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Quantization error in numerical analysis
numerical analysis, computational physics, and simulation, discretization error is the error resulting from the fact that a function of a continuous variable
Discretization_error
Conversion of continuous functions into discrete counterparts
field.) The same is true of discretization error and quantization error. Mathematical methods relating to discretization include the Euler–Maruyama method
Discretization
Methods for numerical approximations
from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the
Numerical_analysis
Process of mapping a continuous set to a countable set
molecules). Beta encoder Color quantization Data binning Discretization Discretization error Least count Posterization Pulse-code modulation Quantile
Quantization (signal processing)
Quantization_(signal_processing)
cost. Accuracy depends on both discretization error and solution error. For discretization error, a given mesh is a discrete approximation of the space,
Types_of_mesh
is numerics including mathematically strict error (rounding error, truncation error, discretization error) evaluation, and it is one field of numerical
Validated_numerics
machine learning, discretization refers to the process of converting or partitioning continuous attributes, features or variables to discretized or nominal
Discretization of continuous features
Discretization_of_continuous_features
Partition into two separate parts
multicategorical variables as binary variables is called dichotomization. The discretization error inherent in dichotomization is temporarily ignored for modeling purposes
Dichotomy
Numerical integration algorithm
inherent in the method reduces the level of local errors introduced into the integration by the discretization by removing all odd-degree terms, here the terms
Verlet_integration
Class of numerical techniques
are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation error or discretization error, the difference
Finite_difference_method
Analog of the continuous Laplace operator
Patra, Michael; Karttunen, Mikko (2006). "Stencils with isotropic discretization error for differential operators". Numerical Methods for Partial Differential
Discrete_Laplace_operator
Numerical analysis method
being described as the optimally isotropic form of discretization, displaying reduced overall error, and Patra-Karttunen having been systematically derived
Nine-point_stencil
Reliable digital data delivery methods on unreliable channels
applications in computer science and telecommunications, error detection and correction (EDAC) or error control are techniques that enable reliable delivery
Error detection and correction
Error_detection_and_correction
Flaws caused by insufficient color depth
time stretching, which adds frames. Downsampling Quantization error Discretization error Color quantization Level-set method Chao, Cheng-Kang Ted (15 July
Posterization
Data whose unit can take on only two possible states
dichotomy). Like all discretization, it involves discretization error, but the goal is to learn something valuable despite the error: treating it as negligible
Binary_data
Probabilistic algorithms to simulate quantum many-body systems
Monte Carlo Stochastic series expansion, a method which avoids the discretization error associated with path integral Monte Carlo by approximating the Taylor
Quantum_Monte_Carlo
Method for numerical integration
relative error of about 0.0419 % {\displaystyle 0.0419\%} . The interval of integration can be shortened (thereby reducing discretization error) by noting
Simpson's_rule
Computer approximation for real numbers
representation of π in the first 7 digits. The difference is the discretization error and is limited by the machine epsilon. The arithmetical difference
Floating-point_arithmetic
Lossy audio coding technique
fundamentally inexact, and involves two errors: discretization error, from sampling at intervals, and quantization error, from rounding. The more bits used
Sub-band_coding
resonant frequencies of the sounding object to be modeled to avoid any discretization error at the dominant and audible frequencies. Banded waveguide synthesis
Banded_waveguide_synthesis
Scheme for controlling errors in data over noisy communication channels
theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable
Error_correction_code
Conversion of raster graphics into vector graphics
Discretization error Downsampling Feature detection (computer vision) Edge detection Image scanner Optical character recognition Quantization error Subpaving
Image_tracing
2006. B.F. Zalewski and R.L. Mullen, "Interval Bounds on the Local Discretization Error in Boundary Element Analysis for Domains with Singular Flux", SAE
Interval boundary element method
Interval_boundary_element_method
Numerical method for solving physical or engineering problems
Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc. Each discretization strategy has certain
Finite_element_method
Technique for the generative modeling of a discrete probability distribution
rate matrix of the backward diffusion process. The time-discretization error, due to using discrete time, not continuous time, when integrating through the
Discrete_diffusion_model
