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Polynomial root-finding algorithm
In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically
Laguerre's_method
French mathematician (1834–1886)
analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He
Edmond_Laguerre
Algorithm for finding zeros of functions
Bisection method Euler method Fast inverse square root Fisher scoring Gradient descent Integer square root Kantorovich theorem Laguerre's method Methods of computing
Newton's_method
Sequence of differential equation solutions
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: x y ″
Laguerre_polynomials
starting point of Horner's method for computing the roots. Closely related to Newton's method are Halley's method and Laguerre's method. Both use the polynomial
Polynomial_root-finding
Algorithm in numerical analysis
Handapangoda, C. C.; Premaratne, M.; Yeo, L.; Friend, J. (2008). "Laguerre Runge-Kutta-Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue"
Runge–Kutta–Fehlberg_method
Mathematical method in numerical analysis
analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating
Gauss–Laguerre_quadrature
Numerical methods for matrix eigenvalue calculation
114–139, doi:10.1016/j.laa.2005.06.022 Li, T. Y.; Zeng, Zhonggang (1992), "Laguerre's Iteration In Solving The Symmetric Tridiagonal Eigenproblem - Revisited"
Eigenvalue_algorithm
British applied mathematician (1932–2026)
1007/BF02165404. S2CID 122353612. Bunch, J. R.; Parlett, B. N. (1971). "Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations". SIAM Journal
Beresford_Parlett
Mathematical analysis of frequency content of signals
Transform based on Laguerre function expansion. The Laguerre method can be used to simulate a weakly nonlinear circuit and the Laguerre method can invert a
Multidimensional_transform
stage takes these potential zeros and “polishes” them by applying Laguerre's method to bring them close to the actual zeros of the polynomial. The third
Lindsey–Fox_algorithm
functions belonging to the Laguerre–Pólya class" Archived 2008-10-06 at the Wayback Machine by D. Dryanov and Q. I. Rahman, Methods and Applications of Analysis
Laguerre–Pólya_class
Jenkins–Traub algorithm — fast, reliable, and widely used Laguerre's method Splitting circle method Analysis: Wilkinson's polynomial Numerical continuation
List of numerical analysis topics
List_of_numerical_analysis_topics
The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and
Fokas_method
{\displaystyle \ s\ ,} using a root-finding algorithm such as Newton's method or Laguerre's method for a given time t . {\displaystyle \ t\ ~.} The value
Universal variable formulation
Universal_variable_formulation
Mathematical transformation
oriented sphere. This method was occasionally and implicitly employed by Lie (1871) himself and explicitly introduced by Laguerre (1880). In addition,
Spherical_wave_transformation
Book by Mary L. Boas
Mathematical Methods in the Physical Sciences is a 1966 textbook by mathematician Mary L. Boas intended to develop skills in mathematical problem-solving
Mathematical Methods in the Physical Sciences
Mathematical_Methods_in_the_Physical_Sciences
Generalization of gamma distribution to multiple dimensions
usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy
Wishart_distribution
Formal power series
1 + z𝓑t(z)t, so-termed tree polynomials, the Bell numbers, B(n), the Laguerre polynomials, and the Stirling convolution polynomials. Polynomials are
Generating_function
Imaging technique based on fluorescence
deconvolution based methods but they suffer from truncation and sampling problems. Moreover, application of methods like Laguerre gauss expansion is mathematically
Fluorescence-lifetime imaging microscopy
Fluorescence-lifetime_imaging_microscopy
Mathematical integral transform
_{0}^{\infty }g(y)K_{iy}(x)\sinh(\pi y)y\,dy.} Laguerre previously studied a similar transform regarding Laguerre function as: g ( y ) = ∫ 0 ∞ f ( x ) e − x
