108 lines
2.1 KiB
Python
108 lines
2.1 KiB
Python
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#!/usr/bin/env python3
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# Composite Numerical Integration:
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# Trapezoid and Simpson Rules
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#---------------------------------
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import numpy as np
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import scipy.integrate as integrate
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def trapz(yk, dx):
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""" Trapezoid Integration """
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longsum = 2*np.sum(yk[1:-1])
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return dx/2 * (longsum + yk[0] + yk[-1])
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def test_trapz():
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def test_func(x):
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return np.exp(x**2 - x**3)
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for _ in range(10):
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N = 1e1
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a = np.random.rand(1)
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b = np.random.rand(1)
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xk = np.linspace(a, b, N)
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yk = test_func(xk)
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h = (b-a)/N
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assert np.isclose( trapz(yk, h) , integrate.trapz(yk, dx=h), rtol=1/N)
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def simps(yk, dx):
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""" Integration using Simpson's 1/3 Rule """
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even_part = 2*np.sum(yk[2:-2:2])
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odd_part = 4*np.sum(yk[1:-1:2])
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return dx/3 * ( even_part + odd_part + yk[0] + yk[-1] )
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def main_2a():
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N = 1e3
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a = 0
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b = 2
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xk, h = np.linspace(a, b, N, retstep=True)
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func = lambda x: np.exp(- x**2 - 1 )
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yk = func(xk)
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print("My Integrators")
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print("Trapz:", trapz(yk,h))
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print("Simps:", simps(yk,h))
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print()
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print("Scipy Integrators")
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print("Trapz:", integrate.trapz(yk,dx=h))
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print("Simps:", integrate.simps(yk,dx=h))
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def main_2b():
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import pytest
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pytest.main(__file__)
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def main_2c():
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from matplotlib import pyplot
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N = 10
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funcs = [ lambda x: x, lambda x: x**2, lambda x: x**(1/2) ]
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exact = [ 1/2, 1/3, 3/2 ]
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diffs = np.zeros((np.size(funcs), N, 2))
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Ns = np.logspace(1,4, N)
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for i, f in enumerate(funcs):
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for j, n in enumerate(Ns):
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xk,h = np.linspace(0, 1, n, retstep=True)
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yk = f(xk)
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print(i,j)
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diffs[i][j][0] = exact[j] - trapz(yk,h)
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diffs[i][j][1] = exact[j] - simps(yk,h)
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pyplot.subplots()
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pyplot.title("Difference between the exact and numeric solution")
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pyplot.xlabel("N")
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pyplot.ylabel("difference")
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pyplot.plot(Ns, diffs[0][:][0], label="Trapz f(x)=x")
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pyplot.show()
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if __name__ == "__main__":
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main_2a()
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#main_2b()
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#main_2c()
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