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Method for bounding the errors of numerical computations
Interval arithmetic (also known as interval mathematics, interval analysis or interval computation) is a mathematical technique used to mitigate rounding
Interval_arithmetic
All numbers between two given numbers
an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. For example, interval arithmetic consists
Interval_(mathematics)
Form of entropy encoding used in data compression
achieve. Arithmetic coding approaches this limit closely, especially for long messages. When all symbols are equally likely, each sub-interval has the
Arithmetic_coding
Branch of elementary mathematics
mathematical objects other than numbers, such as interval arithmetic and matrix arithmetic. Arithmetic operations form the basis of many branches of mathematics
Arithmetic
Real numbers with an added point at infinity
half-open intervals are defined by removing the respective endpoints. This redefinition is useful in interval arithmetic when dividing by an interval containing
Projectively extended real line
Projectively_extended_real_line
Method of construction of the real numbers
of intervals approximating r {\displaystyle r} . This allows the basic arithmetic operations on the real numbers to be defined in terms of interval arithmetic
Dedekind_cut
Strategies to make sure approximate calculations stay close to accurate
being derivative of Gustafson's work on unums and interval arithmetic. "Floating decimal point arithmetic control means for calculator: United States Patent
Floating-point error mitigation
Floating-point_error_mitigation
computation. Affine arithmetic is meant to be an improvement on interval arithmetic (IA), and is similar to generalized interval arithmetic, first-order Taylor
Affine_arithmetic
alpha theory, while a typical example of a priori certification is interval arithmetic. A certificate for a root is a computational proof of the correctness
Numerical_certification
Implementation of arithmetic operations
field arithmetic Floating-point arithmetic Arbitrary-precision arithmetic Interval arithmetic Symmetric level-index arithmetic Matrix arithmetic In the
Computer_arithmetic
Mathematical analysis
extensions of Heyting arithmetic by types including N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} , constructive second-order arithmetic, or strong enough
Constructive_analysis
IEEE standard for floating-point arithmetic
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the
IEEE_754
Real numbers with a multi-valued logical classification
approaches: (1) interval arithmetic approach; and (2) the extension principle approach. A fuzzy number is equal to a fuzzy interval. The degree of fuzziness
Fuzzy_number
German mathematician
in numerical analysis, including the computer implementation of interval arithmetic. After graduation from high school in Freising, Kulisch studied mathematics
Ulrich_Kulisch
Computer approximation for real numbers
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of
Floating-point_arithmetic
Branch of mathematical logic
higher-order arithmetic: on one hand, when restricted to countable covers/the language of second-order arithmetic, the compactness of the unit interval is provable
Reverse_mathematics
INTLAB (INTerval LABoratory) is an interval arithmetic library using MATLAB and GNU Octave, available in Windows and Linux, macOS. It was developed by
INTLAB
Approach to static program analysis
yielding so-called interval arithmetics. Let us now consider the following very simple program: y = x; z = x - y; With reasonable arithmetic types, the result
Abstract_interpretation
Algorithm for finding zeros of functions
implies that N(Y) is well defined and is an interval (see interval arithmetic for further details on interval operations). This naturally leads to the following
Newton's_method
Variant of floating-point numbers in computers
proposed using interval arithmetic with a pair of unums, what he called a ubound, providing the guarantee that the resulting interval contains the exact
Unum_(number_format)
Free and open-source calculator software
solving of equations involving unknowns, uncertainty propagation using interval arithmetic, plotting using Gnuplot, unit and currency conversion and dimensional
Qalculate!
