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1: 3.9 Acceleration of Convergence
§3.9(iii) Aitken’s Δ 2 -Process
When applied repeatedly, Aitken’s process is known as the iterated Δ 2 -process. … … Shanks’ transformation is a generalization of Aitken’s Δ 2 -process. … Aitken’s Δ 2 -process is the case k = 1 . …
2: 20 Theta Functions
Chapter 20 Theta Functions
3: 28.1 Special Notation
Ce ν ( z , q ) , Se ν ( z , q ) , Fe n ( z , q ) , Ge n ( z , q ) ,
in n = fe n , ceh n = Ce n , inh n = Fe n ,
Abramowitz and Stegun (1964, Chapter 20)
4: 2.11 Remainder Terms; Stokes Phenomenon
Similar improvements are achievable by Aitken’s Δ 2 -process, Wynn’s ϵ -algorithm, and other acceleration transformations. … For example, using double precision d 20 is found to agree with (2.11.31) to 13D. …
5: 28.35 Tables
  • Ince (1932) includes eigenvalues a n , b n , and Fourier coefficients for n = 0 or 1 ( 1 ) 6 , q = 0 ( 1 ) 10 ( 2 ) 20 ( 4 ) 40 ; 7D. Also ce n ( x , q ) , se n ( x , q ) for q = 0 ( 1 ) 10 , x = 1 ( 1 ) 90 , corresponding to the eigenvalues in the tables; 5D. Notation: a n = 𝑏𝑒 n 2 q , b n = 𝑏𝑜 n 2 q .

  • Kirkpatrick (1960) contains tables of the modified functions Ce n ( x , q ) , Se n + 1 ( x , q ) for n = 0 ( 1 ) 5 , q = 1 ( 1 ) 20 , x = 0.1 ( .1 ) 1 ; 4D or 5D.

  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues a n ( q ) , b n + 1 ( q ) for n = 0 ( 1 ) 4 , q = 0 ( 1 ) 50 ; n = 0 ( 1 ) 20 ( a ’s) or 19 ( b ’s), q = 1 , 3 , 5 , 10 , 15 , 25 , 50 ( 50 ) 200 . Fourier coefficients for ce n ( x , 10 ) , se n + 1 ( x , 10 ) , n = 0 ( 1 ) 7 . Mathieu functions ce n ( x , 10 ) , se n + 1 ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , x = 0 ( 5 ) 90 . Modified Mathieu functions Mc n ( j ) ( x , 10 ) , Ms n + 1 ( j ) ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , j = 1 , 2 , x = 0 ( .2 ) 4 . Precision is mostly 9S.

  • 6: 8 Incomplete Gamma and Related
    Functions
    7: 28 Mathieu Functions and Hill’s Equation
    8: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 9: 23 Weierstrass Elliptic and Modular
    Functions
    10: 36 Integrals with Coalescing Saddles