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Gaussian Quadrature
In numerical analysis, 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 Gauss–Adrien-Marie Legendre, Legendre quadrature rule is not typically used for integrable functions with endpoint singularity (math), singularities. Instead, if the integrand can be written as f(x) = \left(1 - x\right)^\alpha \left(1 + x\right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Gaussquad Trapezoidal
Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are Similarity (psychology), similar to the other, which are Difference (philosophy), different, and to what degree. Where characteristics are different, the differences may then be evaluated to determine which thing is best suited for a particular purpose. The description of similarities and differences found between the two things is also called a comparison. Comparison can take many distinct forms, varying by field: To compare things, they must have characteristics that are similar enough in relevant ways to merit comparison. If two things are too different to compare in a useful way, an attempt to compare them is colloquially referred to in English as "comparing apples and oranges." Comparison is widely used in society, in science and the arts. General usage Comparison is a natural act ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions. Definition and representation Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval [-1,1]. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similar Matrices
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two possibly different bases, with being the change-of-basis matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in . Motivating example When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tridiagonal Matrix
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main diagonal). For example, the following matrix is tridiagonal: :\begin 1 & 4 & 0 & 0 \\ 3 & 4 & 1 & 0 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 \\ \end. The determinant of a tridiagonal matrix is given by the ''continuant'' of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be done with the Lanczos algorithm. Properties A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. In particular, a tridiagonal matrix is a direct sum of ''p'' 1-by-1 and ''q'' 2-by-2 matrices such that — the dimension of the tridiagonal. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monic Polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one that can be written as :x^n+c_x^+\cdots+c_2x^2+c_1x+c_0, with n \geq 0. Uses Monic polynomials are widely used in algebra and number theory, since they produce many simplifications and they avoid divisions and denominators. Here are some examples. Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: ''Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.'' Vieta's formulas are simpler in the case of monic polynomials: ''The th elementary symmetric function of the roots of a monic polynomial of degree equals (-1)^ic_, where c_ is the coefficient of the th po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ... to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Polynomials
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree of a polynomial, degree that polynomial interpolation, interpolates a given set of data. Given a data set of graph of a function, coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes formulas, Newton–Cotes method of numerical integration, Shamir's Secret Sharing, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: xy'' + (1 - x)y' + ny = 0,\ y = y(x) 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 Gauss–Laguerre quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , ..., are a polynomial sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Polynomials
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind T_n are defined by T_n(\cos \theta) = \cos(n\theta). Similarly, the Chebyshev polynomials of the second kind U_n are defined by U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big). That these expressions define polynomials in \cos\theta is not obvious at first sight but can be shown using de Moivre's formula (see below). The Chebyshev polynomials are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
