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Mathematical concept
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation
Approximation_error
Sigmoid shape special function
conditions are given by the Heaviside step function. The error function and its approximations can be used to estimate results that hold with high probability
Error_function
Something roughly the same as something else
An approximation is anything that is intentionally similar but not exactly equal to something else. The word approximation is derived from Latin approximatus
Approximation
Incorrect or inaccurate action
of techniques to represent (store) and compute approximations to mathematical numerical values. Errors arise from a trade-off between efficiency (space
Error
Approximation of a function by a polynomial
versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named
Taylor's_theorem
Small angle approximation in geometric optics
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system
Paraxial_approximation
Mathematical approximation of a function
are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced
Taylor_series
Angular measurement, thousandth of a radian
thousandth of the radius when using the simplified formula. The approximation error by using the simplified linear formula will increase as the angle
Milliradian
Statistical method
assumption is not quite correct, and introduces some error. To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction
Yates's correction for continuity
Yates's_correction_for_continuity
Computational error due to rounding numbers
operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many
Round-off_error
Theory of getting acceptably close inexact mathematical calculations
characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application. A closely related topic is the approximation of functions
Approximation_theory
Numerical integration process
quasi-Monte Carlo method are beneficial in these situations. The approximation error of the quasi-Monte Carlo method is bounded by a term proportional
Quasi-Monte_Carlo_method
Comparisons in quantitative sciences
tolerance. Another application is in the computation of approximation errors when the relative error of a measurement is required.[citation needed] Minimum
Relative_change
Oscillatory error in Fourier series
undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier
Gibbs_phenomenon
Approximation for factorials
mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate
Stirling's_approximation
Upper bound on rounding error in floating-point arithmetic
Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point number systems. This value characterizes
Machine_epsilon
Algorithms for calculating square roots
{\displaystyle a_{i}} s gives a suitable approximation of the square root, with X n {\displaystyle X_{n}} being the approximation error. For example, in the decimal
Square_root_algorithms
Simplification of the basic trigonometric functions
smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation cos θ ≈ 1 − 1 2 θ 2 {\displaystyle
Small-angle_approximation
Root-finding algorithm
came within an acceptable error range of the actual result. Common software methods in the early 1990s drew approximations from a lookup table. The key
Fast_inverse_square_root
Expressions for approximation accuracy
accuracy of the approximation improves as the order increases, but the order does not directly indicate the percent error of the approximation. See Taylor's
Order_of_approximation
Approximation of powers of some binomials
f(x)\approx f(0)+f'(0)(x-0)=1+\alpha x.} By Taylor's theorem, the error in this approximation is equal to α ( α − 1 ) x 2 2 ⋅ ( 1 + ζ ) α − 2 {\textstyle {\frac
Binomial_approximation
Mathematical method that minimizes maximum error
A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function that
Minimax approximation algorithm
Minimax_approximation_algorithm
Algorithm for approximating nonlinear relationships
approximates functions with discontinuities maintains explicit bounds on approximation error controls local derivatives of the approximating function delivers
In_situ_adaptive_tabulation
Technique in numerical linear algebra
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization
Low-rank_approximation
Methods of calculating definite integrals
from the approximation. An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function
Numerical_integration
Method for estimating new data within known data points
measuring the error. In the simplest case this leads to least squares approximation. Approximation theory studies how to find the best approximation to a given
Interpolation
Ability of numerical algorithms to remain accurate under small changes of inputs
fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called
Numerical_stability
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Form of encryption that allows computation on ciphertexts
A., Polyakov Y. Approximate Homomorphic Encryption with Reduced Approximation Error, In CT-RSA 2022 (Springer) Li, Baily; Micciancio, Daniele (2020)
Homomorphic_encryption
Number in base-10 numeral system
the number of digits after the decimal separator, one can make the approximation errors as small as one wants, when one has a method for computing the new
Decimal
Class of algorithms that find approximate solutions to optimization problems
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems
Approximation_algorithm
Formula to estimate the sine function
In mathematics, Bhāskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the
Bhāskara I's sine approximation formula
Bhāskara_I's_sine_approximation_formula
Concepts from statistical hypothesis testing
Type I error, or a false positive, is the incorrect rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false
Type_I_and_type_II_errors
considering homomorphic operations, the evaluation errors are also included in the approximation error. Basic homomorphic operations, addition and multiplication
HEAAN
Time-frequency analysis
signal with small approximation error. Some matching pursuit algorithms are proposed in reference papers to minimize approximation error when given the amount
Basis expansion time-frequency analysis
Basis_expansion_time-frequency_analysis
Speed of convergence of a mathematical sequence
{\displaystyle \mu } will involve the asymptotic limit of the ratio of an approximation error term above to an asymptotic order q {\displaystyle q} power of a
Rate_of_convergence
Use of numerical analysis to estimate derivatives of functions
known as a first-order divided difference). To obtain an error estimate for this approximation, one can use Taylor expansion of f ( x ) {\displaystyle
Numerical_differentiation
Loss of precision in numerical analysis
{\text{cm}}} . These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 0.2% of the true lengths
Catastrophic_cancellation
Probability theorem
approximately a Poisson distribution and the above inequality bounds the approximation error in terms of the total variation distance. By setting pi = λn/n, we
Le_Cam's_theorem
Statistical method
compare the lasso's error with the approximation error of a sparse linear predictor plus a term accounting for estimation error. Koltchinskii, Lounici
Lasso_(statistics)
Strategies to make sure approximate calculations stay close to accurate
true value; mid-rad: an approximation and an error bound (called midpoint and radius of the interval); triplex: an approximation, a lower bound and an upper
Floating-point error mitigation
Floating-point_error_mitigation
Branch of mathematics
formulating and analyzing mathematical models, constructing approximations, and estimating their errors. Typical analytical questions concern how functions,
Mathematical_analysis
Noise that reduces quantization error
alternative to Error-diffusion dithering Electrostatic Halftoning is modeled after the principles of Electrostatics, which has a low approximation error and creates
Dither
Multidimensional data algorithm
atoms one at a time in order to maximally (greedily) reduce the approximation error. This is achieved by finding the atom that has the highest inner
Matching_pursuit
Solution method for linear differential equations
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially
WKB_approximation
Varying methods used to calculate pi
Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning
Approximations_of_pi
Special mathematical functions defined on the surface of a sphere
functions admit faster approximation by spherical polynomials, while conversely, sufficiently rapid decay of the approximation error implies smoothness.
