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Computing the fixed point of a function
Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function. In its most common form, the given
Fixed-point_computation
Element mapped to itself by a mathematical function
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation
Fixed_point_(mathematics)
Computer format for representing real numbers
contrasted to the more complicated and computationally demanding floating-point representation. In the fixed-point representation, the fraction is often
Fixed-point_arithmetic
Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f
Brouwer_fixed-point_theorem
Higher-order function Y for which Y f = f (Y f)
In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function (i.e., a function that takes a
Fixed-point_combinator
More formally, FIXP contains search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits over basis
FIXP
Root-finding algorithm
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f {\displaystyle
Fixed-point_iteration
Theorem about metric spaces
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem)
Banach_fixed-point_theorem
Algorithms for zeros of functions
the number of queries is given. List of root finding algorithms Fixed-point computation Broyden's method – Quasi-Newton root-finding method for the multivariable
Root-finding_algorithm
Logical formulation of recursion
In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development
Fixed-point_logic
Theorem in order and lattice theory
search. On the other hand, determining whether a given fixed point is unique is computationally hard: For d=2, for componentwise lattice and a value-oracle
Knaster–Tarski_theorem
Fixed-point theorem for set-valued functions
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued
Kakutani_fixed-point_theorem
Computer approximation for real numbers
which works as a fixed-point number. Expectations from mathematics may not be realized in the field of floating-point computation. For example, it is
Floating-point_arithmetic
Mathematical model of computer
that can compute with exact real numbers instead of the binary fixed-point or floating-point numbers used by most actual computers. The real RAM was formulated
Real_RAM
Measure of computer performance
or compute in computing, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate measure
Floating point operations per second
Floating_point_operations_per_second
integers, fixed-point numbers, and floating-point numbers, but not rational numbers and arbitrary-precision numbers. The number of digits being fixed means
Fixed-precision_arithmetic
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 space
Floating-point error mitigation
Floating-point_error_mitigation
Iterative method in numerical analysis
{\displaystyle f} is computationally expensive. Anderson acceleration is a method to accelerate the convergence of the fixed-point sequence. Define the
Anderson_acceleration
Economical computational problem
Market equilibrium computation (also called competitive equilibrium computation or clearing-prices computation) is a computational problem in the intersection
Market equilibrium computation
Market_equilibrium_computation
Theorem in computability theory
fixed-point free. The fixed-point theorem shows that no total computable function is fixed-point free, but there are many non-computable fixed-point-free
Kleene's_recursion_theorem
Academic subfield of computer science
mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation using an algorithm, how efficiently
Theory_of_computation
Theorem in order theory and lattice theory
theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem. Suppose
Kleene_fixed-point_theorem
Solution to x * e^x = 1
converge to Ω as n approaches infinity. This is because Ω is an attractive fixed point of the function e−x. It is much more efficient to use the iteration Ω
Omega_constant
Method in computer arithmetic
Block floating point (BFP) is a method used to provide an arithmetic approaching floating point while using a fixed-point processor. BFP assigns a group
Block_floating_point
Implementation of arithmetic operations
representation of a number is fixed (fixed-point, floating-point and interval arithmetic), the main concern is to control the computational error, as far as possible;
Computer_arithmetic
Branch of computer science
which object a query ray intersects first. If the search space is fixed, the computational complexity for this class of problems is usually estimated by:
Computational_geometry
Computer hardware technology that uses quantum mechanics
classical mechanical device, possibly with a fixed slow-down in physical time. If a classical computation uses randomness, this can be modeled as access
Quantum_computing
13-month calendar where every date is fixed to a day of the week
The International Fixed Calendar (also known as the Cotsworth plan, the Cotsworth calendar, the Eastman plan or the Yearal) was a proposed reform of the
International_Fixed_Calendar
Asano, Jiří Matoušek, and Takeshi Tokuyama in 2007. Formally, it is a fixed point of a certain function. Its existence or uniqueness are not clear in advance
Zone_diagram
Theorem on triangulation graph colorings
invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division
Sperner's_lemma
24-bit output. All computations are performed with fixed-point integer arithmetic, making it ideal for systems without a floating-point unit. The implementation
MPEG_Audio_Decoder
Newton-like root-finding algorithm that does not use derivatives
process applied to fixed-point iteration. Viewed in this way, Steffensen's method naturally generalizes to efficient fixed-point calculation in general
Steffensen's_method
Calculations where numbers' precision is only limited by computer memory
available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of bits related to the size of the processor
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Computation model defining an abstract machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table
Turing_machine
Subfield of cryptography
Secure multi-party computation (also known as secure computation, multi-party computation (MPC) or privacy-preserving computation) is a subfield of cryptography
Secure multi-party computation
Secure_multi-party_computation
128-bit computer number format
Shewchuk, Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates, Discrete & Computational Geometry 18: 305–363, 1997. Knuth,
