quadrature spectrum - ορισμός. Τι είναι το quadrature spectrum
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Τι (ποιος) είναι quadrature spectrum - ορισμός

NUMERICAL INTEGRATION
Gaussian integration; Gaussian numerical integration; Gauss quadrature; Gauss legendre quadrature; Gaussian Quadrature; Gauss–Lobatto quadrature; Gauss-Lobatto quadrature
  • 2}} – 3''x'' + 3}}), the 2-point Gaussian quadrature rule even returns an exact result.
  • ''n'' {{=}} 5)}}

Spectrum (functional analysis)         
TERM USED IN FUNCTIONAL ANALYSIS
Approximate eigenvalue; Operator spectrum; Spectrum of an operator; Compression spectrum; Spectral representation; Point spectrum; Continuous spectrum (functional analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if T-\lambda I is not invertible, where I is the identity operator.
spectrum         
  • blue rightism]]) coding
CONTINUOUS RANGE OF VALUES, SUCH AS WAVELENGTHS IN PHYSICS
Energy spectrum; Spectracular; Energy spectra; Spectroscopic observations; Spectral density (physical science); Spectrum of disease; Spectrum (physics)
(spectra, or spectrums)
1.
The spectrum is the range of different colours which is produced when light passes through a glass prism or through a drop of water. A rainbow shows the colours in the spectrum.
N-SING: the N
2.
A spectrum is a range of a particular type of thing.
Politicians across the political spectrum have denounced the act...
The term 'special needs' covers a wide spectrum of problems.
N-COUNT: usu sing, with supp
3.
A spectrum is a range of light waves or radio waves within particular frequencies.
Vast amounts of energy, from X-rays right through the spectrum down to radio waves, are escaping into space...
N-COUNT
spectrum         
  • blue rightism]]) coding
CONTINUOUS RANGE OF VALUES, SUCH AS WAVELENGTHS IN PHYSICS
Energy spectrum; Spectracular; Energy spectra; Spectroscopic observations; Spectral density (physical science); Spectrum of disease; Spectrum (physics)
n.
Image, appearance, representation.

Βικιπαίδεια

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 n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, …, n. 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 [−1, 1], so the rule is stated as

1 1 f ( x ) d x i = 1 n w i f ( x i ) , {\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i}),}

which is exact for polynomials of degree 2n − 1 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 f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].

The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as

f ( x ) = ( 1 x ) α ( 1 + x ) β g ( x ) , α , β > 1 , {\displaystyle f(x)=\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x),\quad \alpha ,\beta >-1,}

where g(x) is well-approximated by a low-degree polynomial, then alternative nodes xi' and weights wi' will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e.,

1 1 f ( x ) d x = 1 1 ( 1 x ) α ( 1 + x ) β g ( x ) d x i = 1 n w i g ( x i ) . {\displaystyle \int _{-1}^{1}f(x)\,dx=\int _{-1}^{1}\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x)\,dx\approx \sum _{i=1}^{n}w_{i}'g\left(x_{i}'\right).}

Common weights include 1 1 x 2 {\textstyle {\frac {1}{\sqrt {1-x^{2}}}}} (Chebyshev–Gauss) and 1 x 2 {\displaystyle {\sqrt {1-x^{2}}}} . One may also want to integrate over semi-infinite (Gauss-Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature).

It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.