Kontorovich–Lebedev_transform
Methods of calculating definite integrals
integrals on the whole real line and Gauss-Laguerre quadrature for integrals on the positive reals. Monte Carlo methods can also be used, or a change of variables
Numerical_integration
Overseas department and region of France
vice-president of CACEM and member of the Executive Council of Martinique Didier Laguerre, Mayor of Fort-de-France, CACEM and Councillor to the Assembly of Martinique
Martinique
Form of Gaussian quadrature
M. (1964). "Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials". Math. Comp
Gauss–Hermite_quadrature
Relative distance of a point from a circle
minimizing the power is constant. More generally, French mathematician Edmond Laguerre defined the power of a point with respect to any algebraic curve in a similar
Power_of_a_point
Solution of a confluent hypergeometric equation
non-positive integer, then Kummer's function (if it is defined) is a generalized Laguerre polynomial. Just as the confluent differential equation is a limit of the
Confluent hypergeometric function
Confluent_hypergeometric_function
Technique in fluorescence microscopy
005123. PMID 12962391. Török, P.; Munro, P. R. T. (2004). "The use of Gauss-Laguerre vector beams in STED microscopy". Optics Express. 12 (15): 3605–3617. Bibcode:2004OExpr
STED_microscopy
Family of solutions to related differential equations
developed a method to find the zeros of the function. Leonhard Euler in 1736, found a link between other functions (now known as Laguerre polynomials)
Bessel_function
Mathematical model combining space and time
conformal group in four dimensions. The Lorentz group is isomorphic to the Laguerre group transforming planes into planes, it is isomorphic to the Möbius group
Spacetime
Blue dye also used as a medication
PMC 5979000. PMID 29690878. Hu X, Laguerre V, Packert D, Nakasone A, Moscinski L (2015). "A Simple and Efficient Method for Preparing Cell Slides and Staining
Methylene_blue
Quantum mechanical model
,} where Ln are the Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner
Quantum_harmonic_oscillator
Polynomial sequence
the nth-order Hermite function is related to the nth-order Laguerre polynomial. The Laguerre polynomials are L n ( x ) := ∑ k = 0 n ( n k ) ( − 1 ) k k
Hermite_polynomials
Annual selection of best U.S. team
national championship honors this year. Jenkins, Dan (September 11, 1967). Laguerre, André (ed.). "This Year The Fight Will Be In The Open". Sports Illustrated
College football national championships in NCAA Division I FBS
College_football_national_championships_in_NCAA_Division_I_FBS
Description of a quantum-mechanical system
{\displaystyle L_{n-\ell -1}^{2\ell +1}(\cdots )} are the generalized Laguerre polynomials of degree n − ℓ − 1 {\displaystyle n-\ell -1} , n , ℓ , m {\displaystyle
Schrödinger_equation
System of complete and orthogonal polynomials
the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line [ 0 , ∞ ) {\displaystyle
Legendre_polynomials
Device that emits light via optical amplification
with the transverse modes often approximated using Hermite–Gaussian or Laguerre-Gaussian functions. Some high-power lasers use a flat-topped profile known
Laser
Chinese-American engineer and professor (1904-1989)
transforms of Laguerre functions" in 1932. Lee, Y. W. (1930). Synthesis of electrical networks by means of the Fourier transforms of Laguerre's functions
Lee_Yuk-wing
Light-conducting fiber
Ventzislav; Wang, Siyang; Zhang, Xiaoshi (2022). "Dataset for Analytical Lah–Laguerre optical formalism for perturbative chromatic dispersion". Figshare. doi:10
Optical_fiber
2015 film by Pete Docter
March 28, 2026. "Inside Out". Box Office Mojo. Retrieved January 24, 2021. Laguerre-Lewis, Kayla (January 11, 2023). "Inside Out Has A Cute Monsters, Inc.