Quadruple-precision floating-point format
Quadruple-precision_floating-point_format
Terminology used in computer graphics
In computer graphics, fixed-function is a term primarily used to describe 3D graphics APIs and GPUs designed prior to the advent of programmable shaders
Fixed-function (computer graphics)
Fixed-function_(computer_graphics)
Determining where a point is in relation to a coplanar polygon
In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon
Point_in_polygon
Computation modulo a fixed integer
result of a computation does not depend on whether the division by m is performed after each operation, only once at the end of the computation, or at the
Modular_arithmetic
Replacing a number with a simpler value
almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical
Rounding
Hypothetical concept in astrophysics
(for a fixed rate of subjective experience). To make thoughts possible with ever-decreasing energy, the beings must stretch out their computational steps
Dyson's_eternal_intelligence
Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz. The
Knaster–Kuratowski–Mazurkiewicz lemma
Knaster–Kuratowski–Mazurkiewicz_lemma
Cost accounting model
of CVP analysis is the point where total revenues equal total costs (both fixed and variable costs). At this break-even point, a company will experience
Cost–volume–profit_analysis
Branch of mathematics relating to posets
Computation then is modeled by applying monotone functions repeatedly on elements of the domain in order to refine a result. Reaching a fixed point is
Domain_theory
IEEE standard for floating-point arithmetic
an alternative approach at computation with rational numbers (especially beneficial when the exponent range is known, fixed, or bound at compile time)
IEEE_754
Concept in genetic algorithm theory
"Schema Theory for Genetic Programming with One-Point Crossover and Point Mutation". Evolutionary Computation. 6 (3): 231–252. doi:10.1162/evco.1998.6.3.231
Defining_length
the terminology that DenoteS is a fixed point of progressionS. Furthermore, this fixed point is least among all fixed points of progressionS. An important
Denotational semantics of the Actor model
Denotational_semantics_of_the_Actor_model
C library for arbitrary-precision floating-point arithmetic
Precision Floating-Point Reliable Library (GNU MPFR) is a GNU portable C library for arbitrary-precision binary floating-point computation with correct rounding
GNU_MPFR
Procedural, imperative computer programming language
computation, scientific computing, and system programming. It supports recursion, structured programming, linked data structure handling, fixed-point
PL/I
Type of recurrent dynamical network
implemented as memory models using fixed-point attractors. However, they have been largely impractical for computational purposes because of difficulties
Attractor_network
Extension of propositional modal logic
many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic. The (propositional, modal)
Modal_μ-calculus
Scientific area at the interface between computer science and mathematics
The usual number systems used in numerical computation are floating point numbers and integers of a fixed, bounded size. Neither of these is convenient
Computer_algebra
Type of astronomical bodies
In astronomy, the fixed stars (Latin: stellae fixae) are the lights (luminary points), mainly stars visible to the naked eye, that appear not to move
Fixed_stars
Ballistic Research Laboratories Electronic Scientific Computer
second. A fixed-point addition took 5 microseconds, a floating-point addition took 5 to 10 microseconds, a multiplication (fixed- or floating-point) took
BRLESC
Topics referred to by the same term
(linguistics), a word, phrase, or sentence Fixed expression, a form of words with a specific meaning Idiom, a type of fixed expression Metaphorical expression
Expression
Part of a number in scientific notation
Automatic Computation (1st ed.). New Jersey, USA: Prentice-Hall, Englewood Cliffs. ISBN 0-13-165779-8. Sterbenz, Pat H. (1974-05-01). Floating-Point Computation
Significand
Numerical analysis series acceleration method
x_{n+1}=f(x_{n})} for some function f {\displaystyle f} converging to a fixed point, the accelerated sequence's convergence is quadratic. In this case, the
Aitken's delta-squared process
Aitken's_delta-squared_process
Microsoft programming language
compositional computations called computation expressions. Sequence expressions, asynchronous computations and queries are particular kinds of computation expressions
F Sharp (programming language)
F_Sharp_(programming_language)
Variant of nearest neighbor search
In computational geometry, the fixed-radius near neighbor problem is a variant of the nearest neighbor search problem. In the fixed-radius near neighbor
Fixed-radius_near_neighbors
Internal representation of numeric values in a digital computer
with the computation: <sign> × (1 + <fractional significand>) × 2<exponent> − 127 leading to the following range of numbers: Such floating-point numbers
Computer_number_format
Simple polynomial map exhibiting chaotic behavior
point is a fixed point. In mathematical terms, a fixed point is It means a point that does not change when the map is applied. The fixed point will be denoted
Logistic_map
Complex Analysis, Fixed-points and Iterations of Holomorphic Mappings
unique point z in the closure of D such that the iterates of f tend to z uniformly on compact subsets of D. If z lies in D, it is the unique fixed point of
Denjoy–Wolff_theorem
Algorithmic runtime requirements for common math procedures
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Decimal representation of real numbers in computing
floating-point representation over decimal fixed-point and integer representation is that it supports a much wider range of values. For example, while a fixed-point
Decimal_floating_point
Critical point where a periodic solution arises
above computation of the Jacobian can be significantly simplified by working in the tangent plane, tangent to the fixed point. The fixed point is located
Hopf_bifurcation
Model that describes the programmable interface of a computer processor
another memory location or the result of a computation, or to retrieve stored data to perform a computation later. Read or write data from hardware devices
Instruction_set_architecture
16-bit computer number format
and specialized floating point formats with only 8 bits or less are increasingly used to further accelerate certain computations. If the hardware has instructions