Lattice gauge theory action
operators through Symanzik improvement, significantly reducing discretization errors. The action was introduced by Kenneth Wilson in his seminal 1974
Wilson_action
Statistics concept
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element
Errors_and_residuals
Speed of convergence of a mathematical sequence
estimates of the error. In practical applications, when one discretization method gives a desired accuracy with a larger discretization scale parameter
Rate_of_convergence
Probabilistic numerical ODE solver
are modeled as probability distributions that also account for the discretization error introduced through the numerical approximation. This probabilistic
ODE_filter
element method — based on a discretization of the space of solutions gradient discretisation method — based on both the discretization of the solution and of
List of numerical analysis topics
List_of_numerical_analysis_topics
Fermion discretization with four doublers
staggered fermions (also known as Kogut–Susskind fermions) are a fermion discretization that reduces the number of fermion doublers from sixteen to four. They
Staggered_fermion
Error-correcting codes
information theory and coding theory, Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon
Reed–Solomon_error_correction
Strategies to make sure approximate calculations stay close to accurate
Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed
Floating-point error mitigation
Floating-point_error_mitigation
Control loop feedback mechanism
control signal based on error, without discrete modulation. In this model: Term P is proportional to the current value of the SP−PV error e ( t ) {\displaystyle
PID_controller
Foundational law of classical magnetism
numerical solution may not satisfy Gauss's law for magnetism due to the discretization errors of the numerical methods. However, in many cases, e.g., for magnetohydrodynamics
Gauss's_law_for_magnetism
Singular perturbation problem dealing with confinement of Brownian particles
limit theorem and using a large number of samples. There is also a discretization error due to the finite size approximation of the step size in approximating
Narrow_escape_problem
Wiener process with reflecting spatial boundaries
ISBN 978-1118014950. Asmussen, S.; Glynn, P.; Pitman, J. (1995). "Discretization Error in Simulation of One-Dimensional Reflecting Brownian Motion". The
Reflected_Brownian_motion
Computational error due to rounding numbers
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic
Round-off_error
Deviation from the apparently intended form of an utterance
A speech error, commonly referred to as a slip of the tongue (Latin: lapsus linguae, or occasionally self-demonstratingly, lipsus languae) or misspeaking
Speech_error
Optimization algorithm for artificial neural networks
cross-entropy (XC, log loss), while for regression it is usually squared error loss (SEL). L {\displaystyle L} : the number of layers W l = ( w j k l )
Backpropagation
Quantum error correction code
communication, a stabilizer code is a class of quantum codes for performing quantum error correction. The toric code, and surface codes more generally, are types
Stabilizer_code
Choice between two or more discrete alternatives
utilities. The scale of utility is often defined by the variance of the error term in discrete choice models. This variance may differ depending on the characteristics
Discrete_choice
Code used in quantum error correction
flip errors in any single qubit. It can also correct two bit flips as long as the errors occur in separate blocks. Due to discretization of errors it can
Shor_code
Method for estimating new data within known data points
within on a triangle or tetrahedron Brahmagupta's interpolation formula Discretization Fractal interpolation Imputation (statistics) Lagrange interpolation
Interpolation
Branch of numerical analysis
methods. Mortar methods are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains
Numerical methods for partial differential equations
Numerical_methods_for_partial_differential_equations
Regression models accounting for possible errors in independent variables
In statistics, an errors-in-variables model or a measurement error model is a regression model that accounts for measurement errors in the independent
Errors-in-variables_model
Figure of merit for analog-to-digital conversion
difference between the original continuous value and its discretization, and the mean square quantization error (given some probability distribution on the input
Mean square quantization error
Mean_square_quantization_error
two generates a compensating error voltage which tends to move the output voltage towards the design specification. Discrete Transistors Operational amplifiers
Error_amplifier_(electronics)
Processing of natural language by a computer
articles in the financial section of a newspaper. Grammatical error correction Grammatical error detection and correction involves a great band-width of problems
Natural_language_processing
Machine learning framework
networks is discretization invariance and discretization convergence. Unlike conventional neural networks, which are fixed on the discretization of training
Neural_operators
Method for numerical differential equations
