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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
Incorrect or inaccurate action
approximation error. In applying corrections to the trajectory or course being steered, cybernetics can be seen as the most general approach to error
Error
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Formula to estimate the sine function
the approximation formula are visually indistinguishable and are nearly identical. One of the accompanying figures gives the graph of the error function
Bhāskara I's sine approximation formula
Bhāskara_I's_sine_approximation_formula
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
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
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
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
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
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
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
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
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)
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
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
Formulas for numerical integration
error in Abramowitz and Stegun, an early reference book. The exponent of the step size h in the error term gives the rate at which the approximation error
Newton–Cotes_formulas
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
Branch of mathematics
also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers
Mathematical_analysis
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
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
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
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
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
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
Procedural method for the design and implementation of an experiment
results. Approximation error is common to all measurements. These errors can be absolute errors from limitations of the equipment or propagation errors from
Scientific_protocol
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
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
considering homomorphic operations, the evaluation errors are also included in the approximation error. Basic homomorphic operations, addition and multiplication
HEAAN
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
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
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
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
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
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
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
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
Statistical property
The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution
Standard_error
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
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
Method of data analysis
Miranda, Y.-A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural
Principal_component_analysis
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
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
Basic integral in elementary calculus
sums that are suitably close to the limit can be used as numerical approximations. Bernhard Riemann introduced the integral in work presented to the faculty
Riemann_integral
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
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
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
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
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
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
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
sources of error that limit the accuracy of DDAs: Rounding/truncation errors due to the limited precision of the registers, Approximation errors due to the
Digital_differential_analyzer
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
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
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
Process of using data analysis for predicting population data from sample data
1950s, advanced statistics uses approximation theory and functional analysis to quantify the error of approximation. In this approach, the metric geometry
Statistical_inference
Geometric shape
{\displaystyle \mathbf {k} } of control points which result in the least approximation error for a given number of cubic segments. Considering only the 90-degree
Composite_Bézier_curve
perimeter of an ellipse. Throughout history, a large number of closed-form approximations and expressions in terms of integrals or series have been given for
Perimeter_of_an_ellipse
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
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
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
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
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
Substituting rare characters with more common characters
digraph, or a character string. An approximation is different from a typographical error in that an approximation is intentional and aims to preserve
Typographic_approximation
Measure of complexity of real-valued functions
Learning Research 3 463–482 Giorgio Gnecco, Marcello Sanguineti (2008) Approximation Error Bounds via Rademacher's Complexity. Applied Mathematical Sciences
Rademacher_complexity
Function for integral Fourier-like transform
as a Gaussian. The choice of windowing function will affect the approximation error relative to the true Fourier transform. A given resolution cell's
Wavelet
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
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
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)
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
{\displaystyle \delta } . This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step
Runge–Kutta_method_(SDE)
formulation, is a numerical method to compute finite difference approximations. Such approximations tend to be more accurate for their stencil size (i.e. their
Compact_finite_difference
'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
important statement in approximation theory. It provides a bound for the projection error, controlling the error of approximation by a linear subspace based
Lebesgue's_lemma
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
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
APPROXIMATION ERROR
APPROXIMATION ERROR
Girl/Female
Shakespearean
The Comedy of Errors' Adriana's servant.
Boy/Male
Shakespearean
The Comedy of Errors' Father to the twin brothers Antipholus of Ephesus, and Antipholus of Syracuse.
Girl/Female
Hindu, Indian
Without Error
Female
English
Anglicized form of Hebrew Abiyshag, ABISHAG means "my father is a wanderer" or "father of error." In the bible, this is the name of a young girl who cared for David in his old age.Â
Female
Hebrew
(×ֲבִיש×Ö·×’) Hebrew name ABIYSHAG means "my father is a wanderer" or "father of error." In the bible, this is the name of a young girl who cared for David in his old age. Also spelled Avishag.
Boy/Male
Tamil
Errorless
Female
Arthurian
, error for Nineve (q.v.).
Female
Hebrew
(×ֲבִיש×Ö·×’) Variant spelling of Hebrew Abiyshag, AVISHAG means "my father is a wanderer" or "father of error." In the bible, this is the name of a young girl who cared for David in his old age.Â
Boy/Male
Hindu
Errorless
Boy/Male
Shakespearean
The Comedy of Errors' A schoolmaster.
