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Form of interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through
Polynomial_interpolation
Polynomials used for interpolation
equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Lagrange polynomials are also related to matrix eigenspace
Lagrange_polynomial
Method for estimating new data within known data points
is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation. However
Interpolation
Polynomial interpolation using derivative values
interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation
Hermite_interpolation
Mathematical expression
Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes
Newton_polynomial
Method of curve fitting
In mathematics, linear interpolation (sometimes lerp) is a method of curve fitting using linear polynomials to construct new data points within the range
Linear_interpolation
Mathematical method
numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is
Spline_interpolation
Pair of polynomial sequences
are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon
Chebyshev_polynomials
Failure of convergence in interpolation
interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It was discovered by
Runge's_phenomenon
Interpolation with trigonometric polynomials
In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes
Trigonometric_interpolation
Interpolation on functions of more than one variable
Nearest-neighbor interpolation n-linear interpolation (see bi- and trilinear interpolation and multilinear polynomial) n-cubic interpolation (see bi- and
Multivariate_interpolation
Generalizations of Nyquist-Shannon sampling theorem for reconstructing signals
( z ) {\displaystyle p_{n}(z)} using the interpolating polynomials of Lagrange interpolation: I k ( z ) = ( z − z 0 ) ( z − z 1 ) ⋯ ( z − z k − 1 ) (
Nonuniform_sampling
Statistics concept
fitting Line regression Local polynomial regression Polynomial and rational function modeling Polynomial interpolation Response surface methodology Smoothing
Polynomial_regression
Mathematical function defined piecewise by polynomials
defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar
Spline_(mathematics)
Cryptographic algorithm created by Adi Shamir
exploits the Lagrange interpolation theorem, specifically that k {\displaystyle k} points on the polynomial uniquely determines a polynomial of degree less than
Shamir's_secret_sharing
Constants related to interpolation errors
b]} containing all the interpolation nodes. The process of interpolation maps the function f {\displaystyle f} to a polynomial p {\displaystyle p} . This
Lebesgue_constant
Roots of the Chebyshev polynomials of the first kind
grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the projection of a set of
Chebyshev_nodes
Type of mathematical expression
desired by a polynomial function. Practical methods of approximation include polynomial interpolation and the use of splines. Polynomials are frequently
Polynomial
Extension of cubic spline interpolation
by bilinear interpolation or nearest-neighbor interpolation. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines
Bicubic_interpolation
Type of polynomial
\nabla ^{2}f=0} . The value of the polynomial at an arbitrary point can be found by repeated linear interpolation along each coordinate axis. Equivalently
Multilinear_polynomial
Technique for polynomial interpolation
for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given n + 1 points, there is a unique polynomial of degree
Neville's_algorithm
constant Hermite interpolation Birkhoff interpolation Abel–Goncharov interpolation Spline interpolation — interpolation by piecewise polynomials Spline (mathematics)
List of numerical analysis topics
List_of_numerical_analysis_topics
Algorithm to approximate functions
the initial approximation because of their role in the theory of polynomial interpolation. For the initialization of the optimization problem for function
Remez_algorithm
Topics referred to by the same term
about non-linear operators Riesz–Thorin interpolation theorem about linear operators Polynomial interpolation in analysis This disambiguation page lists
Interpolation_theorem
Type of cryptanalytic attack
In its simplest version an interpolation attack expresses the ciphertext as a polynomial of the plaintext. If the polynomial has a relative low number
Interpolation_attack
Matrix of geometric progressions
making the Vandermonde matrix invertible. The polynomial interpolation problem is to find a polynomial p ( x ) = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n {\displaystyle
Vandermonde_matrix
Mathematical theorem in the study of analysis
theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass
Stone–Weierstrass_theorem
mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial P ( x ) {\displaystyle
Birkhoff_interpolation
Type of polynomial used in Numerical Analysis
numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of Bernstein basis polynomials. The idea is named after mathematician
Bernstein_polynomial
Error-correcting codes
a systematic Reed–Solomon code. One method uses Lagrange interpolation to compute polynomial p m {\displaystyle p_{m}} such that p m ( a i ) = m i for
Reed–Solomon_error_correction
Selberg's polynomials, and the Stirling interpolation polynomials as special cases. The general difference polynomial sequence is given by p n ( z ) = z n
Difference_polynomials
Algorithm used for polynomial interpolation
Aitken interpolation is an algorithm used for polynomial interpolation that was derived by the mathematician Alexander Aitken. It is similar to Neville's
Aitken_interpolation
This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. Degree: The maximum exponents
List_of_polynomial_topics
Computational method
mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the
Factorization_of_polynomials
Hermite interpolating basis polynomial corresponding to node λ i {\displaystyle \lambda _{i}} satisfying the interpolation conditions: h i , 0 ( j )
Frobenius_covariant
Polynomial sequence
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets
Hermite_polynomials
Cubic function used for interpolation
{\displaystyle R=0,} thus P = Q . {\displaystyle P=Q.} We can write the interpolation polynomial on the unit interval (for an arbitrary interval see the rescaled
Cubic_Hermite_spline
Element of a basis for a function space
associated interpolation point and 0 at all the other interpolation points. If the corresponding data values are y0, …, yn, the unique polynomial of degree
Basis_function
Code added to allow recovery of lost data
f(i) given. The linear construction above can be generalized to polynomial interpolation. Additionally, points are now computed over a finite field. First
Erasure_code
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
Birkhoff interpolation: an extension of polynomial interpolation Cubic interpolation Hermite interpolation Lagrange interpolation: interpolation using Lagrange
List_of_algorithms
Indian mathematician
1993, p. 312. R.P. Agarwal and P.J.Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Academic Publishers, Dordrecht,
Ravi_Agarwal
In polynomial interpolation of two variables, the Padua points are the first known example (and up to now the only one) of a unisolvent point set (that
Padua_points
Mathematical method that minimizes maximum error
Phillips, George M. (2003). "Best Approximation". Interpolation and Approximation by Polynomials. CMS Books in Mathematics. Springer. pp. 49–11. doi:10
Minimax approximation algorithm
Minimax_approximation_algorithm
In mathematics, Abel–Goncharov interpolation determines a polynomial such that various higher derivatives are the same as those of a given function at
Abel–Goncharov_interpolation
Concept in mathematics
Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They
Trigonometric_polynomial
Problem of determining whether polynomials are identical
Michael F., "Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields", SIAM J. Comput., Vol 19, No.6, pp. 1059-1063
Polynomial_identity_testing
Changing the sampling rate of a discrete signal
using polynomial interpolation. Farrow filter Archived 2018-10-01 at the Wayback Machine Using Farrow filter on the basis of piecewise cubic polynomial interpolation
Sample-rate_conversion
calculates the in-between frames through use of (usually) piecewise polynomial interpolation to draw images semi-automatically. For all applications of this
Interpolation (computer graphics)
Interpolation_(computer_graphics)
Second-order polynomial interpolation formula
Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta
Brahmagupta's interpolation formula
Brahmagupta's_interpolation_formula
Algorithm for computing polynomial coefficients
(x_{n},y_{n})} , the method calculates the coefficients of the interpolation polynomial of these points in the Newton form. It is sometimes denoted by
Divided_differences
Algorithms for zeros of functions
work by interpolation. This consists in using the last computed approximate values of the root for approximating the function by a polynomial of low degree
Root-finding_algorithm
Depicting depth with levels of darkness
and bilinear interpolation of the normals. Hence, second-degree polynomial interpolation was used. This type of biquadratic interpolation was further elaborated
Shading
Family of interpolation and clamping functions
Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, video game engines, and machine learning
Smoothstep
Resampling method
conversion by factor R ∈ R + {\displaystyle \mathbb {R} ^{+}} include polynomial interpolation and the Farrow structure. Harris 2004. "6.1". p 128. Crochiere
Downsampling (signal processing)
Downsampling_(signal_processing)
Sequence in computer science
parallel polynomial interpolation. In particular, it can be used to compute the divided difference coefficients of the Newton form of the interpolation polynomial