N-th root of the product of n numbers
real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of n {\displaystyle
Geometric_mean
numerical analysis. For computation, interval arithmetic is most often used, where all results are represented by intervals. Validated numerics were used by
Validated_numerics
Type of average of a collection of numbers
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum of a collection
Arithmetic_mean
French computer scientist
scientist known for her research on computer arithmetic, including floating-point arithmetic and interval arithmetic. She is a researcher for the French Institute
Nathalie_Revol
American mathematician
mathematician and author who has published in global optimization theory and interval arithmetic. Hansen was born on July 16, 1927 in Rochester, Washington. After
Eldon_Hansen
Constraint logic programming language
on relational interval arithmetic developed at Bell-Northern Research in the 1980s and 1990s. Embedding relational interval arithmetic in a logic programming
BNR_Prolog
Computer programming condition
The term arithmetic underflow (also floating-point underflow, or just underflow) is a condition in a computer program where the result of a calculation
Arithmetic_underflow
Date and time representation system widely used in computing
of seconds elapsed since 1970-01-01T00:00:10 TAI. This makes time interval arithmetic much easier. Time values from these systems do not suffer the ambiguity
Unix_time
Digit necessary to represent a quantity
Guard digit Guesstimate IEEE 754 (IEEE floating-point standard) Interval arithmetic Kahan summation algorithm Precision (computer science) Round-off
Significant_figures
Symbol combining both + and - signs
form m = c ± d, where c is f(a) and d is the range b updated using interval arithmetic. The symbols ± and ∓ are used in chess annotation to denote a moderate
Plus–minus_sign
Range to estimate an unknown parameter
According to frequentist inference, a confidence interval (CI) is a range of values which is likely to contain (in repeated sampling) the true value of
Confidence_interval
Distance between unordered pitch classes
example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on
Interval_class
Upper bound on rounding error in floating-point arithmetic
rounding in floating point number systems. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of
Machine_epsilon
Chaotic model of atmospheric convection
To prove this result, Tucker used rigorous numerics methods like interval arithmetic and normal forms. First, Tucker defined a cross section Σ ⊂ { x 3
Lorenz_system
Distinction between nominal, ordinal, interval and ratio variables
According to Stevens, the mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical
Level_of_measurement
Number that is not a ratio of integers
fundamental theorem of arithmetic (unique prime factorization). A stronger result is the following: Every rational number in the interval ( ( 1 / e ) 1 / e
Irrational_number
Replacing a number with a simpler value
same limiting value (0, +∞, or −∞). Directed rounding is used in interval arithmetic and is often required in financial calculations. If x is positive
Rounding
Estimate of an interval in which future observations will fall
inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain
Prediction_interval
Branch of mathematics
best one found so far by the algorithm. Interval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by
Global_optimization
Computer algebra system
implemented are general functions such as f(x), arbitrary-precision and interval arithmetic, as well as matrices. Mathomatic is capable of solving, differentiating
Mathomatic
American mathematician
American mathematician, known for his pioneering work in the field of interval arithmetic. Moore received an AB degree in physics from the University of California
Ramon_E._Moore
Numeric quantity representing the center of a collection of numbers
known as the quasi-arithmetic mean. For an injective function f : I → R {\displaystyle f\colon I\rightarrow \mathbb {R} } on an interval I ⊂ R {\displaystyle
Mean
Methods of calculating definite integrals
interval arithmetic to produce computer proofs and verified calculations. Several methods exist for approximate integration over unbounded intervals.
Numerical_integration
Mathematical proof at least partially generated by computer
and propagating round-off and truncation errors using for example interval arithmetic. More precisely, one reduces the computation to a sequence of elementary
Computer-assisted_proof
Linear algebra library for C++ programs
arbitrary integer formats (e.g. unsigned short), types for interval arithmetic (e.g. boost::interval) from the Boost C++ Libraries, quaternions (e.g. boost::quaternion)
Matrix_Template_Library
Measure of variation in statistics
measure of the amount of variation of the values of a variable about its (arithmetic) average. A low standard deviation indicates that the values of a set
Standard_deviation
Multi-modular arithmetic
is, in an interval of length M, exactly one integer having any given set of modular values. Using a residue numeral system for arithmetic operations
Residue_number_system
Concept in Bayesian statistics
In Bayesian statistics, a credible interval is an interval used to characterize a probability distribution. It is defined such that an unobserved parameter
Credible_interval
Mathematical construct
The principle is to evaluate f(x) using interval arithmetic (this is the forward step). The resulting interval is intersected with [y]. A backward evaluation
Interval_contractor
Arbitrary-precision arithmetic Interval arithmetic — represent every number by two floating-point numbers guaranteed to have the unknown number between them Interval contractor
List of numerical analysis topics
List_of_numerical_analysis_topics
Fraction with denominator a power of two
dyadic rationals, they are also used for exact real computing using interval arithmetic, and are central to some theoretical models of computable numbers
Dyadic_rational
Type of musical scale and characteristic behaviors
therefore there are only seven tonoi. Pythagoras also construed the intervals arithmetically (if somewhat more rigorously, initially allowing for 1:1 = Unison
Mode_(music)
Methods for numerical approximations
theory Computational science Computational physics Gordon Bell Prize Interval arithmetic List of numerical analysis topics Local linearization method Numerical
Numerical_analysis
floating-Point computing and numerical methods for Microsoft Excel. INTLAB – interval arithmetic library for MATLAB. List of computer algebra systems List of information
List_of_numerical_libraries
System of logic in computer science
systems. Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily Interval arithmetic—is much
Type-2_fuzzy_sets_and_systems
Type of statistical probability
A tolerance interval (TI) is a statistical interval within which, with some confidence level, a specified sampled proportion of a population falls. "More
Tolerance_interval
C library for arbitrary-precision floating-point arithmetic
numbers in a whole program or expression; this is not its goal. Interval arithmetic packages like Arb, MPFI, or Real RAM implementations like iRRAM,
GNU_MPFR
Measured values that are relatively normal for a particular medical test
the sample mean (also called mean or arithmetic mean). To account for these estimations, the 95% prediction interval (95% PI) is calculated as: 95% PI =
Reference_range
Generalization of means
quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean
Quasi-arithmetic_mean
Second edition of the IEEE 754 floating-point standard
IEEE 754r) is a revision of the IEEE 754 standard for floating-point arithmetic. It was published in August 2008 and is a significant revision to, and
IEEE_754-2008_revision
Number represented as a0+1/(a1+1/...)