Spherical_harmonics
Statistical property
The standard error (SE) of a statistic is the standard deviation of its sampling distribution. It is the square root of the variance of an estimator of
Standard_error
Discrete Fourier transform algorithm
the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate
Fast_Fourier_transform
Measure of similarity between curves
This approximation unconditionally yields larger values than the corresponding (continuous) Fréchet distance. However, the approximation error is bounded
Fréchet_distance
Effect of variables' uncertainties on the uncertainty of a function based on them
example, the bias on the error calculated for log(1+x) increases as x increases, since the expansion to x is a good approximation only when x is near zero
Propagation_of_uncertainty
Theory of stochastic processes
features of f. The resulting error is necessarily smaller than the error of a linear approximation which selects the M approximation vectors independently of
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
substance's properties found in a localized lab. Accuracy and precision Error Approximation error Wilbram; Staley; Matta; Waterman (2005). Chemistry. New Jersey:
Accepted and experimental value
Accepted_and_experimental_value
Measure of linear correlation
standard error = SE = 1 n − 3 , {\displaystyle ={\text{SE}}={\frac {1}{\sqrt {n-3}}},} where n is the sample size. The approximation error is lowest
Pearson correlation coefficient
Pearson_correlation_coefficient
Linear algebra aspects of graph theory
(September 2016). "Spectral Graph Wavelets and Filter Banks With Low Approximation Error". IEEE Transactions on Signal and Information Processing over Networks
Spectral_graph_theory
'Best' approximation of a function by a rational function of given order
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique
Padé_approximant
Mathematical function defined piecewise by polynomials
determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful
Spline_(mathematics)
Approximation method in statistics
the idea that this is a good approximation in many cases. The Gauss–Markov theorem. In a linear model in which the errors have expectation zero conditional
Least_squares
Observation that in many real-life datasets, the leading digit is likely to be small
17–34. Dümbgen, L; Leuenberger, C (2008). "Explicit bounds for the approximation error in Benford's Law". Electronic Communications in Probability. 13:
Benford's_law
Sufficiency theorem for reconstructing signals from samples
approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as interpolation error.
