with initial guesses
x1 = 1.4 and
x1 = 1.42.
|->
Babylonian |
Babylonian |
Method X |
|
>-
|
x1 = 1.4
|
x1 = 1.42
|
x1 = 1.4
|
x1 = 1.42
|-
|
x2 = 1.4142857...
|
x2 = 1.41422535...
|
x2 = 1.4016
|
x2 = 1.42026896
|-
|
x3 = 1.414213564...
|
x3 = 1.41421356242...
|
x3 = 1.4028614...
|
x3 = 1.42056...
|-
|
|
| ...
| ...
|-
|
|
|
x1000000 = 1.41421...
|
x28 = 7280.2284...
|}
Observe that the Babylonian method converges fast regardless of the initial guess, whereas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 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 on, if we use a machine that keeps on the first four floating-point digits, a good example on loss of significance is given by these two equivalent functions
- If we compare the results of
-
- and
- by looking to the two above results, we realize that loss of significance which is also called Subtractive Cancelation has a huge effect on the results, even though both functions are equivalent; to show that they are equivalent simply we need to start by f(x) and end with g(x), and so
-
- The true value for the result is 11.174755..., which is exactly g(500) = 11.1748 after rounding the result to 4 decimal digits.
- Now imagine that lots of terms like these functions are used in the program; the error will increase as one proceeds in the program, unless one uses the suitable formula of the two functions each time one evaluates either f(x), or g(x); the choice is dependent on the parity of x.
- The example is taken from Mathew; Numerical methods using matlab, 3rd ed.
Areas of study
The field of numerical analysis is divided into different disciplines according to the problem that is to be solved.
Computing values of functions
|
Interpolation: We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm.
Extrapolation: If the gross domestic productGross domestic product refers to the market value of all final goods and services produced within a country in a given period. GDP per capita is often considered an indicator of a country's standard of living....
of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year.
Regression: In linear regression, given n points, we compute a line that passes as close as possible to those n points.
Optimization: Say you sell lemonade at a lemonade standLemonade Stand is a basic economics game created in 1973 by Bob Jamison of the Minnesota Educational Computing Consortium. Charlie Kellner ported the game to the Apple II platform in February 1979...
, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of $0.01, you will sell one glass of lemonade less per day. If you could charge $1.485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging $1.49 per glass will yield the maximum income of $220.52 per day.
Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.>
One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner schemeIn numerical analysis, the Horner scheme , named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. Horner's method describes a manual process by which one may approximate the roots of a polynomial equation...
, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errorA round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...
s arising from the use of floating pointIn computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
arithmetic.
Interpolation, extrapolation, and regression
InterpolationIn the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
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?
ExtrapolationIn mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...
is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.
RegressionIn statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
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), we want to determine the unknown function. The least squaresThe method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
-method is one popular 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. Standard direct methods, i.e., methods that use some matrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
are Gaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
, LU decompositionIn linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...
, Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
for symmetric (or hermitian) and positive-definite matrixIn linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
, and QR decompositionIn linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
for non-square matrices. Iterative methodIn computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
s such as the Jacobi methodIn numerical linear algebra, the Jacobi method is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process...
, Gauss–Seidel method, successive over-relaxationIn numerical linear algebra, the method of successive over-relaxation 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 iterative process.It was devised simultaneously by David...
and conjugate gradient methodIn mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
are usually preferred for large systems.
Root-finding algorithmA root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f = 0, for a given function f. Such an x is called a root of the function f....
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 differentiableIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
and the derivative is known, then Newton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
is a popular choice. LinearizationIn mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
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 decompositionIn linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
s. For instance, the spectral image compressionThe objective of image compression is to reduce irrelevance and redundancy of the image data in order to be able to store or transmit data in an efficient form.- Lossy and lossless compression :...
algorithm is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.
Optimization
Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraintIn mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
s.
The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programmingLinear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
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 multipliersIn mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
can be used to reduce optimization problems with constraints to unconstrained optimization problems.
Evaluating integrals
Numerical integration, in some instances also known as numerical quadratureQuadrature — historical mathematical term which means calculating of the area. Quadrature problems have served as one of the main sources of mathematical analysis.- History :...
, asks for the value of a definite integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's ruleIn numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:...
) or Gaussian quadratureIn 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....