Prefix_sum
Sequence of data points over time
all relevant dates. Alternatively polynomial interpolation or spline interpolation is used where piecewise polynomial functions are fitted in time intervals
Time_series
Method for estimating new data outside known data points
extended beyond the end of the known data. Polynomial extrapolation is typically done by means of Lagrange interpolation or using Newton's method of finite differences
Extrapolation
terms in the denominator to account for nearby poles. While a polynomial interpolation or extrapolation only yields good results if the nearest pole is
Bulirsch–Stoer_algorithm
Japanese mathematician (c. 1642–1708)
material in these works consisted of algebra with numerical methods, polynomial interpolation and its applications, and indeterminate integer equations. Seki's
Seki_Takakazu
Oscillatory error in Fourier series
this is commonly referred to as the Longo phenomenon. In the polynomial interpolation setting, the Gibbs phenomenon can be mitigated using the S-Gibbs
Gibbs_phenomenon
Method for dividing a secret among multiple parties
the secret divided by k − 1. This scheme makes use of repeated polynomial interpolation and has potential applications in secure information dispersal
Secret_sharing
Italian mathematician and professor
His scientific interests deal mainly with interpolation and approximation of functions and data by polynomials and radial basis functions (RBFs)). Stefano
Stefano_De_Marchi
Method of interpolating functions on a 2D grid
mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation. It is usually
Bilinear_interpolation
regular 3D grid of control points moved to arbitrary positions, with polynomial interpolation of the space between them. Degenerate triangles Zero area triangle
Glossary_of_computer_graphics
Concept in applied mathematics
efficiently, we first determine X ( z ) {\displaystyle X(z)} directly by polynomial interpolation: X ^ [ k ] = X ( z k ) , k = 0 , 1 , … , N − 1. {\displaystyle
Non-uniform discrete Fourier transform
Non-uniform_discrete_Fourier_transform
Automatic mechanical calculator
Gaussian reduction an N−1th degree polynomial interpolation of the function is found. With the optimized polynomial, the initial values can be calculated
Difference_engine
Discrete analog of a derivative
differences of polynomials". divisbyzero.com. February 13, 2018. Fraser, Duncan C. (January 1, 1909). "On the Graphic Delineation of Interpolation Formulæ"
Finite_difference
Class of iterative numerical methods for solving differential equations
{\displaystyle b_{j}} can be determined as follows. Use polynomial interpolation to find the polynomial p of degree s − 1 {\displaystyle s-1} such that p (
Linear_multistep_method
Mathematical approximation of a function
of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function
Taylor_series
Algorithm for multiplying large numbers
polynomial multiplication described by Marco Bodrato. The algorithm has five main steps: Splitting Evaluation Pointwise multiplication Interpolation Recomposition
Toom–Cook_multiplication
Cubic plane curve
analysis, when approximating functions using polynomial interpolation with equally spaced interpolation points, it may be the case for some functions
Witch_of_Agnesi
Algebraic structure
mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]
Ring_of_polynomial_functions
Polynomial function of degree 3
f(x)=ax^{3}+bx^{2}+cx+d,} with a ≠ 0 {\displaystyle a\neq 0} , that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and
Cubic_function
Construction in transcendental number theory
function ex not with a polynomial but with a rational function, that is a quotient of two polynomials. In particular he chose polynomials A(x) and B(x) such
Auxiliary_function
multi-point evaluation and interpolation similar to the fast Fourier transform can help speed them up for large degrees of the polynomial. A free implementation
Polynomial_root-finding
Numerical integration method
Fourier series Residue calculus Euler–Maclaurin summation formula Polynomial interpolation It is argued that the speed of convergence of the trapezoidal rule
Trapezoidal_rule
Mathematical operation that predicts future values of a discrete-time signal
values, a polynomial interpolation is a linear combination of the known values. If the discrete time signal is estimated to obey a polynomial of degree
Linear_prediction
Geometry of the location of polynomial roots
companion matrix of the polynomial on a basis related to Lagrange interpolation to define discs centered at the interpolation points, each containing
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Consistency among data between source and target data stores
Starobinski; S. Agarwal. "Fast PDA Synchronization Using Characteristic Polynomial Interpolation" (PDF). IEEE INFOCOM 2002. doi:10.1109/INFCOM.2002.1019402. Y.