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Numerical integration method
embedded rule). The difference between these two approximations is used to estimate the calculational error of the integration. Like the Gaussian quadrature
Gauss–Kronrod quadrature formula
Gauss–Kronrod_quadrature_formula
Topics referred to by the same term
Theft Auto V Error in numerical analysis: Machine epsilon, an error bound in computer arithmetic absolute value of an approximation error The epsilon operator
Epsilon_(disambiguation)
Algorithm to approximate functions
is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the
Remez_algorithm
Branch of elementary mathematics
subtleties; explicitly keeping track of an estimate or upper bound of the approximation error is a more sophisticated approach. In the example, the person's height
Arithmetic
Technique in mathematical modeling
problems, often the requirements of a reduced order model are: A small approximation error compared to the full order model. Conservation of the properties
Model_order_reduction
Approximation method in statistics
so the numerical derivative is not subject to approximation error by being too large, or round-off error by being too small. Some information is given
Non-linear_least_squares
Algorithm in numerical analysis
due to the approximation that a M + 1 {\displaystyle a_{M+1}} and later terms are zero. Finally, it is confirmed that the approximation error of the procedure
Miller's_recurrence_algorithm
Type of triangle
{\displaystyle b=h\tan \theta \ } yields the desired result. The error of this approximation is less than 10% for angles 31° or less. Applications of the
Skinny_triangle
ABS methods Error analysis (mathematics) Approximation Approximation error Catastrophic cancellation Condition number Discretization error Floating point
List of numerical analysis topics
List_of_numerical_analysis_topics
Analytical expression in statistics
The approximation is justified by the Bernstein–von Mises theorem, which states that, under regularity conditions, the error of the approximation tends
Laplace's_approximation
named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function u {\displaystyle \textstyle u} by a polynomial of
Bramble–Hilbert_lemma
Mathematical theorem in the study of analysis
For differentiable functions, Jackson's inequality bounds the error of approximations by polynomials of a given degree: if f {\displaystyle f} has a
Stone–Weierstrass_theorem
Error made by the human user of a complex system
technical attitude towards user error: Don't think of the user as making errors; think of the actions as approximations of what is desired. Terms like
User_error
Statistical regression model
more interpretable than a general regression surface at the cost of approximation errors. Problems with AM, like many other machine-learning methods, include
Additive_model
Theorem in probability theory
the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured
Berry–Esseen_theorem
Rational-number approximation of a real number
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus
Diophantine_approximation
Topics referred to by the same term
A typographic approximation of 2×, or multiplication by 2 "two power"/"two times" magnification A typographical or transcription error of 2x, or Power
2X
Numerical method for solving physical or engineering problems
a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and
Finite_element_method
Differential equation that is linear with respect to the unknown function
coefficients), evaluation to a high precision with certified bound of the approximation error, limits, localization of singularities, asymptotic behavior at infinity
Linear_differential_equation
Statistics function
{\displaystyle b=5.334} with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by a = 0.339 {\displaystyle a=0.339} and b
Q-function
Statistical technique
shape of X and Y. Using the Eckart–Young theorem, the approximation minimising the norm of the error is such that matrices U {\displaystyle U} and V {\displaystyle
Total_least_squares
Technique to find image offset
interpolation method choice may be larger than any numerical or approximation error in the particular method. Subpixel methods are also particularly
Phase_correlation
Approximation of the definite integral of a function
rule and its Kronrod extension is often used as an estimate of the approximation error. In some applications, it is desirable to have quadrature rules that
Gaussian_quadrature
Numerical integration method
left and right Riemann sums and is sometimes defined this way. The approximation becomes more accurate as the resolution of the partition increases (that
Trapezoidal_rule
Statistics and machine learning technique
S2CID 14357246. Clarke, B., Bayes model averaging and stacking when model approximation error cannot be ignored, Journal of Machine Learning Research, pp 683-712
Ensemble_learning
Measure of the error of an estimator
squared error can serve as a good approximation to a loss function occurring naturally in an application. Like variance, mean squared error has the disadvantage
Mean_squared_error
Method of problem-solving
behavior. Lloyd Morgan, however, had watched and recorded the series of approximations by which the dog had gradually learned the response, and could demonstrate
Trial_and_error
On the flip side, A-CEEI has several disadvantages: There is an approximation error in the items that are allocated - some items might be in excess demand
Approximate Competitive Equilibrium from Equal Incomes
Approximate_Competitive_Equilibrium_from_Equal_Incomes
Mathematical function with convex lower level sets
polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated); however, such theoretically "efficient" methods use "divergent-series"
Quasiconvex_function
Study of kind and quantity of error
evaluation of forward errors is desired in validated numerics. Backward error analysis involves the analysis of the approximation function z ′ = f ′ (
Error_analysis_(mathematics)
Term in numerical analysis
convergence and, in general, all numerical errors correctly. The size of the error of a first-order accurate approximation is directly proportional to h {\displaystyle
Order_of_accuracy
Machine learning technique useful for dimensionality reduction
of quadratic bending and stretching energy with the least squares approximation error. The oriented and scalable map (OS-Map) generalises the neighborhood
Self-organizing_map
Degradation of AI models trained on synthetic data
functional approximation errors sampling errors learning errors Importantly, it happens in even the simplest of models, where not all of the error sources
Model_collapse
Digit necessary to represent a quantity
trillion-digit approximation has 102 trillion significant digits. In practical applications, far fewer digits are used. The everyday approximation 3.14 has
Significant_figures
Probability distribution
(2005). Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small relative error on the whole
Normal_distribution
Type of statistical analysis
{\displaystyle n} goes to infinity, that is, the approximation error converges to zero. Usually, the approximation is measured in terms of L 2 {\displaystyle
Nonparametric_statistics
Approach to finding numerical solutions of ordinary differential equations
error recorded in the last column of the table is the difference between the exact solution at t = 4 {\displaystyle t=4} and the Euler approximation.
Euler_method
Method for numerical integration
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of
Simpson's_rule
Type of sideband modulation
truncation of the spectrum and nonlinear phase distortion from the approximation errors of the practical implementation of the required Hilbert transform
Controlled-envelope single-sideband modulation
Controlled-envelope_single-sideband_modulation
Type of analog-to-digital converter
A successive-approximation ADC (or SAR ADC) is a type of analog-to-digital converter (ADC) that digitizes each sample from a continuous analog waveform
Successive-approximation_ADC
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