. 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 CarloMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
or quasi-Monte Carlo methodIn numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral that is based on low-discrepancy sequences...
s (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse gridSparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally found by the Russian mathematician Smolyak and are based on a sparse tensor product construction...
s.
Differential equations
Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, both ordinary differential equations and partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s.
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 methodThe finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
, a finite differenceA finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
method, or (particularly in engineering) a finite volume methodThe finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]....
. The theoretical justification of these methods often involves theorems from functional analysisFunctional 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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. 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 NetlibNetlib 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 a large number of separate programs and libraries...
repository contains various collections of software routines for numerical problems, mostly in FortranFortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
and CC is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
. Commercial products implementing many different numerical algorithms include the IMSLIMSL is a commercial collection of software libraries of numerical analysis functionality that are implemented in the computer programming languages of C, Java, C#.NET, and Fortran...
and NAGThe Numerical Algorithms Group is a software company which provides methods for the solution of mathematical and statistical problems, and offers services to users of HPC systems. Its products and services are employed by tens of thousands of users from Global 500 companies, universities,...
libraries; a free alternative is the GNU Scientific LibraryIn computing, the GNU Scientific Library is a software library written in the C programming language for numerical calculations in applied mathematics and science...
.
There are several popular numerical computing applications such as MATLABMATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
, S-PLUSS-PLUS is a commercial implementation of the S programming language sold by TIBCO Software Inc..It features object-oriented programming capabilities and advanced analytical algorithms.-Historical timeline:...
, LabVIEWLabVIEW is a system design platform and development environment for a visual programming language from National Instruments. LabVIEW provides engineers and scientists with the tools needed to create and deploy measurement and control systems.The graphical language is named "G"...
, and IDL as well as free and open source alternatives such as FreeMatFreeMat 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, C++, and Fortran code, further parallel distributed...
, ScilabScilab is an open source, cross-platform numerical computational package and a high-level, numerically oriented programming language. Itcan be used for signal processing, statistical analysis, image enhancement, fluid dynamics simulations, numerical optimization, and modeling and simulation of...
, GNU OctaveGNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
(similar to Matlab), IT++ (a C++ library), RR is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
(similar to S-PLUS) and certain variants of PythonPython is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
. 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 systemA computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
s such as MathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results.
Also, any spreadsheetA spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...
software can be used to solve simple problems relating to numerical analysis.
See also
- Scientific computing
- List of numerical analysis topics
- Gram-Schmidt process
- Numerical differentiation
In numerical analysis, numerical differentiation describes algorithms for estimating the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.-Finite difference formulae:...
- Symbolic-numeric computation
In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems.-References:*, Dongming Wang, Lihong Zhi, Springer, 2007, ISBN 3764379839...
- Analysis of algorithms
To analyze an algorithm is to determine the amount of resources necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length...
- Numerical Recipes
Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul Teukolsky, William Vetterling and Brian Flannery. In various editions, the books have been in print since 1986...
External links
Journals
Software and Code
Online Texts
Online Course Material
- Numerical Methods, Stuart Dalziel University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
- Lectures on Numerical Analysis, Dennis Deturck and Herbert S. Wilf University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
- Numerical methods, John D. Fenton University of Karlsruhe
- Numerical Methods for Science, Technology, Engineering and Mathematics, Autar Kaw University of South Florida
The University of South Florida, also known as USF, is a member institution of the State University System of Florida, one of the state's three flagship universities for public research, and is located in Tampa, Florida, USA...
- Numerical Analysis Project, John H. Mathews California State University, Fullerton
California State University, Fullerton is a public university located in Fullerton, California. It is the largest institution in the CSU System by enrollment, it offers long-distance education and adult-degree programs...
- Numerical Methods - Online Course, Aaron Naiman Jerusalem College of Technology
The Jerusalem College of Technology , , is an Orthodox Jewish college. JCT's main campus are situated in the Givat Mordechai neighbourhood of Jerusalem. Other branches are located in the Givat Shaul neighbourhood of Jerusalem and Ramat Gan...
- Numerical Methods for Physicists, Anthony O’Hare Oxford University
- Lectures in Numerical Analysis, R. Radok Mahidol University
Mahidol University is a public research university in Bangkok, Thailand. Established back in 1888 as School of Medical Practitioners, Siriraj Hospital and reorganized in 1943 as University of Medical Sciences . The university originally focused on Health Sciences but also expanded to other...
- Introduction to Numerical Analysis for Engineering, Henrik Schmidt Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...