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Clenshaw–Curtis Quadrature
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the Integrand#Terminology and notation, integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = \cos \theta and use a discrete cosine transform (DCT) approximation for the cosine series. Besides having fast-converging accuracy comparable to Gaussian quadrature rules, Clenshaw–Curtis quadrature naturally leads to nested quadrature rules (where different accuracy orders share points), which is important for both adaptive quadrature and multidimensional quadrature (cubature). Briefly, the function (mathematics), function f(x) to be integrated is evaluated at the N extrema or roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function. This polynomial is then integrated exactly. In practice, the integration weights for the value of the function at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ('' quadrature'' or ''squaring''), as in the quadrature of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliasing
In signal processing and related disciplines, aliasing is a phenomenon that a reconstructed signal from samples of the original signal contains low frequency components that are not present in the original one. This is caused when, in the original signal, there are components at frequency exceeding a certain frequency called Nyquist frequency, f_s / 2, where f_s is the sampling frequency ( undersampling). This is because typical reconstruction methods use low frequency components while there are a number of frequency components, called aliases, which sampling result in the identical sample. It also often refers to the distortion or artifact that results when a signal reconstructed from samples is different from the original continuous signal. Aliasing can occur in signals sampled in time, for instance in digital audio or the stroboscopic effect, and is referred to as temporal aliasing. Aliasing in spatially sampled signals (e.g., moiré patterns in digital images) is referre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine Wave
A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple harmonic motion''; as rotation, it corresponds to ''uniform circular motion''. Sine waves occur often in physics, including wind waves, sound waves, and light waves, such as monochromatic radiation. In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same frequency (but arbitrary phase (waves), phase) are linear combination, linearly combined, the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Grid
Sparse is a computer software tool designed to find possible coding faults in the Linux kernel. Unlike other such tools, this static analysis tool was initially designed to only flag constructs that were likely to be of interest to kernel developers, such as the mixing of pointers to user and kernel address spaces. Sparse checks for known problems and allows the developer to include annotations in the code that convey information about data types, such as the address space that pointers point to and the locks that a function acquires or releases. Linus Torvalds started writing Sparse in 2003. Josh Triplett was its maintainer from 2006, a role taken over by Christopher Li in 2009 and by Luc Van Oostenryck in November 2018. Sparse is released under the MIT License. Annotations Some of the checks performed by Sparse require annotating the source code using the __attribute__ GCC extension, or the Sparse-specific __context__ specifier. Sparse defines the following list of attri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Kronrod Quadrature Formula
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order (the latter called an ''embedded'' rule). The difference between these two approximations is used to estimate the calculational error of the integration. These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss. Description The problem in numerical integration is to approximate definite integrals of the form :\int_a^b f(x)\,dx. Such integrals can be approximated, for example, by ''n''-point Gaussian quadrature :\int_a^b f(x)\,dx \approx \sum_^n w_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nested Quadrature Rule
''Nested'' is the seventh studio album by Bronx-born singer, songwriter, and pianist Laura Nyro. It was released in 1978 on Columbia Records. Following on from her extensive tour to promote 1976's ''Smile (Laura Nyro album), Smile'', which resulted in the 1977 live album ''Season of Lights'', Nyro retreated to her new home in Danbury, Connecticut, where she lived after spending her time in the spotlight in New York City. Nyro had a studio built at her home and recorded there the songs that comprised ''Nested''. The songs deal with themes such as motherhood and womanhood, and Nyro is notably more relaxed in her singing on the album. The instrumentation is laid back and smooth, similar to that of ''Smile'', but perhaps less jazz-inspired and more melodic. Nyro was assisted in production by Roscoe Harring, while Dale and Pop Ashby were chief engineers. Critics praised the album as a melodic return to form, and Nyro supported the album with a solo tour when she was heavily pregnant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see Order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Nodes
In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the Projection (linear algebra), projection of a set of equispaced points on the unit circle onto the real interval [-1, 1], the circle's diameter. There are two kinds of Chebyshev nodes. The ''Chebyshev nodes of the first kind'', also called the Chebyshev–Gauss nodes or Chebyshev zeros, are the Zero of a function, zeros of a Chebyshev polynomial of the first kind, . The corresponding ''Chebyshev nodes of the second kind'', also called the Chebyshev–Lobatto nodes or Chebyshev extrema, are the Maximum and minimum, extrema of , which are also the zeros of a Chebyshev polynomial of the second kind, , along with the two endpoints of the interval. Both types of numbers are commonly referred to as ''Chebyshev nodes'' or ''Chebyshev points'' in literature. They are named aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipót Fejér
Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography He was born in Pécs, Austria-Hungary, into the Jewish family of Victoria Goldberger and Samuel Weiss. His maternal great-grandfather Samuel Nachod was a doctor and his grandfather was a renowned scholar, author of a Hebrew-Hungarian dictionary. Leopold's father, Samuel Weiss, was a shopkeeper in Pecs. In primary schools Leopold was not doing well, so for a while his father took him away to home schooling. The future scientist developed his interest in mathematics in high school thanks to his teacher Sigismund Maksay. Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin, where he was taught by Hermann Schwarz. In 1902 he earned his doctorate from University of Budapest (today Eötvös Loránd University). From 190 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Functions
In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is even if ''n'' is an even integer, and it is odd if ''n'' is an odd integer. Even functions are those real functions whose graph is self-symmetric with respect to the and odd functions are those whose graph is self-symmetric with respect to the origin. If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function. Early history The concept of even and odd functions appears to date back to the early 18th century, with Leonard Euler playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Approximation
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
