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Gauss–Laguerre Quadrature
In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: :\int_^ e^ f(x)\,dx. In this case :\int_^ e^ f(x)\,dx \approx \sum_^n w_i f(x_i) where ''x''''i'' is the ''i''-th root of Laguerre polynomial ''L''''n''(''x'') and the weight ''w''''i'' is given byEquation 25.4.45 in 10th reprint with corrections. :w_i = \frac . The following Python code with the SymPy library will allow for calculation of the values of x_i and w_i to 20 digits of precision: from sympy import * def lag_weights_roots(n): x = Symbol("x") roots = Poly(laguerre(n, x)).all_roots() x_i = t.evalf(20) for rt in roots w_i = rt / ((n + 1) * laguerre(n + 1, rt)) ** 2).evalf(20) for rt in roots return x_i, w_i print(lag_weights_roots(5)) For more general functions To integrate the function f we apply the following transformation : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmond Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He laid the foundations of a geometry of oriented spheres (Laguerre geometry and Laguerre plane), including the Laguerre transformation or transformation by reciprocal directions. Works Selection * * * * Théorie des équations numériques', Paris: Gauthier-Villars. 1884 on Google Books * * Oeuvres de Laguerrepubl. sous les auspices de l'Académie des sciences par MM. Charles Hermite, Henri Poincaré, et Eugène Rouché.'' (Paris, 1898-1905) (reprint: New York : Chelsea publ., 1972 ) Extensive lists More than 80 articleson Nundam.org.p See also * Isotropic line * ''q''-Laguerre polynomials * Big ''q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for . The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken as , so the rule is stated as :\int_^1 f(x)\,dx \approx \sum_^n w_i f(x_i), which is exact for polynomials of degree or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if is well-approximated by a polynomial of degree or less on . The Gaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Polynomial
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues formula, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover
Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidstone. The town is the administrative centre of the Dover District and home of the Port of Dover. Archaeological finds have revealed that the area has always been a focus for peoples entering and leaving Great Britain, Britain. The name derives from the River Dour that flows through it. In recent times the town has undergone transformations with a high-speed rail link to London, new retail in town with St James' area opened in 2018, and a revamped promenade and beachfront. This followed in 2019, with a new 500m Pier to the west of the Harbour, and new Marina unveiled as part of a £330m investment in the area. It has also been a point of destination for many illegal migrant crossings during the English Channel migrant crossings (2018-present) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues formula, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Tables And Other Aids To Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Computational And Applied Mathematics
The ''Journal of Computational and Applied Mathematics'' is a peer-reviewed scientific journal covering computational and applied mathematics. It was established in 1975 and is published biweekly by Elsevier. The editors-in-chief are Yalchin Efendiev (Texas A&M University), Taketomo Mitsui (Nagoya University), Michael Kwok-Po Ng (Hong Kong Baptist University) and Fatih Tank (Ankara University). According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 2.872. References External links * {{Mathematics-journal-stub Biweekly journals Applied mathematics Mathematics journals Elsevier academic journals Publications established in 1975 English-language journals Computational mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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