|
Gauss–Hermite Quadrature
In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature 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 ''n'' is the number of sample points used. The ''x''''i'' are the roots of the physicists' version of the Hermite polynomial ''H''''n''(''x'') (''i'' = 1,2,...,''n''), and the associated weights ''w''''i'' are given by Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'', 10th printing with corrections (1972), Dover, . Equation 25.4.46. :w_i = \frac . Example with change of variable Consider a function ''h(y)'', where the variable ''y'' is Normally distributed: y \sim \mathcal(\mu,\sigma^2). The expectation of ''h'' corresponds to the following integral: E(y)= \int_^ \frac \exp \left( -\frac \right) h(y) dy As this does not exactly correspond to the Hermite polynomial, we need to change variables: x = \frac \Leftrightarrow ... [...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] |
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] |
