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Boole's Rule
In mathematics, Boole's rule, named after George Boole, is a method of numerical integration. Formula Simple Boole's Rule It approximates an integral: : \int_^ f(x)\,dx by using the values of at five equally spaced points: : \begin & x_0 = a\\ & x_1 = x_1 + h \\ & x_2 = x_1 + 2h \\ & x_3 = x_1 + 3h \\ & x_4 = x_1 +4h = b \end It is expressed thus in Abramowitz and Stegun: : \int_^ f(x)\,dx = \frac\bigl 7f(x_0) + 32 f(x_1) + 12 f(x_2) + 32 f(x_3) + 7f(x_4) \bigr+ \text where the error term is : -\,\frac for some number between and where . It is often known as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun. The following constitutes a very simple implementation of the method in Common Lisp which ignores the error term: (defun integrate-booles-rule (f x1 x5) "Calculates the Boole's rule numerical integral of the function F in the closed interval extending from inclusive X1 to inclusive X5 without error term inclusi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Titius–Bode Law
The Titius–Bode law (sometimes termed just Bode's law) is a formulaic prediction of spacing between planets in any given solar system. The formula suggests that, extending outward, each planet should be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit. It is named after Johann Daniel Titius and Johann Elert Bode. Later work by Blagg and Richardson significantly revised the original formula, and made predictions that were subsequently validated by new discoveries and observations. It is these re-formulations that offer "the best phenomenological representations of distances with which to investigate the theoretical significance of Titius–Bode type Laws". Original formulation The law relates the semi-major axis ~a_n~ of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10: :~a = 4 + x~ where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
