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:
“When we speak the word “life,” it must be understood we are not referring to life as we know it from its surface of fact, but to that fragile, fluctuating center which forms never reach.”
—Antonin Artaud (1896–1948)
“Of the three forms of pride, that is to say pride proper, vanity, and conceit, vanity is by far the most harmless, and conceit by far the most dangerous. The meaning of vanity is to think too much of our bodily advantages, whether real or unreal, over others; while the meaning of conceit is to believe we are cleverer, wiser, grander, and more important than we really are.”
—John Cowper Powys (1872–1963)