Other Forms
The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand, and allowing an interval other than . That is, the problem is to calculate
for some choices of a, b, and ω. For a = −1, b = 1, and ω(x) = 1, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for Abramowitz and Stegun (A & S).
| Interval | ω(x) | Orthogonal polynomials | A & S | For more information, see ... |
|---|---|---|---|---|
| Legendre polynomials | 25.4.29 | Section Gauss–Legendre quadrature, above | ||
| (−1, 1) | Jacobi polynomials | 25.4.33 | Gauss–Jacobi quadrature | |
| (−1, 1) | Chebyshev polynomials (first kind) | 25.4.38 | Chebyshev–Gauss quadrature | |
| Chebyshev polynomials (second kind) | 25.4.40 | Chebyshev–Gauss quadrature | ||
| [0, ∞) | Laguerre polynomials | 25.4.45 | Gauss–Laguerre quadrature | |
| [0, ∞) | Generalized Laguerre polynomials | Gauss–Laguerre quadrature | ||
| (−∞, ∞) | Hermite polynomials | 25.4.46 | Gauss–Hermite quadrature |
Read more about this topic: Gaussian Quadrature
Famous quotes containing the word forms:
“How superbly brave is the Englishman in the presence of the awfulest forms of danger & death; & how abject in the presence of any & all forms of hereditary rank.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed.”
—Thomas Jefferson (1743–1826)