Half-precision floating-point format
Half-precision_floating-point_format
Subfield of computer science and mathematics
foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The ACM's Special Interest Group on Algorithms and Computation Theory
Theoretical_computer_science
Study of computation
Fundamental areas of computer science Computer science is the study of computation, information, and automation. Included broadly in the sciences, computer
Computer_science
Computational method in Bayesian statistics
Approximate Bayesian computation (ABC) constitutes a class of computational methods rooted in Bayesian statistics that can be used to estimate the posterior
Approximate Bayesian computation
Approximate_Bayesian_computation
Concept in numerical analysis
environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs
Adaptive_mesh_refinement
Ability to solve a problem by an effective procedure
of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked
Computability
Method in numerical analysis
Euclidean n-space is a finite-dimensional Banach space. A steady state, or fixed point, of a parameterized family of flows or maps are of this form, and by
Numerical_continuation
Economical computational problem
Nash equilibrium (NE) computation is a class of computational problems in the intersection of game theory and computer science. The input to this problem
Nash_equilibrium_computation
American mathematician (1945-2020)
advanced the application of numerical techniques to the computation of Brouwer's fixed point, contributing significantly to the development of modern
Tien-Yien_Li
Distance estimation problems in computational geometry
points for a fixed 'k'. Shortest path among obstacles Distance of closest approach Franco P. Preparata and Michael Ian Shamos (1985). Computational Geometry
Proximity_problems
Vectors mapped to 0 by a linear map
The following is a simple illustration of the computation of the kernel of a matrix (see § Computation by Gaussian elimination, below for methods better
Kernel_(linear_algebra)
Model of concurrent computation
mathematical model of concurrent computation that treats an actor as the basic building block of concurrent computation. In response to a message it receives
Actor_model
Typed functional language
Application (of a term of type σ → τ to a term of type σ) λ-abstraction The Y fixed point combinator (making terms of type σ out of terms of type σ → σ) The successor
Programming Computable Functions
Programming_Computable_Functions
around each image point. Other examples are computation of local derivatives of the image data. It is also rather common to use a fixed but non-linear function
Neighborhood_operation
Conceptual framework used in numerical analysis of surfaces and shapes
surfaces and shapes. LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these
Level-set_method
Software feature
Incremental computing, also known as incremental computation, is a software feature which, whenever a piece of data changes, attempts to save time by
Incremental_computing
Mathematical-logic system
calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding
Lambda_calculus
Methods that imitate, replicate or use natural processes
Natural computing, also called natural computation, is a terminology introduced to encompass three classes of methods: 1) those that take inspiration
Natural_computing
Class of data processing algorithms
(ISLs) or Stencil computations are a class of numerical data processing solution which update array elements according to some fixed pattern, called a
Iterative_Stencil_Loops
Computation machine that uses continuously varying data technology
An analog computer or analogue computer is a type of computation machine (computer) that uses physical phenomena such as electrical, mechanical, or hydraulic
Analog_computer
Solar cell power extraction method
conditions. The array's operating point is thus kept near MPP by regulating the array voltage and matching it to the fixed reference voltage V ref = k V oc
Maximum_power_point_tracking
Algorithm
bone models, etc. The Iterative Closest Point algorithm keeps one point cloud, the reference or target, fixed, while transforming the other, the source
Iterative_closest_point
Algorithms for calculating square roots
construct a series of increasingly accurate approximations. Most square root computation methods are iterative: after choosing a suitable initial estimate of
Square_root_algorithms
Concept in game theory
In game theory, a focal point (or Schelling point) is a solution that people tend to choose by default in the absence of communication in order to avoid
Focal_point_(game_theory)
Solution concept of a non-cooperative game
the Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the
Nash_equilibrium
Simple Turing complete logic
from ν), then its fixed point ββ expresses the whole recursive computation, since using the same function ββ for the "rest of computation" call (with ββν
SKI_combinator_calculus
Aspect of cryptography
transport Non-repudiation methods Secret sharing methods Secure multi-party computation For example, Transport Layer Security (TLS) is a cryptographic protocol
Cryptographic_protocol
Branch of mathematical logic
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of
Descriptive_complexity_theory
equilibrium computation. The function f is called convex-concave if it is a convex function of x for any fixed y, and a concave function of y for any fixed x.
Min-max_optimization
usually based on numerical computation with approximate floating-point numbers, while symbolic computation emphasizes exact computation with expressions containing
Glossary_of_computer_science
Rendering optimization technique
sophisticated variant called fixed foveated rendering doesn't utilise eye tracking and instead assumes a fixed focal point. Research into foveated rendering
Foveated_rendering
Variant of floating-point numbers in computers
floating point (TFP) is a format similar to floating point, but with variable-sized entries for the significand and exponent instead of the fixed-length
Tapered_floating_point
Mathematical theory about infinitely iterated function composition
compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another
Infinite compositions of analytic functions
Infinite_compositions_of_analytic_functions
Unsolved problem in computer science
studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given
P_versus_NP_problem
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