substitute these discrete elements in lieu of the continuous elements in (2). More precisely, the GDM starts by defining a Gradient Discretization (GD), which
Gradient discretisation method
Gradient_discretisation_method
Asymptotic variances under heteroskedasticity
standard errors (or simply robust standard errors), Eicker–Huber–White standard errors (also Huber–White standard errors or White standard errors), to recognize
Heteroskedasticity-consistent standard errors
Heteroskedasticity-consistent_standard_errors
Approach to finding numerical solutions of ordinary differential equations
which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional
Euler_method
Q_{l-i+1}^{(d-1)}\right)f} The index to Q {\displaystyle Q} is the level of the discretization. If a 1-dimension integration on level i {\displaystyle i} is computed
Sparse_grid
Methods for solving differential equations
index h {\displaystyle h} should show the relation to an underlying discretization given by ( x k ) k {\displaystyle \left(x_{k}\right)_{k}} . Note here
Discontinuous_Galerkin_method
can be solved on a computer. Errors creep in during the discretization process. The nature and characteristics of the errors must be controlled in order
Numerical methods in fluid mechanics
Numerical_methods_in_fluid_mechanics
Mathematical function
elementary antiderivatives; the integral of the Gaussian function is the error function: ∫ exp ( − x 2 ) d x = π 2 erf x + C . {\displaystyle \int
Gaussian_function
Discrete Fourier transform algorithm
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform
Fast_Fourier_transform
sometimes called as an example of a compatible discretization technique, and bears similarities with discrete exterior calculus, although they are distinct
Finite element exterior calculus
Finite_element_exterior_calculus
Method of encoding digital data on multiple carrier frequencies
run slower by applying more robust modulation or error coding to those subcarriers. The term discrete multitone modulation (DMT) denotes OFDM-based communication
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
Numerical analysis procedure
must satisfy the discretized equation exactly, the error ϵ j n {\displaystyle \epsilon _{j}^{n}} must also satisfy the discretized equation. Here we
Von Neumann stability analysis
Von_Neumann_stability_analysis
Limit on data transfer rate
channel, it is possible (in theory) to communicate discrete data (digital information) nearly error-free up to a computable maximum rate through the channel
Noisy-channel_coding_theorem
Algorithm that estimates unknowns from a series of measurements over time
linear dynamic systems discretized in the time domain. They are modeled on a Markov chain built on linear operators perturbed by errors that may include Gaussian
Kalman_filter
parameter between 0 and 1. Substitution of θ = 0 gives the explicit discretization of the unsteady conductive heat transfer equation. T i f − T i f − 1
Numerical solution of the convection–diffusion equation
Numerical_solution_of_the_convection–diffusion_equation
Mathematical problem in cryptography
In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. It is based on the idea
Learning_with_errors
Probability distribution
bell curve", "normal (distribution)", "Gaussian", and "Error, law of error, theory of errors, etc.". Amari, Shun'ichi; Nagaoka, Hiroshi (2000). Methods
Normal_distribution
AI that learns decision rules from data
domain knowledge, data types(discrete or continuous) and in combinations. Repeated incremental pruning to produce error reduction (RIPPER) is a propositional
Rule-based_machine_learning
Elliptic partial differential equation
the problem of surface reconstruction, it is necessary to find a good discretization of the vector field V. The basic approach is to bound the data with
Poisson's_equation
This article provides an error analysis of time discretization applied to spatially discrete approximation of the stationary and nonstationary Navier-Stokes
Stokes approximation and artificial time
Stokes_approximation_and_artificial_time
Minimum amount of a physical entity involved in an interaction
can exist only in certain discrete values. Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom
Quantum
Errors arising in numerical integration
Truncation errors in numerical integration are of two kinds: local truncation errors – the error caused by one iteration, and global truncation errors – the
Truncation error (numerical integration)
Truncation_error_(numerical_integration)
Learning technique
programs to analyse the learner's input and diagnose errors. Davies (2002) writes: "Discrete error analysis and feedback were a common feature of traditional
Computer-assisted language learning
Computer-assisted_language_learning
Family of implicit and explicit iterative methods
iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These
Runge–Kutta_methods
Middle quantile of a data set or probability distribution
or outliers are untrustworthy, i.e., may be measurement or transcription errors. For example, consider the multiset 1, 2, 2, 2, 3, 14. The median is 2 in
Median
Concept relating to waves and signals
peaks). The smaller peaks and valleys may be due to measurement errors rather than discrete spectral lines. Spectrum (disambiguation) § Physics OpenStax
