This worksheet will investigate Newton's method to find $\sqrt[3]{m}$.
The recurrence is
\[ x_{n+1} = \frac{2x_n}{3} + \frac{m}{3x_n^2} \]
which leads to
\[ a_{n+1} = 2a_n^3 + m b_n^3 \]
and
\[ b_{n+1} = 3 a_n^2 b_n\]
|
666.666667333333 666.666667333333 |
155/144 155/144 |
940195/894348 940195/894348 |
1.15742112496514 1.15742112496514 |
746496 746496 |
1430745813712011259 1430745813712011259 |
1430703422454144384 1430703422454144384 |
\[ x_{n+1} = \frac{2x_n}{3} + \frac{2}{3x_n^2} \]
so
\[ a_{n+1} = 2 a_n^3 + 2 b_n^3 \]
and
\[ b_{n+1} = 2 a_n^2 b_n \]
(not relatively prime
4.64158883361278 4.64158883361278 |
ratio cubed is 2.96296296297185e23 1 0.00310692181377248 1 1.06088520360338e-8 ratio cubed is 8.77914951991660e22 2 0.192304826298903 2 1.59200311573855e-8 ratio cubed is 2.60122948738270e22 3 0.738186910711232 3 2.39104615006580e-8 ratio cubed is 7.70734662928206e21 4 1.14431026360066 4 3.60029076269501e-8 ratio cubed is 2.28365826052802e21 5 1.31993830836785 5 5.46265673355810e-8 ratio cubed is 6.76639484600894e20 6 1.38273984189257 6 8.48047926703125e-8 ratio cubed is 2.00485773215080e20 7 1.40381018775935 7 1.41022151517848e-7 ratio cubed is 5.94031920637273e19 8 1.41073037066657 8 2.88207737873006e-7 ratio cubed is 1.76009457966600e19 9 1.41298645523345 9 1.09339735727062e-6 ratio cubed is 5.21509505086221e18 10 1.41371986992094 10 0.0000265346064861493 ratio cubed is 1.54521334840362e18 11 1.41395795334425 11 0.168552045594713 ratio cubed is 4.57840992119590e17 12 1.41403516570552 12 1.92006370544324e10 ratio cubed is 1.35656590257656e17 13 1.41406018485401 13 1.26147297918662e43 ratio cubed is 2.96296296297185e23 1 0.00310692181377248 1 1.06088520360338e-8 ratio cubed is 8.77914951991660e22 2 0.192304826298903 2 1.59200311573855e-8 ratio cubed is 2.60122948738270e22 3 0.738186910711232 3 2.39104615006580e-8 ratio cubed is 7.70734662928206e21 4 1.14431026360066 4 3.60029076269501e-8 ratio cubed is 2.28365826052802e21 5 1.31993830836785 5 5.46265673355810e-8 ratio cubed is 6.76639484600894e20 6 1.38273984189257 6 8.48047926703125e-8 ratio cubed is 2.00485773215080e20 7 1.40381018775935 7 1.41022151517848e-7 ratio cubed is 5.94031920637273e19 8 1.41073037066657 8 2.88207737873006e-7 ratio cubed is 1.76009457966600e19 9 1.41298645523345 9 1.09339735727062e-6 ratio cubed is 5.21509505086221e18 10 1.41371986992094 10 0.0000265346064861493 ratio cubed is 1.54521334840362e18 11 1.41395795334425 11 0.168552045594713 ratio cubed is 4.57840992119590e17 12 1.41403516570552 12 1.92006370544324e10 ratio cubed is 1.35656590257656e17 13 1.41406018485401 13 1.26147297918662e43 |
It appears that for $m=2$, when one of $a,b$ is equal to 1, and the other is very large, then
$b_n$ grows like $(a_0\sqrt{2})^{3^n}$ or $(b_0\sqrt{2})^{3^n}$!
|
3.45386246502860 3.45386246502860 |
2.03964890265551 2.03964890265551 |
2 2 |
|
0 1 1 1 2 3 2 6 9 3 15 27 4 37 81 5 114 243 6 321 729 7 996 2187 8 2863 6561 9 8699 19683 10 26217 59049 0 1 1 1 2 3 2 6 9 3 15 27 4 37 81 5 114 243 6 321 729 7 996 2187 8 2863 6561 9 8699 19683 10 26217 59049 |
1.25992104989487 1.25992104989487 |
1.44224957030741 1.44224957030741 |
|