Approximation
Encyclopedia
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to number
s, it is also frequently applied to such things as mathematical functions
, shape
s, and physical law
s.
Approximations may be used because incomplete information
prevents use of exact representations. Many problems in physics are either too complex to solve analytically, or impossible to solve using the available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.
For instance, physicists often approximate the shape of the Earth
as a sphere
even though more accurate representations are possible, because many physical behaviours—e.g. gravity—are much easier to calculate for a sphere than for other shapes.
It is difficult to exactly analyze the motion of several planets orbiting a star, for example, due to the complex interactions of the planets' gravitational effects on each other, so an approximate solution is effected by performing iteration
s. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained. The use of perturbations
to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions.
As another example, in order to accelerate the convergence rate of evolutionary algorithms, fitness approximation
—that leads to build model of the fitness function to choose smart search steps—is a good solution.
The type of approximation used depends on the available information
, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.
is carried out with a constant interaction between scientific laws (theory) and empirical measurement
s, which are constantly compared to one another.
The approximation also refers to using a simpler process. This model is used to make predictions easier. The most common versions of philosophy of science
accept that empirical measurement
s are always approximations—they do not perfectly represent what is being measured. The history of science
indicates that the scientific laws commonly felt to be true at any time in history are only approximations to some deeper set of laws. For example, attempting to resolve a model
using outdated physical laws alone incorporates an inherent source of error, which should be corrected by approximating the quantum effects not present in these laws.
Each time a newer set of laws is proposed, it is required that in the limiting
situations in which the older set of laws were tested against experiment
s, the newer laws are nearly identical to the older laws, to within the measurement
uncertainties of the older measurements. This is the correspondence principle
.
Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. It also is used when a number is not rational
, such as the number π
, which often is shortened to 3.14159, or √2 to 1.414.
Numerical approximations sometimes result from using a small number of significant digits
. Approximation theory
is a branch of mathematics, a quantitative part of functional analysis
. Diophantine approximation
deals with approximations of real number
s by rational number
s.
Related to approximation of functions is the asymptotic
value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum (k/2)+(k/4)+(k/8)+...(k/2^n) is asymptotically equal to k. Unfortunately no consistent notation is used throughout mathematics and some texts will use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around.
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s, it is also frequently applied to such things as mathematical functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, shape
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
s, and physical law
Physical law
A physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...
s.
Approximations may be used because incomplete information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
prevents use of exact representations. Many problems in physics are either too complex to solve analytically, or impossible to solve using the available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.
For instance, physicists often approximate the shape of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
as a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
even though more accurate representations are possible, because many physical behaviours—e.g. gravity—are much easier to calculate for a sphere than for other shapes.
It is difficult to exactly analyze the motion of several planets orbiting a star, for example, due to the complex interactions of the planets' gravitational effects on each other, so an approximate solution is effected by performing iteration
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
s. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained. The use of perturbations
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions.
As another example, in order to accelerate the convergence rate of evolutionary algorithms, fitness approximation
Fitness approximation
In function optimization, fitness approximation is a method for decreasing the number of fitness function evaluations to reach a target solution...
—that leads to build model of the fitness function to choose smart search steps—is a good solution.
The type of approximation used depends on the available information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.
Science
The scientific methodScientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
is carried out with a constant interaction between scientific laws (theory) and empirical measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
s, which are constantly compared to one another.
The approximation also refers to using a simpler process. This model is used to make predictions easier. The most common versions of philosophy of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
accept that empirical measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
s are always approximations—they do not perfectly represent what is being measured. The history of science
History of science
The history of science is the study of the historical development of human understandings of the natural world and the domains of the social sciences....
indicates that the scientific laws commonly felt to be true at any time in history are only approximations to some deeper set of laws. For example, attempting to resolve a model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
using outdated physical laws alone incorporates an inherent source of error, which should be corrected by approximating the quantum effects not present in these laws.
Each time a newer set of laws is proposed, it is required that in the limiting
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
situations in which the older set of laws were tested against experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
s, the newer laws are nearly identical to the older laws, to within the measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
uncertainties of the older measurements. This is the correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
.
Mathematics
| ≐ | general approximation |
| ≈ | asymptotic analysis |
Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. It also is used when a number is not rational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
, such as the number π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, which often is shortened to 3.14159, or √2 to 1.414.
Numerical approximations sometimes result from using a small number of significant digits
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
. Approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
is a branch of mathematics, a quantitative part of functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. Diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
deals with approximations of real number
Real number
In 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 by rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s.
Related to approximation of functions is the asymptotic
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum (k/2)+(k/4)+(k/8)+...(k/2^n) is asymptotically equal to k. Unfortunately no consistent notation is used throughout mathematics and some texts will use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around.
See also
- Approximation errorApproximation errorThe approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because#the measurement of the data is not precise due to the instruments...
- Congruence relationCongruence relationIn abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...
- EstimationEstimationEstimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...
- Fitness approximationFitness approximationIn function optimization, fitness approximation is a method for decreasing the number of fitness function evaluations to reach a target solution...
- Fermi estimate
- Linear approximationLinear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...
- Newton's methodNewton'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...
- Numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
- Orders of approximationOrders of approximationIn science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...
- Runge–Kutta methodsRunge–Kutta methodsIn numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta.See the article...
- Successive Approximation ADCSuccessive Approximation ADCA successive approximation ADC is a type of analog-to-digital converter that converts a continuous analog waveform into a discrete digital representation via a binary search through all possible quantization levels before finally converging upon a digital output for each conversion.-Block...
- Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
- Least squaresLeast 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...