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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. … Again, p n and ε n converge quadratically to M ( a 0 , g 0 ) and 0, respectively, and Q n converges to 0 faster than quadratically. …
2: 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 . …
22.20.2 max ( | a n M | , | b n M | , | c n | ) (const.) × 2 2 n ,
K = π 2 M ( 1 , k ) ,
using the arithmetic-geometric mean. …
3: 19.22 Quadratic Transformations
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
The AGM, M ( a 0 , g 0 ) , of two positive numbers a 0 and g 0 is defined in §19.8(i). …
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 ) ,
As n , p n and ε n converge quadratically to M ( a 0 , g 0 ) and 0, respectively, and Q n converges to 0 faster than quadratically. …
4: 1.7 Inequalities
§1.7(iii) Means
1.7.7 H G A ,
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, and the discriminant is positive, that is c 3 27 d 2 > 0 , 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 3 27 d 2 < 0 , or 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: 15.17 Mathematical Applications
    Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean19.22(ii)). … …
    7: 1.2 Elementary Algebra
    §1.2(iv) Means
    The geometric mean G and harmonic mean H of n positive numbers a 1 , a 2 , , a n are given by
    1.2.18 G = ( a 1 a 2 a n ) 1 / n ,
    1.2.26 lim r 0 M ( r ) = G .
    8: 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.
  • 9: 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.
  • 10: 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.