Inside_Out
Orbital mechanics term
S2CID 189832421. Conway, Bruce A. (1986). "An improved algorithm due to Laguerre for the solution of Kepler's equation". 24th Aerospace Sciences Meeting
Kepler's_equation
Formula for the Legendre polynomials
e^{-(x-u)^{2}}\\&=e^{2xu-u^{2}}\end{aligned}}} Source: For associated Laguerre polynomials L n ( α ) ( x ) = x − α e x n ! d n d x n ( e − x x n + α )
Rodrigues'_formula
Type of angular momentum in light
beams that are in a helical mode carry nonzero OAM. As an example, any Laguerre-Gaussian mode with rotational mode number l ≠ 0 {\displaystyle l\neq 0}
Orbital angular momentum of light
Orbital_angular_momentum_of_light
Complex-valued function
functions Laguerre polynomials § Hardy–Hille formula Hardy, G. H. (1932-07-01). "Addendum: Summation of a Series of Polynomials of Laguerre*". Journal
Mehler_kernel
Approximation of the definite integral of a function
{1-x^{2}}}} . One may also want to integrate over semi-infinite (Gauss–Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature). It can be
Gaussian_quadrature
Optical phenomenon
holograms (CGHs) are the calculated interferogram between a plane wave and a Laguerre-Gaussian beam which is transferred to film. The CGH resembles a common
Optical_vortex
Plane curve
Variation of the paper strip method 1 Animation of the variation of the paper strip method 1 Method 2 The second method starts with a strip of paper of
Ellipse
International rugby league tournament
Team Method of qualification Date of qualification Total times qualified Last time qualified Current consecutive appearances Previous best performance
2026 Men's Rugby League World Cup qualification
2026_Men's_Rugby_League_World_Cup_qualification
American mathematician and philosopher (1894–1964)
is the Hermite-Laguerre expansion. This was developed in detail in Nonlinear Problems in Random Theory. Wiener applied Hermite-Laguerre expansion to nonlinear
Norbert_Wiener
Curve from a cone intersecting a plane
Hartmann, Erich, Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes (PDF), retrieved 20 September 2014 (PDF; 891 kB)
Conic_section
Type of function
) d x . {\displaystyle \langle f,g\rangle =\int w(x)f(x)g(x)\,dx.} For Laguerre polynomials on ( 0 , ∞ ) {\displaystyle (0,\infty )} the weight function
Orthogonal_functions
American animated anthology series
Need You by Sean Godsey and Mike Rosenthal, I Love You Jocelyn by Tracey Laguerre (Art and Animation Director for brands like Google, DreamWorks Animation
What_a_Cartoon!
Area of combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various
Algebraic_combinatorics
Mathematical constant
ISSN 0945-3245. Csordas, G.; Ruttan, A.; Varga, R. S. (1991-06-01). "The Laguerre inequalities with applications to a problem associated with the Riemann
De_Bruijn–Newman_constant
Atom of the element hydrogen
In other places, the Laguerre polynomial includes a factor of ( n + ℓ ) ! {\displaystyle (n+\ell )!} , or the generalized Laguerre polynomial appearing
Hydrogen_atom
Geometry of the location of polynomial roots
developed, mainly for the method of continued fractions for real-root isolation. If all roots of a polynomial are real, Laguerre proved the following lower
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Kostka–Foulkes polynomial Mikhail Kravchuk: Kravchuk polynomial Edmond Laguerre: Laguerre polynomials Johann Heinrich Lambert: Lambert W function Gabriel Lamé:
List of eponyms of special functions
List_of_eponyms_of_special_functions
"New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials". pp. 2, 6. arXiv:1911.00186 [math.NT]. Comtet, Louis (1974)
List of logarithmic identities
List_of_logarithmic_identities
American judge (born 1951)
2009 bombing in the Bronx, New York. The suspects were Onta Williams, Laguerre, David Williams, and James Cromitie. The case involved FBI agent Robert
Colleen_McMahon
Novel type of computer memory
NASA's Jet Propulsion Laboratory (JPL) on improving methods of read out, including a novel method of non-destructive readout using pulses of UV radiation
Ferroelectric_RAM
Scientific instruments
