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55 lines
1.8 KiB
Python
55 lines
1.8 KiB
Python
"""
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Author: P Shreyas Shetty
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Implementation of Newton-Raphson method for solving equations of kind
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f(x) = 0. It is an iterative method where solution is found by the expression
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x[n+1] = x[n] + f(x[n])/f'(x[n])
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If no solution exists, then either the solution will not be found when iteration
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limit is reached or the gradient f'(x[n]) approaches zero. In both cases, exception
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is raised. If iteration limit is reached, try increasing maxiter.
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"""
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import math as m
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def calc_derivative(f, a, h=0.001):
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"""
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Calculates derivative at point a for function f using finite difference
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method
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"""
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return (f(a + h) - f(a - h)) / (2 * h)
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def newton_raphson(f, x0=0, maxiter=100, step=0.0001, maxerror=1e-6, logsteps=False):
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a = x0 # set the initial guess
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steps = [a]
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error = abs(f(a))
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f1 = lambda x: calc_derivative(f, x, h=step) # noqa: E731 Derivative of f(x)
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for _ in range(maxiter):
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if f1(a) == 0:
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raise ValueError("No converging solution found")
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a = a - f(a) / f1(a) # Calculate the next estimate
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if logsteps:
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steps.append(a)
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if error < maxerror:
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break
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else:
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raise ValueError("Iteration limit reached, no converging solution found")
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if logsteps:
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# If logstep is true, then log intermediate steps
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return a, error, steps
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return a, error
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if __name__ == "__main__":
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from matplotlib import pyplot as plt
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f = lambda x: m.tanh(x) ** 2 - m.exp(3 * x) # noqa: E731
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solution, error, steps = newton_raphson(
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f, x0=10, maxiter=1000, step=1e-6, logsteps=True
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)
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plt.plot([abs(f(x)) for x in steps])
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plt.xlabel("step")
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plt.ylabel("error")
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plt.show()
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print(f"solution = {{{solution:f}}}, error = {{{error:f}}}")
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