Data_synchronization
homogenous polynomials and the hypersurface of dimension d with many known lists of exceptions. In which case, the classic polynomial interpolation that is
Alexander–Hirschowitz_theorem
About simultaneous modular congruences
case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one: P i ( X ) = X − x i
Chinese_remainder_theorem
Linear map or polynomial function of degree one
is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing such a linear
Linear_function
Signal processing method
the extrema are evenly spaced in the pass and stop band. Perform polynomial interpolation and re-estimate positions of the local extrema. Move extrema to
Parks–McClellan filter design algorithm
Parks–McClellan_filter_design_algorithm
German applied mathematician
Periodische Lagrange- und Hermite-Spline-Interpolation, concerned polynomial interpolation using Lagrange polynomials and Hermite splines, and was supervised
Gerlind_Plonka
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting
Successive parabolic interpolation
Successive_parabolic_interpolation
process modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting. A polynomial function is one
Polynomial and rational function modeling
Polynomial_and_rational_function_modeling
Numerical integration method
expensive, it may be preferable to replace the polynomial interpolation of Richardson with the rational interpolation proposed by Bulirsch & Stoer (1967). To
Romberg's_method
Surface-approximation method
In the mathematical subfield numerical analysis, tricubic interpolation is a method for obtaining values at arbitrary points in 3D space of a function
Tricubic_interpolation
Multiplication algorithm
problem to product problem, through FFT. By finding the FFT of the polynomial interpolation of each C k {\displaystyle C_{k}} , one can determine the desired
Schönhage–Strassen_algorithm
Field in logic and theoretical computer science
(2010). "Effectively polynomial simulations" (PDF). ICS: 370–382. Bonet, M.L.; Pitassi, Toniann; Raz, Ran (2000). "On Interpolation and Automatization for
Proof_complexity
Numerical technique
u_{p}(y)} be the corresponding Lagrange basis polynomials. One can show that the interpolating polynomial 1 y − x = ∑ i = 1 p 1 t i − x u i ( y ) + ϵ p
Fast_multipole_method
Method of multivariate interpolation on a 3-dimensional regular grid
D=2} , to dimension D = 3 {\displaystyle D=3} . These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires
Trilinear_interpolation
Polynomial in numerical analysis
Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the roots of a polynomial: the
Wilkinson's_polynomial
Concept in cryptography
modulo q. Any t + 1 share holders can recover the secret s by using polynomial interpolation modulo q, but any set of at most t share holders cannot. (In fact
Verifiable_secret_sharing
Topics referred to by the same term
Tarring Neville, East Sussex, England Neville's algorithm, used for polynomial interpolation The Neville Brothers, American band Naville, a surname Nevil (disambiguation)
Neville
Approximation method
{\displaystyle \kappa } is sufficiently smooth, we can approximate it by polynomial interpolation to obtain κ ~ ( x , y ) = ∑ ν = 1 k κ ( x , ξ ν ) ℓ ν ( y ) , {\displaystyle
Hierarchical_matrix
Type of error-correcting code
decoding of Reed-Muller codes is polynomial interpolation. The key concept behind a Reed-Muller code is a multivariate polynomial of degree d {\displaystyle
Locally_decodable_code
Polynomial sequence
polynomials because of their use in interpolation and spline theory; see Schoenberg. The Type B Eulerian numbers and polynomials satisfy many similar identities
Eulerian_number
Line of digital and stage pianos
Privia was produced from 2003 to 2006. It utilized the Zygotech Polynomial Interpolation (ZPI) synthesis sound engine, as used in Casio's former flagship
Privia
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