Hermite-Gaussian beams (TEMxy), Laguerre-Gaussian (LG) beams (TEMpl) and Bessel beams. Optical tweezers based on Laguerre-Gaussian beams have the unique
Optical_tweezers
Wigner distribution function in physics as opposed to in signal processing
L m ( x ) {\displaystyle L_{m}(x)} denotes the m {\displaystyle m} -th Laguerre polynomial. This may follow from the expression for the static eigenstate
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
English art restorer and forger (1917–1984)
than an original Keating counterfeit." Major restoration of the Louis Laguerre murals depicting the Battle of Blenheim in the West Staircase at Marlborough
Tom_Keating
Plane curve: conic section
Moebius-, Laguerre- and Minkowski Planes, S. 33, (PDF; 757 kB) Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski
Hyperbola
Development of linear transformations forming the Lorentz group
hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group. In physics, Lorentz transformations
History of Lorentz transformations
History_of_Lorentz_transformations
Hungarian-American mathematician and educator (1887–1985)
1073/pnas.36.1.49. PMC 1063130. PMID 16588947. Integer-valued polynomial Laguerre–Pólya class Multivariate Pólya distribution Pólya's characterization theorem
George_Pólya
Probability distribution
indicates the square of the Laguerre polynomial L 1 / 2 ( ⋅ ) {\displaystyle L_{1/2}(\cdot )} , not the generalized Laguerre polynomial L 1 / 2 ( 2
Rice_distribution
Möbius transformation generalized to rings other than the complex numbers
\exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1.} Laguerre transformations Linear-fractional programming H-infinity methods in control theory N. J. Young (1984) "Linear
Linear fractional transformation
Linear_fractional_transformation
Proposed technological device
doughnut-shaped Laguerre-Gaussian laser beam, which has a high-intensity ring of light that surrounds a dark core along the beam axis. This method confines particles
Tractor_beam
Real numbers adjoined with a nil-squaring element
analysis Perturbation theory Infinitesimal Screw theory Dual-complex number Laguerre transformations Grassmann number Automatic differentiation Alexander J
Dual_number
Mapping involving integration between function spaces
kernel Circular convolution Circulant matrix Differential equations Kernel method List of transforms List of operators List of Fourier-related transforms
Integral_transform
1968), writer; author of Uñas pintadas de azul/Blue Fingernails Enrique A. Laguerre (1906–2005), writer; nominated in 1998, for the Nobel Prize in literature
List_of_Puerto_Ricans
Canonical solutions of the general Legendre equation
polynomials Spherical harmonics Whipple's transformation of Legendre functions Laguerre polynomials Hermite polynomials Courant & Hilbert 1953, V, §10. Abramowitz
Associated Legendre polynomials
Associated_Legendre_polynomials
Indian scientist
applications of Baker’s method to Diophantine equations and Ramanujan’s T-function. Shorey's contribution to irreducibility of Laguerre polynomials is extensive
Tarlok_Nath_Shorey
Model for the potential energy of a diatomic molecule
Morse potential can be found using operator methods. One approach involves applying the factorization method to the Hamiltonian. To write the stationary
Morse_potential
Locus of the zeros of a polynomial of degree two
Interactive Java 3D models of all quadric surfaces Lecture Note Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes, p. 117
Quadric
Mathematical description of quantum state
is also known as the Hartree–Fock method. The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater. Schrödinger
Wave_function
Branch of mathematics
geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties
Algebraic_geometry
Plane curve: conic section
Moebius-, Laguerre- and Minkowski-planes, p. 36. E. Hartmann, Lecture Note Planar Circle Geometries, an Introduction to Möbius-, Laguerre- and Minkowski
Parabola
Algebraic structure in linear algebra
elements in R2 and R4; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations. In 1857, Cayley
Vector_space
Equation for radii of tangent circles
