def square_root_of_expr(expr):
"""
If expression is product of even powers then every power is divided
by two and the product is returned. If some terms in product are
not even powers the sqrt of the absolute value of the expression is
returned. If the expression is a number the sqrt of the absolute
value of the number is returned.
"""
if expr.is_number:
if expr > 0:
return(sqrt(expr))
else:
return(sqrt(-expr))
else:
expr = trigsimp(expr)
(coef, pow_lst) = sqf_list(expr)
if coef != S(1):
if coef.is_number:
coef = square_root_of_expr(coef)
else:
coef = sqrt(abs(coef)) # Product coefficient not a number
for p in pow_lst:
(f, n) = p
if n % 2 != 0:
return(sqrt(abs(expr))) # Product not all even powers
else:
coef *= f ** (n / 2) # Positive sqrt of the square of an expression
return coef
def square_root_of_expr(expr):
"""
If expression is product of even powers then every power is divided
by two and the product is returned. If some terms in product are
not even powers the sqrt of the absolute value of the expression is
returned. If the expression is a number the sqrt of the absolute
value of the number is returned.
"""
if expr.is_number:
if expr > 0:
return(sqrt(expr))
else:
return(sqrt(-expr))
else:
expr = trigsimp(expr)
(coef, pow_lst) = sqf_list(expr)
if coef != S(1):
if coef.is_number:
coef = square_root_of_expr(coef)
else:
coef = sqrt(abs(coef)) # Product coefficient not a number
for p in pow_lst:
(f, n) = p
if n % 2 != 0:
return(sqrt(abs(expr))) # Product not all even powers
else:
coef *= f ** (n / 2) # Positive sqrt of the square of an expression
return coef
def test3():
x, a, b, c = map(Symbol, ['x', 'a', 'b', 'c'])
#f = x**3 - 8.99085235595703*x**2 + 14.963430343676*x + 24.954282699633
f = x**3 - 8.990852355957031 * x**2 + 14.96343034367601 * x + 24.95428269963304
#f = (x + 1) * (x - 4.995426177978516)**2
f = expand(f)
print('f : ', f)
f_ = diff(f)
print('f_ : ', f_)
a0 = gcd(f, f_)
print('a0 : ', a0)
b1 = quo(f, a0)
print('b1 : ', b1)
c1 = quo(f_, a0)
print('c1 : ', c1)
d1 = c1 - diff(b1)
print('d1 : ', d1)
a1 = gcd(b1, d1)
print('a1 : ', a1)
b2 = quo(b1, a1)
print('b2 : ', b2)
L = sqf_list(f)
#L = sqf(f)
print(L)
def _square_free(Phi,var):
"""
Given a polynomial `\Phi \in \mathbb{L}[Z]` returns a collection
of pairs `\{(\Psi,r)\}` with `\Psi \in \mathbb{L}[Z]` and `r` a
positive integer such that each `\Psi` is square-free and `\Phi =
\prod \Psi^r` and the `\Psi`'s are pairwise coprime.
ALGORITHM:
Such a decomposition can be obtained with derivations and gcd
computations without any factorization algorithm. It's implemented
in sympy as ``sympy.sqf`` and ``sympy.sqf_list``
"""
return sympy.sqf_list(Phi,var)[1]
def square_root_of_expr(expr):
if expr.is_number:
if expr > 0:
return(sqrt(expr))
else:
return(sqrt(-expr))
else:
expr = trigsimp(expr)
(coef, pow_lst) = sqf_list(expr)
if coef != S(1):
if coef.is_number:
coef = square_root_of_expr(coef)
else:
coef = sqrt(abs(coef))
for p in pow_lst:
(f, n) = p
if n % 2 != 0:
return(sqrt(abs(expr)))
else:
coef *= f ** (n / 2)
return coef
def test2():
x, a, b, c = map(Symbol, ['x', 'a', 'b', 'c'])
#f = 2*x**2 + 5*x**3 + 4*x**4 + x**5
f = (x + 2) * x**2 * (x + 1)**2
f = expand(f)
#print('ex : ', expand((x + 2) * (x**2 + x)**2))
print('f : ', f)
f_ = diff(f)
print('f_ : ', f_)
a0 = gcd(f, f_)
print('a0 : ', a0)
b1 = quo(f, a0)
print('b1 : ', b1)
c1 = quo(f_, a0)
print('c1 : ', c1)
d1 = c1 - diff(b1)
print('d1 : ', d1)
a1 = gcd(b1, d1)
print('a1 : ', a1)
b2 = quo(b1, a1)
print('b2 : ', b2)
c2 = quo(d1, a1)
print('c2 : ', c2)
d2 = c2 - diff(b2)
print('d2 : ', d2)
a2 = gcd(b2, d2)
print('a2 : ', a2)
b3 = quo(b2, a2)
print('b3 : ', b3)
L = sqf_list(f)
#sqf(f)
print(L)
import sympy as s from sympy.abc import x, y f = (x+1)**2 * (x+3)*(x+4) f = s.expand(f) # I expect: [(x**2+7x + 12), 1] , [(x+1), 2] # but it factors (x+3) , (x+4) print(s.sqf_list(f))