Surname or Lastname
English
English : of uncertain derivation. The first recorded instance seems to be William Cleike (Yorkshire 1176), but this may well be an error for Clerke. In subsequent records the name is concentrated in Devon; it seems to have been originally a habitational name connected with a piece of land in the parish of Ermington near Plymouth, first recorded in 1278 as Clekeland(e), and still known as Clickland; the names John de Clakelond and Robert Cleaklond occur in this parish in 1332 and 1337 respectively. The place name may be from Old English cleaca ‘stepping stone’, ‘boundary stone’ (of Celtic origin) + land ‘territory’. Compare Clack.Americanized spelling of German Glück (see Gluck).
Boy/Male
Shakespearean
The Comedy of Errors' Twin brothers, both named Dromio, attendants on the twin Antipholuses....
Girl/Female
Indian
Goddess Aadisakti: She who Maintains the Rules of Justice without the Slightest Error
Boy/Male
Shakespearean
The Comedy of Errors' A merchant of Syracuse.
Boy/Male
Shakespearean
The Comedy of Errors' Duke of Ephesus.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Error-less
Boy/Male
Shakespearean
The Comedy of Errors' A merchant.
Boy/Male
Shakespearean
The Comedy of Errors' Twin brothers, both named Antipholus, sons to Aemelia and Aegion....
APPROXIMATION ERROR
APPROXIMATION ERROR
Surname or Lastname
English
English : probably a variant of Reading 1, from the place name + the Middle English suffix -tune ‘settlement’. However, the surname is quite common in Lancashire and Yorkshire, and so perhaps a northern place named as the ‘settlement (Old English tūn) associated with Rēad(a)’ is to be sought.
Girl/Female
Indian, Modern, Punjabi, Sikh
Given by God
Girl/Female
Hindu, Indian
Well-wisher
Girl/Female
Buddhist, Gujarati, Indian, Kannada
The Holder of Intellect
Boy/Male
English
Jewel name; bridge protector.
Boy/Male
Indian
The majestic, The revered, The sublime
Girl/Female
Tamil
Of good caste
Girl/Female
Hindu, Indian
Goddess Durga
Girl/Female
Indian
Wise
Boy/Male
Hindu, Indian
Glorious
APPROXIMATION ERROR
APPROXIMATION ERROR
APPROXIMATION ERROR
APPROXIMATION ERROR
APPROXIMATION ERROR
n. pl.
A group of ganoid fishes, including the living genera Ceratodus and Lepidosiren, which present the closest approximation to the Amphibia. The air bladder acts as a lung, and the nostrils open inside the mouth. See Ceratodus, and Illustration in Appendix.
n.
A defect; a fault; an error; a blemish; an imperfection; as, the vices of a political constitution; the vices of a horse.
a.
Committing no mistake; incapable or error or failure certain; sure; unfailing; as, the unerring wisdom of God.
n.
The difference between the observed value of a quantity and that which is taken or computed to be the true value; -- sometimes called residual error.
n.
A wandering or deviation from the right course or standard; irregularity; mistake; inaccuracy; something made wrong or left wrong; as, an error in writing or in printing; a clerical error.
a.
Full of error; wrong.
p. pr. & vb. n.
of Approximate
n.
One who encourages and propagates error; one who holds to error.
a.
Resembling, or approximating to, a hemisphere in form.
a.
Approaching; approximate.
n.
The transient approximation of the edges of a natural opening; imperforation.
n.
The act of approximating; a drawing, advancing or being near; approach; also, the result of approximating.
n.
A value that is nearly but not exactly correct.
a.
Pertaining to the first in time of the three subdivisions into which the Tertiary formation is divided by geologists, and alluding to the approximation in its life to that of the present era; as, Eocene deposits.
n.
A continual approach or coming nearer to a result; as, to solve an equation by approximation.
n.
An approach to a correct estimate, calculation, or conception, or to a given quantity, quality, etc.
v. t.
To mention or suggest as an estimate, hypothesis, or approximation; hence, to suppose; -- in the imperative, followed sometimes by the subjunctive; as, he had, say fifty thousand dollars; the fox had run, say ten miles.
adv.
With approximation; so as to approximate; nearly.
n.
The act of violently forcing air out through the nasal passages while the cavity of the mouth is shut off from the pharynx by the approximation of the soft palate and the base of the tongue.
n.
One who, or that which, approximates.