Nevertheless, for almost all numbers on the unit interval, they have the same limit behavior. The arithmetic average diverges: lim n → ∞ 1 n ∑ k = 1 n a k
Simple_continued_fraction
Search algorithm finding the position of a target value within a sorted array
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position
Binary_search
Interval bounded by an upper and a lower limit statistics
In statistics, interval estimation is the use of sample data to estimate an interval of possible values of a (sample) parameter of interest. This is in
Interval_estimation
Family of programming languages
1967/1968 Karlsruhe, Germany ALGOL 60 (1963) with triplex numbers for interval arithmetic Chinese Algol 1972 China Chinese characters, expressed via the Symbol
ALGOL
Concept in topology
Stefan (2015). "Effective topological degree computation based on interval arithmetic". Mathematics of Computation. 84 (293): 1265–1290. arXiv:1207.6331
Degree of a continuous mapping
Degree_of_a_continuous_mapping
Probability distribution
{1}{2}}n^{2}\sigma ^{2}}.} Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed
Log-normal_distribution
Inverse of the average of the inverses of a set of numbers
for positive arguments only. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean
Harmonic_mean
Natural number
1088/0026-1394/31/6/013. Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt
1
Dutch-Canadian computer scientist (1937–2023)
verification and correctness, and constraint satisfaction, along with interval arithmetic and interval propagation . He wrote an advice-taking Prolog program for
Maarten_van_Emden
Mathematical computing environment
special mathematical function libraries Complex numbers and interval arithmetic Arithmetic, greatest common divisors and factorization for multivariate
Maple_(software)
Graphical representation of the distribution of numerical data
series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable
Histogram
1995 edition of the Fortran programming language standard
subprogram is MODULE interval_arithmetic TYPE interval REAL lower, upper END TYPE interval INTERFACE OPERATOR(+) MODULE PROCEDURE add_intervals END INTERFACE
Fortran_95_language_features
Sums vector sets A and B by adding each vector in A to each vector in B
morphology Erosion – Basic operation in mathematical morphology Interval arithmetic – Method for bounding the errors of numerical computations Mixed
Minkowski_addition
verify that your answers make sense." Units of measure for variables Interval arithmetic Anonymous functions Frink was named after Professor Frink, recurring
Frink_(programming_language)
Interval of binary floating-point numbers with a common sign and exponent
for the exceptional interval ( 0 , 2 e m i n ) {\displaystyle (0,2^{\mathrm {emin} })} of subnormal numbers. Floating-point arithmetic IEEE 754 Significand
Binade
Overview of ancient Greek music theory
inversely proportional to its length. Pythagoras construed the intervals arithmetically, allowing for 1:1 = Unison, 2:1 = Octave, 3:2 = Fifth, 4:3 = Fourth
Musical system of ancient Greece
Musical_system_of_ancient_Greece
Middle quantile of a data set or probability distribution
no distinct middle value and the median is usually defined to be the arithmetic mean of the two middle values. For example, this data set of 8 numbers
Median
Condition in which the value of a measurement or observation is only partially known
time interval data is to class as left censored intervals when the start time is unknown. In these cases, we have a lower bound on the time interval; thus
Censoring_(statistics)
Study of collection and analysis of data
bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical if a two-sided interval is built violating symmetry around
Statistics
First publication of complete tables of logarithms, 1614
notation works with some examples. He also introduces a form of interval arithmetic to bound any errors that occur in his calculations. Another now familiar
Mirifici Logarithmorum Canonis Descriptio
Mirifici_Logarithmorum_Canonis_Descriptio
Computer format for representing real numbers
