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1: 4.43 Cubic Equations
§4.43 Cubic Equations
2: 16.6 Transformations of Variable
Cubic
3: 1.11 Zeros of Polynomials
Cubic Equations
For the roots α 1 , α 2 , α 3 , α 4 of g ( w ) = 0 and the roots θ 1 , θ 2 , θ 3 of the resolvent cubic equation
1.11.20 θ 1 θ 2 θ 3 = q .
Resolvent cubic is z 3 + 12 z 2 + 20 z + 9 = 0 with roots θ 1 = 1 , θ 2 = 1 2 ( 11 + 85 ) , θ 3 = 1 2 ( 11 85 ) , and θ 1 = 1 , θ 2 = 1 2 ( 17 + 5 ) , θ 3 = 1 2 ( 17 5 ) . …
4: 3.8 Nonlinear Equations
3.8.3 | z n + 1 ζ | < A | z n ζ | p
If p = 2 , then the convergence is quadratic; if p = 3 , then the convergence is cubic, and so on. … The rule converges locally and is cubically convergent. …
5: Bibliography J
  • G. S. Joyce (1973) On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.
  • G. S. Joyce (1994) On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.
  • 6: 19.29 Reduction of General Elliptic Integrals
    These theorems reduce integrals over a real interval ( y , x ) of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over ( 0 , ) containing the square root of a cubic polynomial (compare §19.16(i)). …Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. … In the cubic case ( h = 3 ) the basic integrals are … (This shows why I ( 𝐞 α ) is not needed as a basic integral in the cubic case.) … In the cubic case, in which a 2 = 1 , b 2 = 0 , (19.29.26) reduces further to …
    7: 15.8 Transformations of Variable
    §15.8(v) Cubic Transformations
    Ramanujan’s Cubic Transformation
    This is used in a cubic analog of the arithmetic-geometric mean. …
    8: 23.21 Physical Applications
    The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x 4 x 3 . …
    9: Bibliography R
  • W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.
  • 10: 10.74 Methods of Computation
    Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …