CODEWITHSUNDEEP
pythonmath
Henrik Hillestad Løvold
Henrik Hillestad Løvold

Reputation: 1281

Taylor polynomial calculation

I am currently doing a python exercise for my University studies. I am very stuck at this task:

The taylor polynomial of degree N for the exponential function e^x is given by:

        N
p(x) = Sigma  x^k/k!  
k = 0

Make a program that (i) imports class Polynomial (found under), (ii) reads x and a series of N values from the command line, (iii) creates a Polynomial instance representing the Taylor polynomial, and (iv) prints the values of p(x) for the given N values as well as the exact value e^x. Try the program out with x = 0.5, 3, 10 and N = 2, 5, 10, 15, 25.

Polynomial.py

import numpy

class Polynomial:
def __init__(self, coefficients):
    self.coeff = coefficients

def __call__(self, x):
    """Evaluate the polynomial."""
    s = 0
    for i in range(len(self.coeff)):
        s += self.coeff[i]*x**i
    return s

def __add__(self, other):
    # Start with the longest list and add in the other
    if len(self.coeff) > len(other.coeff):
        result_coeff = self.coeff[:]  # copy!
        for i in range(len(other.coeff)):
            result_coeff[i] += other.coeff[i]
    else:
        result_coeff = other.coeff[:] # copy!
        for i in range(len(self.coeff)):
            result_coeff[i] += self.coeff[i]
    return Polynomial(result_coeff)

def __mul__(self, other):
    c = self.coeff
    d = other.coeff
    M = len(c) - 1
    N = len(d) - 1
    result_coeff = numpy.zeros(M+N+1)
    for i in range(0, M+1):
        for j in range(0, N+1):
            result_coeff[i+j] += c[i]*d[j]
    return Polynomial(result_coeff)

def differentiate(self):
    """Differentiate this polynomial in-place."""
    for i in range(1, len(self.coeff)):
        self.coeff[i-1] = i*self.coeff[i]
    del self.coeff[-1]

def derivative(self):
    """Copy this polynomial and return its derivative."""
    dpdx = Polynomial(self.coeff[:])  # make a copy
    dpdx.differentiate()
    return dpdx

def __str__(self):
    s = ''
    for i in range(0, len(self.coeff)):
        if self.coeff[i] != 0:
            s += ' + %g*x^%d' % (self.coeff[i], i)
    # Fix layout
    s = s.replace('+ -', '- ')
    s = s.replace('x^0', '1')
    s = s.replace(' 1*', ' ')
    s = s.replace('x^1 ', 'x ')
    #s = s.replace('x^1', 'x') # will replace x^100 by x^00
    if s[0:3] == ' + ':  # remove initial +
        s = s[3:]
    if s[0:3] == ' - ':  # fix spaces for initial -
        s = '-' + s[3:]
    return s

def simplestr(self):
    s = ''
    for i in range(0, len(self.coeff)):
        s += ' + %g*x^%d' % (self.coeff[i], i)
    return s


def _test():
    p1 = Polynomial([1, -1])
    p2 = Polynomial([0, 1, 0, 0, -6, -1])
    p3 = p1 + p2
print p1, '  +  ', p2, '  =  ', p3
p4 = p1*p2
print p1, '  *  ', p2, '  =  ', p4
print 'p2(3) =', p2(3)
p5 = p2.derivative()
print 'd/dx', p2, '  =  ', p5
print 'd/dx', p2,
p2.differentiate()
print '  =  ', p5
p4 = p2.derivative()
print 'd/dx', p2, '  =  ', p4

if __name__ == '__main__':
_test()

Now I'm really stuck at this, and I would love to get an explaination! I am supposed to write my code in a separate file. I'm thinking about making an instance of the Polynomial class, and sending in the list in argv[2:], but that doesn't seem to be working. Do I have to make a def to calculate the taylor polynomial for the different values of N before sending it in to the Polynomial class?

Any help is great, thanks in advance :)

Upvotes: 0

Views: 2536

Answers (2)

Palaios
Palaios

Reputation: 57

I solved the task in the following way, though im not sure if it answers question (iv). The output just compares the exact value of e**x to the calculated value from module Polynomial.

from math import factorial, exp
from Polynomial import *
from sys import *

#Reads x and N from the command line on the form [filename.py, x-value, N-value]
x = eval(argv[1])
N = eval(argv[2])

#Creating list of coefficients on the form [1 / i!]
list_coeff = [1./factorial(i) for i in range(N)]

print list_coeff

#Creating an instance of class Polynomial
p1 = Polynomial(list_coeff)

print 'Calculated value of e**%f = %f ' %(x, p1.__call__(x))
print 'Exact value of e**%f = %f'% (x, exp(x))

"""Test Execution
Terminal > python Polynomial_exp.py 0.5 5
[1.0, 1.0, 0.5, 0.16666666666666666, 0.041666666666666664]
Calculated value of e**0.500000 = 1.648438 
Exact value of e**0.500000 = 1.648721
"""

Upvotes: 1

RParadox
RParadox

Reputation: 6851

Not quite finished, but this answers your main question I believe. Put class Polynomial in poly.p and import it.

from poly import Polynomial as p
from math import exp,factorial

def get_input(n):
    ''' get n numbers from stdin '''
    entered = list()
    for i in range(n):
        print 'input number '
    entered.append(raw_input())
    return entered

def some_input():
    return [[2,3,4],[4,3,2]]



get input from cmd line                                                                                                                                   
n = 3                                                                                                                                                     
a = get_input(n)                                                                                                                                          
b = get_input(n)                                                                                                                                          


#a,b = some_input()

ap = p(a)
bp = p(b)


print 'entered : ',a,b

c = ap+bp

print 'a + b = ',c

print exp(3)

x = ap
print x

sum = p([0])
for k in range(1,5):
    el = x
    for j in range(1,k):
        el  el * x
        print 'el: ',el
    if el!=None and sum!=None:
        sum = sum + el
        print 'sum ',sum

output

entered :  [2, 3, 4] [4, 3, 2]
a + b =  6*1 + 6*x + 6*x^2
20.0855369232
2*1 + 3*x + 4*x^2
sum  2*1 + 3*x + 4*x^2
el:  4*1 + 12*x + 25*x^2 + 24*x^3 + 16*x^4
sum  6*1 + 15*x + 29*x^2 + 24*x^3 + 16*x^4
el:  4*1 + 12*x + 25*x^2 + 24*x^3 + 16*x^4
el:  8*1 + 36*x + 102*x^2 + 171*x^3 + 204*x^4 + 144*x^5 + 64*x^6
sum  14*1 + 51*x + 131*x^2 + 195*x^3 + 220*x^4 + 144*x^5 + 64*x^6
el:  4*1 + 12*x + 25*x^2 + 24*x^3 + 16*x^4
el:  8*1 + 36*x + 102*x^2 + 171*x^3 + 204*x^4 + 144*x^5 + 64*x^6
el:  16*1 + 96*x + 344*x^2 + 792*x^3 + 1329*x^4 + 1584*x^5 + 1376*x^6 + 768*x^7 + 256*x^8
sum  30*1 + 147*x + 475*x^2 + 987*x^3 + 1549*x^4 + 1728*x^5 + 1440*x^6 + 768*x^7 + 256*x^8

Upvotes: 2

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