g., a fractional amount of hours as an integer multiple of ten-minute intervals. Fixed-point number representation is often contrasted to the more complicated
Fixed-point_arithmetic
Statistical method
Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates. This technique allows estimation
Bootstrapping_(statistics)
Mathematical function of two positive real arguments
mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence
Arithmetic–geometric_mean
American computer scientist
computational statistics and computational mathematics generally, including interval arithmetic, fuzzy mathematics, probability theory, and probability bounds analysis
Vladik_Kreinovich
Entropy coding method
Nigel N. Martin in a 1979 paper, which effectively rediscovered the FIFO arithmetic code first introduced by Richard Clark Pasco in 1976. Given a stream of
Range_coding
Formula for the average value of a function over its domain
domain, the mean f ¯ {\displaystyle {\bar {f}}} of a function f(x) over the interval [a, b] is defined by f ¯ = 1 b − a ∫ a b f ( x ) d x . {\displaystyle {\bar
Mean_of_a_function
Geometrical object
1016/0005-1098(93)90106-4. Delanoue, N.; Jaulin, L.; Cottenceau, B. (2005). "Using interval arithmetic to prove that a set is path-connected" (PDF). Theoretical Computer
Subpaving
Conditional probability used in Bayesian statistics
various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPDI). But while
Posterior_probability
Statistical methods for comparing samples
the standard normal distribution to obtain p-values or form confidence intervals for the difference in proportions (derived slightly differently). This
Two-proportion_Z-test
Statistical considerations on how many observations to make
eventually obtained, i.e., if a high precision is required (narrow confidence interval) this translates to a low target variance of the estimator. the use of
Sample_size_determination
Probability distribution
the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression
Student's_t-distribution
Branch of numerical optimization
consist of methods which make use of zero-order interval arithmetic. A representative example is interval bisection. First-order methods consist of methods
Deterministic global optimization
Deterministic_global_optimization
Parameter estimation via sample statistics
with interval estimation: interval estimates are typically either confidence intervals, in the case of frequentist inference, or credible intervals, in
Point_estimation
Statistical value representing the center or average of a distribution
the late 1920s. The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for
Central_tendency
range The length of the smallest interval which contains all of the data in a dataset, calculated as the arithmetic difference between the largest and
Glossary of probability and statistics
Glossary_of_probability_and_statistics
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
any meaning for data on an interval scale. For example, most temperature scales (e.g., Celsius, Fahrenheit etc.) are interval scales with arbitrary zeros
Coefficient_of_variation
Sequence of notes that results from inverting the intervals of the overtone series
series is based upon arithmetic multiplication of frequencies, resulting in a harmonic series, the undertone series is based on arithmetic division. The hybrid
Undertone_series
Logical problem studied in computer science
directly in SMT solvers; see, for instance, the decidability of Presburger arithmetic. SMT can be thought of as a constraint satisfaction problem and thus a
Satisfiability modulo theories
Satisfiability_modulo_theories
Concept in musical set theory
of interval: Ordered pitch interval Unordered pitch interval Ordered pitch-class interval Unordered pitch-class interval The ordered pitch interval is
Pitch_interval
INTERVAL ARITHMETIC
INTERVAL ARITHMETIC
Boy/Male
Sikh
Protector of Indra, Variant of Inder
Male
Greek
(ἈπολλÏων) Greek name APOLLYÅŒN means "destroyer." In the New Testament bible, this is the name of the angel-prince of the infernal regions, the minister of death and author of havoc on earth. He is also known by the name AbaddÅn.