4153/CJM-1962-042-6 Rigby, J. F. (1981), "The geometry of cycles, and generalized Laguerre inversion", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The
Descartes'_theorem
Schooley, New York, AIP, 421–427 (1982). Sawada, K.; T. Kotera (1974). "A method for finding N-soliton solutions of the KdV equation and KdV-like equation"
List of scientific equations named after people
List_of_scientific_equations_named_after_people
Mathematical metric in geometry
the measure on a line. Another important insight was the Laguerre formula by Edmond Laguerre (1853), who showed that the Euclidean angle between two lines
Cayley–Klein_metric
Atoms with a single valence electron, so they behave like hydrogen
ℓ + 1 ) {\displaystyle L_{n-\ell -1}^{(2\ell +1)}} are the generalized Laguerre polynomials. a μ = 4 π ε 0 ℏ 2 μ e 2 = ℏ c α μ c 2 = m e μ a 0 {\displaystyle
Hydrogen-like_atom
Q-analog of the ordinary derivative
(1998). Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order difference equation for the rth associated and the Laguerre-Freud
Q-derivative
Quasi-arithmetic mean Classical orthogonal polynomials Hermite polynomials Laguerre polynomials Jacobi polynomials Gegenbauer polynomials Legendre polynomials
List_of_real_analysis_topics
Discipline related to neuroscience and biomedical engineering
Marmarelis, V. Z. (1993). IDENTIFICATION OF NONLINEAR BIOLOGICAL-SYSTEMS USING LAGUERRE EXPANSIONS OF KERNELS. [Article]. Annals of Biomedical Engineering, 21(6)
Neuroprosthetics
Physical quantity carried in photons
OAM can also be generated by converting a Hermite-Gaussian beam into a Laguerre-Gaussian one by using an astigmatic system with two well-aligned cylindrical
Angular_momentum_of_light
Town in Devon, England
plaque. Benjamin Haughton (1865–1924) landscape artist, lived locally André Laguerre (1915–1979), journalist and magazine editor of Sports Illustrated, 1960
Ottery_St_Mary
American Jewish mathematician
49–113, Bibcode:1987AnPhy.176...49K, doi:10.1016/0003-4916(87)90178-3 "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary
Bertram_Kostant
Ethnic group in Benin Republic
(in Spanish: The Sub-Saharan element in the Venezuelan lexicon) Michael Laguerre (1973), The place of Voodoo in the Social Structure of Haiti, Caribbean
Fon_people
Harassment occurring in a public setting
about after many people were outraged at a man attacking a woman (Marie Laguerre) due to her response to his harassment of her. In 2017, the Dutch cities
Street_harassment
507–513 E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), S. 121-123.
Ovoid_(projective_geometry)
Quantum mechanics with supersymmetry
differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The outcome is the spectrum of hydrogen-atom energy states
Supersymmetric quantum mechanics
Supersymmetric_quantum_mechanics
Formalized folk medicine
MEDICINE: NIGERIA The role of traditional medicine" (PDF). The Lancet. Laguerre, Michel S. (1987). Afro-Caribbean folk medicine. New York: Bergin & Garvey
Traditional_medicine
form of PBE. Monte Carlo methods , discretization methods, numerical techniques like Finite Volume Method and moment methods are mainly used to solve
Population_balance_equation
Norwegian physicist (1898–1985)
October 28, 1965) was a Norwegian theoretical physicist known for creating a method for predicting the ground state energy of two-electron atoms and trial wave
Egil_Hylleraas
conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane
Steiner_conic
Formulation of quantum mechanics
equation in the form of a modified Laguerre equation (not Hermite's equation!), the solution of which involves the Laguerre polynomials as F n ( u ) = ( −
Phase-space_formulation
Family of power series in mathematics
{}_{1}F_{1}(-n;b;z)} is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms
Generalized hypergeometric function
Generalized_hypergeometric_function
Canadian mathematician
mathematician at the University of New Brunswick who introduced Weisner's method. He graduated in 1923 from Columbia University with a Ph.D. in mathematics
Louis_Weisner
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