Naive Recursive Algorithm for Polynomial Multiplication in Python

user_10369367 0
Viewed 1545

I am trying to implement the divide-and-conquer algorithm for polynomial multiplication. Here is the pseudocode given in the lecture notes: enter image description here

where A, B are lists of coefficients of each polynomial, n is the size of the problem (degree - 1) and a_l, b_l are indices of the coefficients of interest.

Here is my attempt at implementing it using Python3:

def poly_mult_dc_naive(A, B, n, a, b):
  n = int(n)
  a = int(a)
  b = int(b)
  C = [None] * int(2*n - 1)

  if n == 1:
    C[0] = A[a] * B[b]
    return C[0]

  C[0:n-1] = poly_mult_dc_naive(A, B, n//2, a, b)
  C[n:2*n-1] = poly_mult_dc_naive(A, B, n//2, a + (n // 2), b + (n // 2))

  W = poly_mult_dc_naive(A, B, n/2, a, b + (n // 2))
  V = poly_mult_dc_naive(A, B, n/2, a + n/2, b)
  C[n // 2:n + (n // 2) - 1] += W + V
  return C

However I'm getting strange results. For example let A = [1,2,3,4] B = [4,3,2,1] I get:

[4, None, 8, 3, 6, 12, None, 16, 9, 12, 2, None, 4, 1, 2, None, 8, 3, 4, None, None]

Correct answer is [4, 11, 20, 30, 20, 11, 4]

Could someone please point out where I've gone wrong and how it could be done?

polynomials polynomial-math recursion python
2 Answers

Quick update: I think I've managed to debug my code my using a numpy array for C instead of a list. Here is the updated version:

import numpy as np

def poly_mult_dc_naive(A, B, n: int, a: int, b: int):
  C = np.zeros(2*n - 1, dtype=int) # here I changed it from list to np array
  if n == 1:
    C[0] = A[a] * B[b]
    return C[0]
  C[0:n-1] = poly_mult_dc_naive(A, B, n//2, a, b)
  C[n:2*n-1] = poly_mult_dc_naive(A, B, n//2, a + (n // 2), b + (n // 2))
  W = poly_mult_dc_naive(A, B, n//2, a, b + (n // 2))
  V = poly_mult_dc_naive(A, B, n/2, a + (n//2), b)
  C[(n // 2) : (3*n // 2) - 1] += W + V
  return C

BONUS QUESTION: Does anyone know of a better way I could keep the arguments n, a, and b as int types?

I just feel like having to write:

n = int(n)
a = int(a)
b = int(b)

might not be the most elegant way.

user_103693670
  • There should be no need to coerce n,a,b from float to int. Just use // 2 integer division everywhere (i.e. in the W,V lines). That will "keep" ints as ints.
  • The line C[n // 2:n + (n // 2) - 1] badly needs parenthesizing, it misreads very easily. I'd write C[(n//2) : (3*n//2)-1]
  • But I seriously recommend you use numpy vectors, not Python lists. Adding, multiplying etc. are vectorized.
user_2022290
Related