Girl/Female
Arabic, Bengali, Gujarati, Hindu, Indian, Kannada, Marathi, Muslim, Punjabi, Sikh, Sindhi, Telugu
Heart; Inner Beauty; Fame; Internal Nature; Wisdom
Surname or Lastname
English and French
English and French : nickname for a handsome man (perhaps also ironically for an ugly one), from Old French beu, bel ‘fair’, ‘lovely’ (Late Latin bellus).Hungarian (Bél) : from the old secular Hungarian name Bél, or alternatively from bél ‘internal part’, probably an occupational name for a servant who worked in the household.Czech (BÄ›l) from Czech bÃlý ‘white’.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Punjabi, Sanskrit, Sikh, Traditional
Protector of All; Protector of God Indra; Gods Friends
Boy/Male
Indian
Internal Cleanliness
Surname or Lastname
Irish
Irish : reduced Anglicized form of either of two Gaelic names, Ó DuibhÃn ‘descendant of DuibhÃn’, a byname meaning ‘little black one’, or Ó DaimhÃn ‘descendant of DaimhÃn’, a byname meaning ‘fawn’, ‘little stag’. These are attenuated versions of Ó Dubháin and Ó Damháin, and are the phonetic origin of Anglicizations with an internal v (as opposed to w, as in Dewan, or monosyllabic forms with an o or u) (see Doane).English and French : nickname, of literal or ironic application, from Middle English, Old French devin, divin ‘excellent’, ‘perfect’ (Latin divinus ‘divine’).
Girl/Female
American, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Plucked Flower; Voice of Heart; Woman; Intellect; Behold of Any Beautiful Scene; Internal Beauty
Male
Greek
(á¾Î¹Î´Î·Ï‚) Greek name derived from the word aides, HAIDES means "unseen." In mythology, this is the name of the god of the underworld, brother of Zeus and husband of Persephone. In the Greek bible, Haides is associated with Orcus, the realm of the dead, the infernal regions where disembodied spirits live, a dark and dismal place in the depths of the earth. Only later was Haides described as the grave, death, and hell. Also spelled HadÄ“s.Â
Male
English
Anglicized form of Greek ApollyÅn, APOLLYON means "destroyer." In the New Testament bible, this is the name of the angel-prince of the infernal regions, the minister of death and author of havoc on earth. He is also known by the name Abaddon.
INTERVAL ARITHMETIC
INTERVAL ARITHMETIC
Boy/Male
Armenian, Australian, French, German, Hebrew
Armenian
Girl/Female
Hindu, Indian, Traditional
Flower of the Gods
Girl/Female
Hindu
Boy/Male
Norse
Brother of Gunnlaug.
Boy/Male
African, Arabic, Australian, Muslim
Life; Name of the Second Caliph
Surname or Lastname
English and Scottish
English and Scottish : occupational name for a clerk or copyist (see Scriven).
Girl/Female
Tamil
Joyful
Boy/Male
Hindu
Chanting prayers
Boy/Male
Indian, Punjabi, Sikh
Worshipping God
Boy/Male
Hindu, Indian, Kannada, Tamil, Telugu
Lord Murugan; God
INTERVAL ARITHMETIC
INTERVAL ARITHMETIC
INTERVAL ARITHMETIC
INTERVAL ARITHMETIC
INTERVAL ARITHMETIC
n.
An interval of a fifth; also, a part sung with such intervals.
a.
Pertaining to its own affairs or interests; especially, (said of a country) domestic, as opposed to foreign; as, internal trade; internal troubles or war.
n.
A brief space of time between the recurrence of similar conditions or states; as, the interval between paroxysms of pain; intervals of sanity or delirium.
n.
An interval.
n.
An interval.
n.
Space of time between any two points or events; as, the interval between the death of Charles I. of England, and the accession of Charles II.
n.
Alt. of Intervale
n.
A space between things; a void space intervening between any two objects; as, an interval between two houses or hills.
n.
Intervening space; interval.
n.
An inhabitant of the infernal regions; also, the place itself.
a.
Internal.
v. t.
To interpel.
n.
An interhyal ligament or cartilage.
n.
Difference in pitch between any two tones.
n.
A small interval, less than any in actual practice, but used in the mathematical calculation of intervals.
a.
Of or pertaining to, resembling, or inhabiting, hell; suitable for hell, or to the character of the inhabitants of hell; hellish; diabolical; as, infernal spirits, or conduct.
a.
Derived from, or dependent on, the thing itself; inherent; as, the internal evidence of the divine origin of the Scriptures.
a.
Inward; interior; being within any limit or surface; inclosed; -- opposed to external; as, the internal parts of a body, or of the earth.
n.
Interval; intermission.
a.
Pertaining to, or proceeding by, integration; as, the integral calculus.