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@anthonymorast
Created Oct 3, 2021
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from scipy.misc import derivative
import math
class TaylorSeries():
def __init__(self, function, order, center=0):
self.center = center
self.f = function
self.order = order
self.d_pts = order*2
self.coefficients = []
# number of points (order) for scipy.misc.derivative
if self.d_pts % 2 == 0: # must be odd and greater than derivative order
self.d_pts += 1
self.__find_coefficients()
def __find_coefficients(self):
for i in range(0, self.order+1):
self.coefficients.append(round(derivative(self.f, self.center, n=i, order=self.d_pts)/math.factorial(i), 5))
def print_equation(self):
eqn_string = ""
for i in range(self.order + 1):
if self.coefficients[i] != 0:
eqn_string += str(self.coefficients[i]) + ("(x-{})^{}".format(self.center, i) if i > 0 else "") + " + "
eqn_string = eqn_string[:-3] if eqn_string.endswith(" + ") else eqn_string
print(eqn_string)
def print_coefficients(self):
print(self.coefficients)
def approximate_value(self, x):
"""
Approximates the value of f(x) using the taylor polynomial.
x = point to approximate f(x)
"""
fx = 0
for i in range(len(self.coefficients)):
fx += self.coefficients[i] * ((x - self.center)**i) # coefficient * nth term
return fx
def approximate_derivative(self, x):
"""
Estimates the derivative of a function f(x) from its Taylor series.
Useless since we need the derivative of the actual function to find the series
"""
value = 0
for i in range(1, len(self.coefficients)): # skip the first value (constant) as the derivative is 0
value += self.coefficients[i] * i * ((x - self.center)**(i-1)) # differentiate each term: x^n => n*x^(n-1)
return value
def approximate_integral(self, x0, x1):
"""
Estimates the definite integral of the function using the Taylor series expansion.
More useful, consider e^x * sin(x), easy to differentiate but difficult to integrate.
x0 - lower limit of integration
x1 - upper limit of integration
"""
# integrals can be off by a constant since int(f(x)) = F(x) + C
value = 0
for i in range(len(self.coefficients)):
value += ((self.coefficients[i] * (1/(i+1)) * ((x1 - self.center)**(i+1))) -
(self.coefficients[i] * (1/(i+1)) * ((x0 - self.center)**(i+1)))) # integrate each term: x^n => (1/n+1)*x^(n+1)
return value
def get_coefficients(self):
"""
Returns the coefficients of the taylor series
"""
return self.coefficients
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