Reduced normal form in python

rnf.py

# Naive implementation of reduce normal form with some helper functions
from fractions import *

# helper functions
def row_sub(reduceby, reduceto, index):
    """
    Take row to reduce and row to reduce by with the index of the element to
    reduce to 0. Assume reduce by has value 1 in the index of reduceto.
    """
    row_merged = []
    for i in range(len(reduceby)):  # for each number in reduceby
        # subtract each by reduceby * whatever is above reduceby's leading 1
    	row_merged.append(reduceto[i] - reduceby[i] * reduceto[index])
    return row_merged

def row_reduce(row):
    """
    Reduces row to lead with 1.
    """
    # near_zero = 10 ** (-10)  # threshold for rounding errors
    for i in range(len(row)):
    	if abs(row[i]) > 0:  # first non-zero element
    		row[:] = [x / row[i] for x in row]
    		for x in range(len(row)):
    			row[x] = round(row[x], 15)
    		break

def augment(mat1, mat2):
    """
    Return the augmentation of mat2 onto mat1.
    """
    for i in range(len(mat1)):
    	mat1[i] += mat2[i]

# main RNF function
def rnf(mat):
    """
    Return the reduced normal form of a matrix. No pivoting is used.
    """
    for r in range(len(mat)):  # convert each mat element to decimal, exact value
    	for c in range(len(mat[0])):
    		mat[r][c] = Fraction('{}'.format(mat[r][c]))

    mat.sort(reverse = True)  # leading 0's naturally fall down
    for row in range(len(mat)):  # forward step
    	row_reduce(mat[row])  # to get leading 1
    	leading_one = 0  # need index of leading one for reverse, could have 0's
    	for i in range(len(mat[0])):
    		if abs(mat[row][i]) > 0:
    			leading_one = i 
    			break
    	for rowbelow in range(row + 1, len(mat)):  # all rows below row
            mat[rowbelow] = row_sub(mat[row], mat[rowbelow], leading_one)  # 0 column

    for row in reversed(range(len(mat))):  # reverse step
    	leading_one = 0  # need index of leading one for reverse, could have 0's
    	for i in range(len(mat[0])):
    		if abs(mat[row][i]) > 0:
    			leading_one = i 
    			break
    	for rowabove in reversed(range(row)):  # all rows above row
    		mat[rowabove] = row_sub(mat[row], mat[rowabove], leading_one)

    for r in range(len(mat)):  # convert each element back to float
    	for c in range(len(mat[0])):
    		mat[r][c] = float(mat[r][c])

    return mat