# cubic

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## 1—10 of 20 matching pages

##### 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: 10.74 Methods of Computation
Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …
##### 10: 19.14 Reduction of General Elliptic Integrals
The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …