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1: 19.8 Quadratic Transformations
§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
As n , a n and g n converge to a common limit M ( a 0 , g 0 ) called the AGM (Arithmetic-Geometric Mean) of a 0 and g 0 . …showing that the convergence of c n to 0 and of a n and g n to M ( a 0 , g 0 ) is quadratic in each case. …
19.8.10 p 0 2 = 1 - ( k 2 / α 2 ) .
2: 15.17 Mathematical Applications
Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
3: 22.20 Methods of Computation
§22.20(ii) Arithmetic-Geometric Mean
Then as n sequences { a n } , { b n } converge to a common limit M = M ( a 0 , b 0 ) , the arithmetic-geometric mean of a 0 , b 0 . … The rate of convergence is similar to that for the arithmetic-geometric mean. … using the arithmetic-geometric mean. … Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute am ( x , k ) . …
4: 19.22 Quadratic Transformations
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
19.22.9 4 π R G ( 0 , a 0 2 , g 0 2 ) = 1 M ( a 0 , g 0 ) ( a 0 2 - n = 0 2 n - 1 c n 2 ) = 1 M ( a 0 , g 0 ) ( a 1 2 - n = 2 2 n - 1 c n 2 ) ,
19.22.10 R D ( 0 , g 0 2 , a 0 2 ) = 3 π 4 M ( a 0 , g 0 ) a 0 2 n = 0 Q n ,
19.22.12 R J ( 0 , g 0 2 , a 0 2 , p 0 2 ) = 3 π 4 M ( a 0 , g 0 ) p 0 2 n = 0 Q n ,
5: 23.22 Methods of Computation
  • (a)

    In the general case, given by c d 0 , we compute the roots α , β , γ , say, of the cubic equation 4 t 3 - c t - d = 0 ; see §1.11(iii). These roots are necessarily distinct and represent e 1 , e 2 , e 3 in some order.

    If c and d are real, then e 1 , e 2 , e 3 can be identified via (23.5.1), and k 2 , k 2 obtained from (23.6.16).

    If c and d are not both real, then we label α , β , γ so that the triangle with vertices α , β , γ is positively oriented and [ α , γ ] is its longest side (chosen arbitrarily if there is more than one). In particular, if α , β , γ are collinear, then we label them so that β is on the line segment ( α , γ ) . In consequence, k 2 = ( β - γ ) / ( α - γ ) , k 2 = ( α - β ) / ( α - γ ) satisfy k 2 0 k 2 (with strict inequality unless α , β , γ are collinear); also | k 2 | , | k 2 | 1 .

    Finally, on taking the principal square roots of k 2 and k 2 we obtain values for k and k that lie in the 1st and 4th quadrants, respectively, and 2 ω 1 , 2 ω 3 are given by

    23.22.1 2 ω 1 M ( 1 , k ) = - 2 i ω 3 M ( 1 , k ) = π 3 c ( 2 + k 2 k 2 ) ( k 2 - k 2 ) d ( 1 - k 2 k 2 ) ,

    where M denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 2 possible choices of the square root.

  • 6: Bibliography C
  • D. A. Cox (1984) The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.
  • D. A. Cox (1985) Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.
  • 7: Bibliography
  • G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π , and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.
  • 8: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • 9: 15.8 Transformations of Variable
    This is used in a cubic analog of the arithmetic-geometric mean. …