# cubic

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##### 3: 1.11 Zeros of Polynomials
###### Cubic Equations
For the roots $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ of $g(w)=0$ and the roots $\theta_{1},\theta_{2},\theta_{3}$ of the resolvent cubic equation
1.11.20 $\sqrt{-\theta_{1}}\;\sqrt{-\theta_{2}}\;\sqrt{-\theta_{3}}=-q.$
Resolvent cubic is $z^{3}+12z^{2}+20z+9=0$ with roots $\theta_{1}=-1$, $\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})$, $\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})$, and $\sqrt{-\theta_{1}}=1$, $\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})$. …
##### 4: 3.8 Nonlinear Equations
3.8.3 $\left|z_{n+1}-\zeta\right|
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 $(0,\infty)$ 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 $I(\mathbf{e}_{\alpha})$ 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
###### Ramanujan’s Cubic Transformation
This is used in a cubic analog of the arithmetic-geometric mean. …
##### 8: 23.21 Physical Applications
The Weierstrass function $\wp$ plays a similar role for cubic potentials in canonical form $g_{3}+g_{2}x-4x^{3}$. …
##### 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. …