functions for polynomial

intro

• When have to work with function representing polynomial, e.g. plot polynomial function, the function can be made by summing all terms manually (Imam, 2018).
• To calculate $f(x)$ for single value of $x$ there is at least two ways with different time complexity, $O(n^2)$ by adding each term for $x^j$ with multiplication and summation loops, and $O(n)$ by using Horner’s method (Taparia, 2022).
• Python math library can be used to calculate $x^j$ and reduce previous approach with two loops into single loop (Bhojasia, 2022).
• User-defined class can be constructed to meet only what is needed, coefficients, string representation in LaTeX, and calculate the value for every $x$ using callabes (Klein, 2023).
• By representing a polynomial in its coefficent NumPy functions can be used to calculate the value for every $x$, finding roots of the the polynomial, perform differential and integral to it (Kitchin, 2013).
• There are also some series, e.g. power, Chebyshev, Hermite, Laguerre, Legendre, provided by NumPy (Developers, 2022).
• Here only polynomial functions with constant and variable coefficients are discussed.

constant coefficents

• Suppose there is a polynomial in the form of $$\tag{1} f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4. $$
• There are some approaches to calculate the value of $f(x)$.

approach 1

• Code

  def f(x):
    y0 = 1
    y1 = 2 * x
    y2 = 3 * x**2
    y3 = 4 * x**3
    y4 = 5 * x**4
    y = y0 + y1 + y2 + y3 + y4
    return y
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwc33ef [20231230].
• Output

  x	f(x)
  0	1
  1	15
  2	129
  3	547
  4	1593
  

• This approach is straight forward and very clear, but it works only for one specific polynomial.

approach 2

• Code

  import math
  
  def f(x):
    y0 = 1 * math.pow(x, 0)
    y1 = 2 * math.pow(x, 1)
    y2 = 3 * math.pow(x, 2)
    y3 = 4 * math.pow(x, 3)
    y4 = 5 * math.pow(x, 4)
    y = y0 + y1 + y2 + y3 + y4
    return y
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwcjjud [20231230].
• Output

  x	f(x)
  0	1
  1	15
  2	129
  3	547
  4	1593
  

• This approach is straight forward and also systematic since the power is also clear, but it works only for one specific polynomial.

approach 3

• Code

  import math
  
  def f(x):
    c = [1, 2, 3, 4, 5];
    y0 = c[0] * math.pow(x, 0)
    y1 = c[1] * math.pow(x, 1)
    y2 = c[2] * math.pow(x, 2)
    y3 = c[3] * math.pow(x, 3)
    y4 = c[4] * math.pow(x, 4)
    y = y0 + y1 + y2 + y3 + y4
    return y
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwcwhvq [20231230].
• Output

  x	f(x)
  0	1.0
  1	15.0
  2	129.0
  3	547.0
  4	1593.0
  

• It is easier to modify for certain polynomial compared the previous code, since only one line c = [c0, c1, c2, c3, c4] that has to be changed.

approach 4

• Code

  import math
  
  def f(x):
    c = [1, 2, 3, 4, 5];
    y = 0
    for i in range(len(c)):
      yi = c[i] * math.pow(x, i)
      y = y + yi
    return y
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwd8ve2 [20231230].
• Output

  x	f(x)
  0	1.0
  1	15.0
  2	129.0
  3	547.0
  4	1593.0
  

• It is the same the previous code, with more compact in calculating each term using yi = c[i] * math.pow(x, i).

approach 5

• Code

  def f(x):
    c = [1, 2, 3, 4, 5];
    y = [(c[i] * x**i) for i in range(len(c))]
    return sum(y)
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwe4z2n [20231230].
• Output

  x	f(x)
  0	1
  1	15
  2	129
  3	547
  4	1593
  

• Using Python feature known as list comprehension more lines can be reduced.

approach 6

• Code

  def f(x):
    c = [1, 2, 3, 4, 5];
    y = [cf * x**n for n, cf in enumerate(c)]
    return sum(y)
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwe8ed8 [20231230].
• Output

  x	f(x)
  0	1
  1	15
  2	129
  3	547
  4	1593
  

• With this code, number of lines is still the same but the use of Python enumerate function make the code more clear.

approach 7

• Code

  def f(x):
    return sum([c * x**n for n, c in enumerate(coefs)])
  
  coefs = [1, 2, 3, 4, 5];
  
  print("x\tf(x)")
  for x in range(0, 5):
    print(x, f(x), sep='\t')
  

  url https://onecompiler.com/python/3zxwf2k3c [20231230].
• Output

  x	f(x)
  0	1
  1	15
  2	129
  3	547
  4	1593
  

• Now the function f(x) can be in one line and coefs list is put outside the function.
• Notice that coefs can be modified before the function f(x) is called.

variable coefficients

• By modifying previous approach 7, following code can be obtained.

  def f(x, coefs):
    return sum([c * x**n for n, c in enumerate(coefs)])
  
  
  c1 = [1, 2, 3, 4, 5]
  print("x\tf(x)")
  for x in range(0, 4):
    print(x, f(x, c1), sep='\t')
  
  print()
  
  c2 = [1, 1, 1, 1]
  print("x\tf(x)")
  for x in range(0, 3):
    print(x, f(x, c2), sep='\t')
  

• Output

  x	f(x)
  0	1
  1	15
  2	129
  3	547
  
  x	f(x)
  0	1
  1	4
  2	15
  

• Notice that there are two polynomials, each with coefficients c1 and c2.
• Call the function f(x, c) will calculate polynomial $f(x)$ with coefficients $c$.

challenges

• Modify last code to calculate $f(x) = -1 + x - x^2 + x^3$.
• Calculate other polynomial $f(x) = 32x^7 - x^4 + 0.5$.

refs

• Bhojasia M (2022) “Python Program to Compute a Polynomial Equation”, Sanfoundy, 30 May 2022, url https://www.sanfoundry.com/python-program-compute-polynomial-equation-coefficients-list/#google_vignette [20131227].
• Developers N (2022) “Polynomials”, NumPy, 2008-2022, url https://numpy.org/doc/stable/reference/routines.polynomials.html [20231227].
• Imam A (2018) “Plotting polynomial function in Python”, Medium, 27 Dec 2018, url https://aadhil-imam.medium.com/361230a1e400 [20131227].
• Kitchin J (2013) “Polynomials in python”, The Kitchin Research Group, Chemical Engineering at Carnegie Mellon University, 27 Feb 2013, url https://kitchingroup.cheme.cmu.edu/blog/2013/01/22/Polynomials-in-python/ [20231227].
• Klein B (2023) “14. Polynomial Class”, 7 Dec 2023, url https://python-course.eu/oop/polynomial-class.php [20231227].
• Taparia A (2022) “Python program to Compute a Polynomial Equation”, GeeksforGeeks, 21 Jun 2022, url https://www.geeksforgeeks.org/python-program-to-compute-a-polynomial-equation/ [20231227].