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a8b6bda993
* Remove eval from arithmetic_analysis/newton_raphson.py
* Relocate contents of arithmetic_analysis/
Delete the arithmetic_analysis/ directory and relocate its files because
the purpose of the directory was always ill-defined. "Arithmetic
analysis" isn't a field of math, and the directory's files contained
algorithms for linear algebra, numerical analysis, and physics.
Relocated the directory's linear algebra algorithms to linear_algebra/,
its numerical analysis algorithms to a new subdirectory called
maths/numerical_analysis/, and its single physics algorithm to physics/.
* updating DIRECTORY.md
---------
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
45 lines
1014 B
Python
45 lines
1014 B
Python
import numpy as np
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def runge_kutta(f, y0, x0, h, x_end):
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"""
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Calculate the numeric solution at each step to the ODE f(x, y) using RK4
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https://en.wikipedia.org/wiki/Runge-Kutta_methods
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Arguments:
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f -- The ode as a function of x and y
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y0 -- the initial value for y
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x0 -- the initial value for x
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h -- the stepsize
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x_end -- the end value for x
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>>> # the exact solution is math.exp(x)
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>>> def f(x, y):
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... return y
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>>> y0 = 1
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>>> y = runge_kutta(f, y0, 0.0, 0.01, 5)
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>>> y[-1]
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148.41315904125113
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"""
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n = int(np.ceil((x_end - x0) / h))
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y = np.zeros((n + 1,))
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y[0] = y0
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x = x0
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for k in range(n):
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k1 = f(x, y[k])
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k2 = f(x + 0.5 * h, y[k] + 0.5 * h * k1)
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k3 = f(x + 0.5 * h, y[k] + 0.5 * h * k2)
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k4 = f(x + h, y[k] + h * k3)
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y[k + 1] = y[k] + (1 / 6) * h * (k1 + 2 * k2 + 2 * k3 + k4)
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x += h
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return y
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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