python写的多项式符号乘法

(x - 1)(x² + x + 1) = x³ - 1

import ply.lex as lex # pip install ply
import ply.yacc as yacc

def parse(s):
    t = {}
    tokens = ('NUM', 'VAR'); t_NUM = r'\d+'; t_VAR = r'[x|X]'; literals = ['+', '-', '*', '^']
    def t_error(t): t.lexer.skip(1)
    precedence = (('left', '+', '-'), ('nonassoc', '*'), ('nonassoc', '^'))
    def p_1(p):
        '''poly : poly '+' term
                | poly '-' term'''
        if p[2] == '-': t[p[3]] = -t[p[3]]
    def p_2(p): 'poly : term'
    def p_3(p): "poly : '-' term"; t[p[2]] = -t[p[2]]
    def p_4(p): 'term : NUM'; t[0] = int(p[1]); p[0] = 0;
    def p_5(p): 'term : VAR'; t[1] = 1; p[0] = 1;
    def p_6(p): "term : NUM '*' VAR"; t[1] = int(p[1]); p[0] = 1
    def p_7(p): "term : VAR '^' NUM"; t[int(p[3])] = 1; p[0] = int(p[3])
    def p_8(p): "term : NUM '*' VAR '^' NUM"; t[int(p[5])] = int(p[1]); p[0] = int(p[5])
    def p_error(p): raise Exception()
    lexer = lex.lex()
    yacc.yacc().parse(s)
    return t

class poly:
    @staticmethod # e: exponent, c: coefficient
    def canonical(d): return { e:c for e,c in d.items() if c }
    def __init__(m, s): m.d = poly.canonical(parse(s) if isinstance(s, str) else s)
    def __str__(m):
        first = 1
        s = ''
        for e,c in sorted(m.d.items(), key=lambda ec:ec[0], reverse=1):
            if first: s += '-' if c < 0 else ''
            else: s += ' - ' if c < 0 else ' + '
            c = abs(c)
            if e == 0: s += str(c); continue
            if c != 1: s += str(c) + '*'
            s += 'x' if e == 1 else 'x^' + str(e)
            first = 0
        return s
    def __mul__(a, b):
        d = {}
        for e,c in a.d.items():
            for e2,c2 in b.d.items():
                e3 = e + e2; d[e3] = d.get(e3, 0) + c * c2
        return poly(d)
#print(poly('-x^4 - 3*x^2 - 2*x - 5'))
a = poly('x - 1'); b = poly('x^2 + x + 1')
print('(', a, ') * (', b, ') = ', a * b, sep='')

search(computer algebra), search(unexpected applications of polynomials in combinatorics), search(多项式算法 快速傅里叶变换FFT)

科学计算可分为数值计算和符号计算两类。MATLAB和SymPy好像两者都用上了。比如MATLAB能分解因式，好像会出来0.999999999999这样的系数，但多项式相乘的复杂度是优化前O(n*n)，用上FFT后O(n*log(n))。

# parsetab.py
# This file is automatically generated. Do not edit.
# pylint: disable=W,C,R
_tabversion = '3.10'
_lr_method = 'LALR'
_lr_signature = "left+-nonassoc*nonassoc^NUM VARpoly : poly '+' term\n\t\t\t\t\t\t| poly '-' termpoly : termpoly : '-' termterm : NUMterm : term : NUM '*' VARterm : VAR '^' NUMterm : NUM '*' VAR '^' NUM"
    
_lr_action_items = {'-':([0,1,2,4,5,8,11,12,13,14,16,],[3,7,-3,-5,-6,-4,-1,-2,-7,-8,-9,]),'NUM':([0,3,6,7,10,15,],[4,4,4,4,14,16,]),'VAR':([0,,7,9,],[5,5,5,5,13,]),'$end':([1,2,4,5,8,11,12,13,14,16,],[0,-3,-5,-6,-4,-1,-2,-7,-8,-9,]),'+':([1,2,4,5,8,11,12,13,14,16,],[6,-3,-5,-6,-4,-1,-7,-8,-9,]),'*':([4,],[9,]),'^':([5,13,],[10,15,]),}
_lr_action = {}
for _k, _v in _lr_action_items.items():
   for _x,_y in zip(_v[0],_v[1]):
      if not _x in _lr_action:  _lr_action[_x] = {}
      _lr_action[_x][_k] = _y
del _lr_action_items
_lr_goto_items = {'poly':([0,],[1,]),'term':([0,3,6,7,],[2,8,11,12,]),}
_lr_goto = {}
for _k, _v in _lr_goto_items.items():
   for _x, _y in zip(_v[0], _v[1]):
       if not _x in _lr_goto: _lr_goto[_x] = {}
       _lr_goto[_x][_k] = _y
del _lr_goto_items
_lr_productions = [
  ("S' -> poly","S'",1,None,None,None),
  ('poly -> poly + term','poly',3,'p_1','poly.py',10),
  ('poly -> poly - term','poly',3,'p_1','poly.py',11),
  ('poly -> term','poly',1,'p_2','poly.py',14),
  ('poly -> - term','poly',2,'p_3','poly.py',15),
  ('term -> NUM','term',1,'p_4','poly.py',16),
  ('term -> VAR','term',1,'p_5','poly.py',17),
  ('term -> NUM * VAR','term',3,'p_6','poly.py',18),
  ('term -> VAR ^ NUM','term',3,'p_7','poly.py',19),
  ('term -> NUM * VAR ^ NUM','term',5,'p_8','poly.py',20),
]
Created by PLY version 3.11 (http://www.dabeaz.com/ply) https://ply.readthedocs.io/en/latest/
Grammar
Rule 0 S' -> poly
Rule 1 poly -> poly + term
Rule 2 poly -> poly - term
Rule 3 poly -> term
Rule 4 poly -> - term
Rule 5 term -> NUM
Rule 6 term -> VAR
Rule 7 term -> NUM * VAR
Rule 8 term -> VAR ^ NUM
Rule 9 term -> NUM * VAR ^ NUM
Terminals, with rules where they appear
* : 7 9
+ : 1
- : 2 4
NUM : 5 7 8 9 9
VAR : 6 7 8 9
^ : 8 9
error :
Nonterminals, with rules where they appear
poly : 1 2 0
term : 1 2 3 4
Parsing method: LALR
state 0
(0) S' -> . poly
(1) poly -> . poly + term
(2) poly -> . poly - term
(3) poly -> . term
(4) poly -> . - term
(5) term -> . NUM
(6) term -> . VAR
(7) term -> . NUM * VAR
(8) term -> . VAR ^ NUM
(9) term -> . NUM * VAR ^ NUM
- shift and go to state 3
NUM shift and go to state 4
VAR shift and go to state 5
poly shift and go to state 1
term shift and go to state 2
state 1
(0) S' -> poly .
(1) poly -> poly . + term
(2) poly -> poly . - term
+ shift and go to state 6
- shift and go to state 7
state 2
(3) poly -> term .
+ reduce using rule 3 (poly -> term .)
- reduce using rule 3 (poly -> term .)
$end reduce using rule 3 (poly -> term .)
state 3
(4) poly -> - . term
(5) term -> . NUM
(6) term -> . VAR
(7) term -> . NUM * VAR
(8) term -> . VAR ^ NUM
(9) term -> . NUM * VAR ^ NUM
NUM shift and go to state 4
VAR shift and go to state 5
term shift and go to state 8
state 4
(5) term -> NUM .
(7) term -> NUM . * VAR
(9) term -> NUM . * VAR ^ NUM
+ reduce using rule 5 (term -> NUM .)
- reduce using rule 5 (term -> NUM .)
$end reduce using rule 5 (term -> NUM .)
* shift and go to state 9
state 5
(6) term -> VAR .
(8) term -> VAR . ^ NUM
+ reduce using rule 6 (term -> VAR .)
- reduce using rule 6 (term -> VAR .)
$end reduce using rule 6 (term -> VAR .)
^ shift and go to state 10
state 6
(1) poly -> poly + . term
(5) term -> . NUM
(6) term -> . VAR
(7) term -> . NUM * VAR
(8) term -> . VAR ^ NUM
(9) term -> . NUM * VAR ^ NUM
NUM shift and go to state 4
VAR shift and go to state 5
term shift and go to state 11
state 7
(2) poly -> poly - . term
(5) term -> . NUM
(6) term -> . VAR
(7) term -> . NUM * VAR
(8) term -> . VAR ^ NUM
(9) term -> . NUM * VAR ^ NUM
NUM shift and go to state 4
VAR shift and go to state 5
term shift and go to state 12
state 8
(4) poly -> - term .
+ reduce using rule 4 (poly -> - term .)
- reduce using rule 4 (poly -> - term .)
$end reduce using rule 4 (poly -> - term .)
state 9
(7) term -> NUM * . VAR
(9) term -> NUM * . VAR ^ NUM
VAR shift and go to state 13
state 10
(8) term -> VAR ^ . NUM
NUM shift and go to state 14
state 11
(1) poly -> poly + term .
+ reduce using rule 1 (poly -> poly + term .)
- reduce using rule 1 (poly -> poly + term .)
$end reduce using rule 1 (poly -> poly + term .)
state 12
(2) poly -> poly - term .
+ reduce using rule 2 (poly -> poly - term .)
- reduce using rule 2 (poly -> poly - term .)
$end reduce using rule 2 (poly -> poly - term .)
state 13
(7) term -> NUM * VAR .
(9) term -> NUM * VAR . ^ NUM
+ reduce using rule 7 (term -> NUM * VAR .)
- reduce using rule 7 (term -> NUM * VAR .)
$end reduce using rule 7 (term -> NUM * VAR .)
^ shift and go to state 15
state 14
(8) term -> VAR ^ NUM .
+ reduce using rule 8 (term -> VAR ^ NUM .)
- reduce using rule 8 (term -> VAR ^ NUM .)
$end reduce using rule 8 (term -> VAR ^ NUM .)
state 15
(9) term -> NUM * VAR ^ . NUM
NUM shift and go to state 16
state 16
(9) term -> NUM * VAR ^ NUM .
+ reduce using rule 9 (term -> NUM * VAR ^ NUM .)
- reduce using rule 9 (term -> NUM * VAR ^ NUM .)
$end reduce using rule 9 (term -> NUM * VAR ^ NUM .)