| 单词 | newton's formula |
| 释义 | > as lemmasNewton's formula Newton's equation n. Optics an equation which states the relationship between the focal length of a lens and the distances of an object plane and its image plane from their respective foci; similarly Newton's formula.extracted from Newtonn.Newton's formula 1. Mathematics. Used attributively and in the genitive with reference to various methods of numerical analysis developed by Newton or based on his work, esp. methods for approximating the solution of an equation and for approximating the integral of a function; as Newton formula (also Newton's formula) (see also sense 2d), Newton process (usually with reference to the Newton–Raphson method), Newton's method, Newton's rule. Cf. Newton–Cotes n., Newton–Raphson n.Newton's method of approximating the solution of an equation involved the same reasoning as the Newton–Raphson method, but was procedurally different.Newton's method of approximating the integral of a function over a finite range involved evaluating the function at each of a series of arguments in the range, and finding a polynomial function with the same value at each of the arguments. ΚΠ 1685 J. Wallis Treat. Algebra lxxxv. 318 Another Method of Approximation, by Sir Isaac Newton. 1695 J. Wallis in Philos. Trans. (Royal Soc.) 19 2 Mr Newton's Method of Approximation for the Extraction of Roots.] 1779 Philos. Trans. 1778 (Royal Soc.) 68 382 The theorem..is not general, though applied in the eleventh chapter to the demonstration of Newton's rule for finding impossible roots in all equations. a1832 A. De Morgan Calculus of Functions in Encycl. Metrop. (1845) II. 317/1 This theorem is used in a less defined form throughout every part of the practice of Algebra, from Division and Involution to Newton's rule for approximating to the roots of equations. 1881 Amer. Jrnl. Math. 4 275 Newton's method of approximation consists in adding to an approximate value, a, of a root, the correction k = −f(a)/f′(a). 1907 Ann. Math. 9 8 This formula is called by Lefort ‘Mouton's Interpolation Formula’, but is an immediate consequence of Newton's formula. 1967 E. U. Condon & H. Odishaw Handbk. Physics (ed. 2) i. vii. 102/1 However, there is a drawback to the Newton process: If f(z) has real coefficients and z0 is real, all zi will be real and we cannot get to a complex zero. 1968 L. Fox & D. F. Mayers Computing Methods for Scientists & Engineers viii. 161 The Newton formula might best be used near the ends of a range, whenever the central differences are not available. 1994 D. W. Jordan & P. Smith Math. Techniques xxxv. 603 Set up a Newton routine xn+1 = xn − [f(xn)/f′(xn)], for calculating the roots of f(x) = 0, and find, starting at x = 0.5 and 1.6, the roots to ten significant figures. < as lemmas |
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