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1: 3.8 Nonlinear Equations
Bisection Method
The convergence of iterative methods …
2: 19.22 Quadratic Transformations
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 ,
19.22.15 p 0 2 = a 0 2 ( q 0 2 + g 0 2 ) / ( q 0 2 + a 0 2 ) .
3: 19.8 Quadratic Transformations
When a 0 and g 0 are positive numbers, define …
19.8.2 c n = a n 2 - g n 2 .
19.8.3 c n + 1 = a n - g n 2 = c n 2 4 a n + 1 ,
19.8.4 1 M ( a 0 , g 0 ) = 2 π 0 π / 2 d θ a 0 2 cos 2 θ + g 0 2 sin 2 θ = 1 π 0 d t t ( t + a 0 2 ) ( t + g 0 2 ) .
19.8.6 E ( k ) = π 2 M ( 1 , k ) ( a 0 2 - n = 0 2 n - 1 c n 2 ) = K ( k ) ( a 1 2 - n = 2 2 n - 1 c n 2 ) , - < k 2 < 1 , a 0 = 1 , g 0 = k ,
4: 29.20 Methods of Computation
A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …
5: 19.24 Inequalities
19.24.5 1 a n 2 π R F ( 0 , a 0 2 , g 0 2 ) 1 g n , n = 0 , 1 , 2 , ,
19.24.9 1 2 g 1 2 R G ( a 0 2 , g 0 2 , 0 ) R F ( a 0 2 , g 0 2 , 0 ) 1 2 a 1 2 ,
6: 3.9 Acceleration of Convergence
When applied repeatedly, Aitken’s process is known as the iterated Δ 2 -process. … …
§3.9(v) Levin’s and Weniger’s Transformations
7: 22.20 Methods of Computation
Four iterations of (22.20.1) lead to c 4 = 6.5×10⁻¹² . … If k = k = 1 / 2 , then three iterations of (22.20.1) give M = 0.84721 30848 , and from (22.20.6) K = π / ( 2 M ) = 1.85407 46773 — in agreement with the value of ( Γ ( 1 4 ) ) 2 / ( 4 π ) ; compare (23.17.3) and (23.22.2). … If k = 1 - i , then four iterations of (22.20.1) give K = 1.23969 74481 + i 0.56499 30988 . …
8: 17.12 Bailey Pairs
When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …
9: 18.20 Hahn Class: Explicit Representations
10: Bibliography J
  • G. Julia (1918) Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).