Spectrum_(physical_sciences)
Transform in numerical harmonic analysis
and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet
Discrete_wavelet_transform
Technique used in signal processing and data compression
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies
Discrete_cosine_transform
Numerical method
A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of
Discrete_element_method
Technique used in the field of human reliability assessment (HRA)
The Technique for human error-rate prediction (THERP) is a technique that is used in the field of Human Reliability Assessment (HRA) to evaluate the probability
Technique for human error-rate prediction
Technique_for_human_error-rate_prediction
Type of fallacious argument (logical fallacy)
In propositional logic, affirming the consequent (also known as converse error, fallacy of the converse, or confusion of necessity and sufficiency) is
Affirming_the_consequent
Reconstruction of binary images from a small number of their projections
P. Gritzmann, On Stability, Error Correction, and Noise Compensation in Discrete Tomography, SIAM Journal on Discrete Mathematics 20 (1), pp. 227-239
Discrete_tomography
Scientific study of digital information
source coding/data compression (e.g. for ZIP files), and channel coding/error detection and correction (e.g. for DSL). Its impact has been crucial to
Information_theory
Theorem related to ordinary least squares
estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be
Gauss–Markov_theorem
German mathematician (1930–2025)
principle and discretization methods", Numerische Mathematik, vol. 29, 1978, 425–433. doi:10.1007/BF01432879. Analysis of Discretization Methods in Ordinary
Hans_Jörg_Stetter
Concept in computing
information theory, the error exponent of a channel code or source code over the block length of the code is the rate at which the error probability decays
Error_exponent
that the LM scheme gives a smaller error than Euler-Maruyama. While there are many algorithms that can give reduced error compared to the Euler scheme (see
Leimkuhler–Matthews_method
Maximum frequency of non-aliased component upon sampling
the highest frequency which a sampled data system can reproduce without error. James J. Condon & Scott M. Ransom (2016). Essential Radio Astronomy. Princeton
Nyquist_frequency
Representing a given context-free language in terms of two simpler languages
In formal language theory, the Chomsky–Schützenberger representation theorem is a theorem derived by Noam Chomsky and Marcel-Paul Schützenberger in 1959
Chomsky–Schützenberger representation theorem
Chomsky–Schützenberger_representation_theorem
Signal representation
frequency domain. A discrete frequency domain is a frequency domain that is discrete rather than continuous. For example, the discrete Fourier transform
Frequency_domain
Statistical test comparing two probability distributions
{\displaystyle n=1000} , the corresponding maximum error is about 0.9 % {\displaystyle 0.9~\%} ; this error increases to 2.6 % {\displaystyle 2.6~\%} when
Kolmogorov–Smirnov_test
the difference depends on the system being simulated and the type of discretization that is used. Numerical diffusion Von Neumann stability analysis numerical
Numerical_dispersion
Data pre-processing technique
called data discrete binning or data bucketing, is a data pre-processing technique used to reduce the effects of minor observation errors. The original
Data_binning
French mathematician
Philip (2012). Discretization of Processes. doi:10.1007/978-3-642-24127-7. ISBN 9783642241260. J. JACOD, P. PROTTER: Asymptotic error distributions for
Jean_Jacod
Computer simulations to discover and understand chemical properties
the potential. The timestep must be chosen small enough to avoid discretization errors (i.e., smaller than the period related to fastest vibrational frequency
Molecular_dynamics
Statistical confidence interval for success counts
probit) corresponding to the target error rate α . {\displaystyle \ \alpha ~.} For a 95% confidence level, the error α = 1 − 0.95 = 0.05
Binomial proportion confidence interval
Binomial_proportion_confidence_interval
Method of solving differential equations
Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the finite element method may be recast as
Multigrid_method
Type of filter in signal processing
feedback. This means that any rounding errors are not compounded by summed iterations. The same relative error occurs in each calculation. This also makes
Finite_impulse_response
Effect in signal processing
it misses the maximum value by 1⁄2 bin, and the resultant measurement error is referred to as scalloping loss (inspired by the shape of the peak). For
Spectral_leakage
Quantum Information Science
information science, which has many applications in quantum error-correcting code, discrete AdS/CFT correspondence, AdS/CMT correspondence, and more. It
Absolutely maximally entangled state
Absolutely_maximally_entangled_state
Computational problem possibly useful for post-quantum cryptography
In post-quantum cryptography, ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms
Ring_learning_with_errors
DISCRETIZATION ERROR
DISCRETIZATION ERROR
Boy/Male
Tamil
Errorless
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.
