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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>
65 lines
2.0 KiB
Python
65 lines
2.0 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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from collections.abc import Callable
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DerivativeFunc = Callable[[float], float]
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def calc_derivative(f: DerivativeFunc, a: float, h: float = 0.001) -> float:
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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(
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f: DerivativeFunc,
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x0: float = 0,
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maxiter: int = 100,
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step: float = 0.0001,
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maxerror: float = 1e-6,
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logsteps: bool = False,
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) -> tuple[float, float, list[float]]:
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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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