python写的多项式符号乘法
(x - 1)(x2 + x + 1) = x3 - 1
1 import ply.lex as lex # pip install ply 2 import ply.yacc as yacc 3 4 def parse(s): 5 t = {} 6 tokens = ('NUM', 'VAR'); t_NUM = r'\d+'; t_VAR = r'[x|X]'; literals = ['+', '-', '*', '^'] 7 def t_error(t): t.lexer.skip(1) 8 precedence = (('left', '+', '-'), ('nonassoc', '*'), ('nonassoc', '^')) 9 def p_1(p): 10 '''poly : poly '+' term 11 | poly '-' term''' 12 if p[2] == '-': t[p[3]] = -t[p[3]] 13 def p_2(p): 'poly : term' 14 def p_3(p): "poly : '-' term"; t[p[2]] = -t[p[2]] 15 def p_4(p): 'term : NUM'; t[0] = int(p[1]); p[0] = 0; 16 def p_5(p): 'term : VAR'; t[1] = 1; p[0] = 1; 17 def p_6(p): "term : NUM '*' VAR"; t[1] = int(p[1]); p[0] = 1 18 def p_7(p): "term : VAR '^' NUM"; t[int(p[3])] = 1; p[0] = int(p[3]) 19 def p_8(p): "term : NUM '*' VAR '^' NUM"; t[int(p[5])] = int(p[1]); p[0] = int(p[5]) 20 def p_error(p): raise Exception() 21 lexer = lex.lex() 22 yacc.yacc().parse(s) 23 return t 24 25 class poly: 26 @staticmethod # e: exponent, c: coefficient 27 def canonical(d): return { e:c for e,c in d.items() if c } 28 def __init__(m, s): m.d = poly.canonical(parse(s) if isinstance(s, str) else s) 29 def __str__(m): 30 first = 1 31 s = '' 32 for e,c in sorted(m.d.items(), key=lambda ec:ec[0], reverse=1): 33 if first: s += '-' if c < 0 else '' 34 else: s += ' - ' if c < 0 else ' + ' 35 c = abs(c) 36 if e == 0: s += str(c); continue 37 if c != 1: s += str(c) + '*' 38 s += 'x' if e == 1 else 'x^' + str(e) 39 first = 0 40 return s 41 def __mul__(a, b): 42 d = {} 43 for e,c in a.d.items(): 44 for e2,c2 in b.d.items(): 45 e3 = e + e2; d[e3] = d.get(e3, 0) + c * c2 46 return poly(d) 47 #print(poly('-x^4 - 3*x^2 - 2*x - 5')) 48 a = poly('x - 1'); b = poly('x^2 + x + 1') 49 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 .)