Girl/Female
Indian
Goddess Aadisakti: She who Maintains the Rules of Justice without the Slightest Error
Boy/Male
Shakespearean
The Comedy of Errors' Father to the twin brothers Antipholus of Ephesus, and Antipholus of Syracuse.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Error-less
Boy/Male
Shakespearean
The Comedy of Errors' Twin brothers, both named Antipholus, sons to Aemelia and Aegion....
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
Girl/Female
Hindu, Indian
Without Error
Female
Arthurian
, error for Nineve (q.v.).
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' A schoolmaster.
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
Shakespearean
The Comedy of Errors' Duke of Ephesus.
Boy/Male
Shakespearean
The Comedy of Errors' A merchant of Syracuse.
Boy/Male
Shakespearean
The Comedy of Errors' Twin brothers, both named Dromio, attendants on the twin Antipholuses....
Girl/Female
Shakespearean
The Comedy of Errors' Adriana's servant.
Boy/Male
Hindu
Errorless
Boy/Male
Shakespearean
The Comedy of Errors' A merchant.
DISCRETIZATION ERROR
DISCRETIZATION ERROR
Boy/Male
Sikh
Boy/Male
British, English, Irish
Chief; Head; Man of Israel; Dusky Hound
Surname or Lastname
Scottish
Scottish : variant of Norris.English : variant of Northey.
Boy/Male
Hindi
Solitude.
Girl/Female
Indian
Love; Buetiful; And Sweet
Boy/Male
Biblical
Strong, my sign, a plowshare.
Boy/Male
Arabic, Indian, Muslim, Parsi
Prince; Emperor; King
Boy/Male
Hindu, Indian, Punjabi, Sikh
Head of God; Godly Person; The Highest God
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Innocent
Boy/Male
American, Australian, British, Chinese, Christian, English, French, German, Greek, Hebrew
Man of the Army; Army Man; Noble; Name of a Place During Biblical Period; Hardy Man; Variant of Herman
DISCRETIZATION ERROR
DISCRETIZATION ERROR
DISCRETIZATION ERROR
DISCRETIZATION ERROR
DISCRETIZATION ERROR
n.
To adhere to fixed principles; to maintain moral rectitude; to keep from falling into error or vice.
n.
The quality or state of being wrong; wrongfulness; error; fault.
v. i.
To fall into a crime or an error; to err.
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.
v. t.
To recite (a lesson) or pass (an examination) without an error.
n.
One who encourages and propagates error; one who holds to error.
a.
Committing no mistake; incapable or error or failure certain; sure; unfailing; as, the unerring wisdom of God.
n.
Any cause of stumbling, perplexity, or error.
n.
A defect; a fault; an error; a blemish; an imperfection; as, the vices of a political constitution; the vices of a horse.
n.
Fig.: That which preserves from corruption or error; that which purifies; a corrective; an antiseptic; also, an allowance or deduction; as, his statements must be taken with a grain of salt.
a.
Deviation or departure from truth or fact; state of falsity; error; as, to be in the wrong.
a.
Full of error; wrong.
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.
v. t.
That which fills up, completes, or makes an addition to, something already organized, arranged, or set apart; specifically, a part added to, or issued as a continuation of, a book or paper, to make good its deficiencies or correct its errors.
a.
Of or pertaining to structure; affecting structure; as, a structural error.
a.
A true statement; freedom from error of falsehood; adherence to truth or fact.
n.
Acknowledgement of a fault; confession of error.
n.
A false step; a stumble; a misstep; a loss of footing or balance. Fig.: An error; a failure; a mistake.
a.
Deceived or misled respecting one's self by one's own mistake or error.
n.
An instrument in writing, under seal, in an epistolary form, issued from the proper authority, commanding the performance or nonperformance of some act by the person to whom it is directed; as, a writ of entry, of error, of execution, of injunction, of mandamus, of return, of summons, and the like.