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Python
1.0
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Python
1.0
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Finding roots (or zeroes) of a real-valued function, by using Newton's Method. More...
Functions | |
| def | xx_eq_sinx () |
| Solves the equation \(x^{2} = sin (x)\). More... | |
| def | main (argv=None) |
Finding roots (or zeroes) of a real-valued function, by using Newton's Method.
\[x_{n+1} = x_n - \frac {f(x_n)} {f'(x_n)}.\]
Consider \(x^{2} - sin(x) = 0\), which gives us the following recursive formula:
\begin{eqnarray*} x_{k+1} &=& x_k - \frac {(x_k²-sin(x_k))}{(2*x_k - cos(x_k))}, k >= 0, x_0 > 0 \\ &=& \frac {(x_k² - x_k*cos(x_k) + sin(x_k))}{(2*x_k - cos(x_k))}, \\ x_0 &=& 1. \end{eqnarray*}
Stopping condition: \( | x_{k+1} - x_k | < \epsilon \)
Alternative way, by using the first two terms of the Mclaurin series for sin(x):
\[x^{2} - (x - \frac{x^{3}}{3!}) = 0\]
or
\[6x^{2} - 6x + x^{3} = 0 => x^{2} + 6x - 6 = 0\]
\[x = \sqrt{15} - 3 = 0.872983346\]
Remainder \(< \frac {x^{5}}{5!} < \frac {1}{200} = 0.005\)
| def _04b_xx_eq_sinx.main | ( | argv = None | ) |
References xx_eq_sinx().
| def _04b_xx_eq_sinx.xx_eq_sinx | ( | ) |
1.8.20