CODEWITHSUNDEEP
pythonmathsympytaylor-series
Nitin Sharma
Nitin Sharma

Reputation: 140

Taylor series sympy expression of a python function

I have a very complicated non-linear function f. I want to get taylor series till degree n in a form of sympy expression for the function f at value x. f is a regular python function not a sympy expression. Output of get_polynomial should be a sympy expression.

Is there any function that will get taylor-series of a function?

from math import sin, cos, log, e

def f(x):
    # a very complicated function
    y = sin(x) + cos(x) + log(abs(x)+2)**2/e**2 + sin(cos(x/2)**2) + 1
    return y

def get_polynomial(function, x, degree):
    #    .......
    #    using Taylor Series
    #    .......
    return sympy_expression_for_function_at_value_x

Output:

get_polynomial(sin, 0, 3) ---> 0 + x + 0*x**2 + (1/6)*x**3
get_polynomial(lambda x: e**x, 0, 1) --> 1 + x

In a similar manner I wanna calculate get_polynomial(f, 0, 3)

Upvotes: 0

Views: 2836

Answers (1)

Yakov Dan
Yakov Dan

Reputation: 3347

The following code is close to what you're looking for. What this does it to parse the code the of the function you wish you expand into a Taylor series, convert it into a symbolic representation using Sympy and then compute the Taylor expansion.

One limitation is that you need to have an explicit function definition so you can't use lambda expressions. This can be solved with further work. Otherwise the code does what you ask for. Note that when you define a function, it has to contain a line of the form y = ... for this code to work

from inspect import *
import sympy

def f(x):
    # a very complicated function
    y = sin(x) + cos(x) + log(abs(x)+2)**2/e**2 + sin(cos(x/2)**2) + 1
    return y

def my_sin(x):
    y = sin(x)
    return y

def my_exp(x):
    y = e**x
    return y 


x = sympy.Symbol('x')

def get_polynomial(function, x0, degree):
    # parse function definition code

    lines_list  = getsource(function).split("\n")
    for line in lines_list:
        if '=' in line:
            func_def = line

    elements = func_def.split('=')
    line = ' '.join(elements[1:])
    sympy_function = sympy.sympify(line)

    # compute taylor expansion symbolically 
    i = 0
    taylor_exp = sympy.Integer(0)
    while i <= degree:
        taylor_exp = taylor_exp + (sympy.diff(sympy_function,x,i).subs(x,x0))/(sympy.factorial(i))*(x-x0)**i
        i += 1


    return taylor_exp

print (get_polynomial(my_sin,0,5))
print (get_polynomial(my_exp,0,5))
print (get_polynomial(f,0,5))

Upvotes: 2

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