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Martin Thoma 2014-01-02 16:16:13 +01:00
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documents/math-minimal-distance-to-cubic-function/quadratic-case-2.1.tex
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@ -1,10 +1,8 @@
$4 \alpha^3 + 27 \beta^2 \geq 0$:
One solution of Equation~\ref{eq:simple-cubic-equation-for-quadratic-distance}
is
The first solution of $x^3 + \alpha x + \beta = 0$ is
\[x = \frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}\]
When you insert this in Equation~\ref{eq:simple-cubic-equation-for-quadratic-distance}
you get:\footnote{Remember: $(a-b)^3 = a^3-3 a^2 b+3 a b^2-b^3$}
Let's validate this solution:
\allowdisplaybreaks
\begin{align}
0 &\stackrel{!}{=} \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right )^3 + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
@ -12,27 +10,24 @@ you get:\footnote{Remember: $(a-b)^3 = a^3-3 a^2 b+3 a b^2-b^3$}
- 3 (\frac{t}{\sqrt[3]{18}})^2 \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}
+ 3 (\frac{t}{\sqrt[3]{18}})(\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^2
- (\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^3
+ \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
+ \frac{t \alpha}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha^2 }{t} + \beta\\
&= \frac{t^3}{18}
- \frac{3t^2}{\sqrt[3]{18^2}} \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}
+ \frac{3t}{\sqrt[3]{18}} \frac{\sqrt[3]{\frac{4}{9}} \alpha^2 }{t^2}
- \frac{\frac{2}{3} \alpha^3 }{t^3}
+ \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
+ \frac{t \alpha }{\sqrt[3]{18}} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t} + \beta\\
&= \frac{t^3}{18}
- \frac{\sqrt[3]{18} t \alpha}{\sqrt[3]{18^2}}
+ \frac{\sqrt[3]{12} \alpha^2}{\sqrt[3]{18} t}
- \frac{2 \alpha^3 }{3t^3}
+ \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
+ \frac{t \alpha }{\sqrt[3]{18}} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t} + \beta\\
&= \frac{t^3}{18}
- \frac{t \alpha}{\sqrt[3]{18}}
\color{red}+ \frac{\sqrt[3]{2} \alpha^2}{\sqrt[3]{3} t} \color{black}
- \frac{2 \alpha^3 }{3 t^3}
+ \color{red}\alpha \color{black} \left (\frac{t}{\sqrt[3]{18}} \color{red}- \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \color{black}\right )
+ \beta\\
&= \frac{t^3}{18} \color{blue}- \frac{t \alpha}{\sqrt[3]{18}} \color{black}
- \frac{2 \alpha^3 }{3 t^3}
\color{blue}+ \frac{\alpha t}{\sqrt[3]{18}} \color{black}
+ \beta\\
\color{blue} - \frac{t \alpha}{\sqrt[3]{18}}
\color{red} + \frac{\sqrt[3]{2} \alpha^2}{\sqrt[3]{3} t}
\color{black}- \frac{2 \alpha^3 }{3 t^3}
\color{blue} + \frac{t \alpha }{\sqrt[3]{18}}
\color{red} - \frac{\sqrt[3]{2} \alpha^2 }{\sqrt[3]{3} t}
\color{black}+ \beta\\
&= \frac{t^3}{18} - \frac{2 \alpha^3 }{3 t^3} + \beta\\
&= \frac{t^6 - 12 \alpha^3 + \beta 18 t^3}{18t^3}
\end{align}