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Gauss–Legendre Quadrature
In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval , the rule takes the form: :\int_^1 f(x)\,dx \approx \sum_^n w_i f(x_i) : where * ''n'' is the number of sample points used, * ''w''''i'' are quadrature weights, and * ''x''''i'' are the roots of the ''n''th Legendre polynomial. This choice of quadrature weights ''w''''i'' and quadrature nodes ''x''''i'' is the unique choice that allows the quadrature rule to integrate degree polynomials exactly. Many algorithms have been developed for computing Gauss–Legendre quadrature rules. The Golub–Welsch algorithm presented in 1969 reduces the computation of the nodes and weights to an eigenvalue problem which is solved by the QR algorithm. This algorithm was popular, but significantly more efficient algorithms exist. Algorithms based on the Newton–Raphson method are able to compute quadrature rules for signific ... [...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 to find 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 like economics, 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 simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Biography Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of a banker, Simon Jacobi. His elder brother, Moritz, would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathematics, sciences, etc. As a result of the good education he had received from his uncle, as well as his own remarkable abilities, after less than half a year Jacobi was moved to the senior year despite his young age. However, as the Unive ... [...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] |
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] |
Arbitrary-precision Arithmetic
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are potentially limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision. Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of bits related to the size of the processor register, these implementations typically use variable-length arrays of digits. Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should ... [...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] |
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] |
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] |
Continued Fraction (non-simple)
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fraction is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
