def test_squarefree():
assert Poly(x-1, x).is_squarefree == True
assert Poly((x-1)**2, x).is_squarefree == False
assert Poly(3*x**2, x).as_squarefree() == Poly(3*x, x)
assert Poly(x**2+2*x+1, x).as_squarefree() == Poly(x+1, x)
assert Poly(x**5-x**4-x+1, x).as_squarefree() == Poly(x**4-1, x)
assert poly_sqf(1, x) == [Poly(1, x)]
assert poly_sqf(x, x) == [Poly(x, x)]
assert poly_sqf(3*x**2, x) == [Poly(3, x), Poly(x, x)]
assert poly_sqf(x**2+2*x+1, x) == [Poly(1, x), Poly(x+1, x)]
assert poly_sqf(x**5-x**4-x+1, x) == \
[Poly(x**3 + x**2 + x + 1, x), Poly(x-1, x)]
assert poly_sqf(x**8+6*x**6+12*x**4+8*x**2, x) == \
[Poly(1, x), Poly(x, x), Poly(x**2+2, x)]
# Bronstein, Symbolic Integration, pp. 52
A = Poly(x**4 - 3*x**2 + 6, x)
D = Poly(x**6 - 5*x**4 + 5*x**2 + 4, x)
f, g = D, A - D.diff(x).mul_term(t)
res, R = poly_subresultants(f, g)
S = poly_sqf(Poly(res, t))
assert S == [Poly(45796, t), Poly(1, t), Poly(4*t**2 + 1, t)]
def test_squarefree():
assert Poly(x-1, x).is_squarefree == True
assert Poly((x-1)**2, x).is_squarefree == False
assert Poly(3*x**2, x).as_squarefree() == Poly(3*x, x)
assert Poly(x**2+2*x+1, x).as_squarefree() == Poly(x+1, x)
assert Poly(x**5-x**4-x+1, x).as_squarefree() == Poly(x**4-1, x)
assert poly_sqf(1, x) == [Poly(1, x)]
assert poly_sqf(x, x) == [Poly(x, x)]
assert poly_sqf(3*x**2, x) == [Poly(3, x), Poly(x, x)]
assert poly_sqf(x**2+2*x+1, x) == [Poly(1, x), Poly(x+1, x)]
assert poly_sqf(x**5-x**4-x+1, x) == \
[Poly(x**3 + x**2 + x + 1, x), Poly(x-1, x)]
assert poly_sqf(x**8+6*x**6+12*x**4+8*x**2, x) == \
[Poly(1, x), Poly(x, x), Poly(x**2+2, x)]
# Bronstein, Symbolic Integration, pp. 52
A = Poly(x**4 - 3*x**2 + 6, x)
D = Poly(x**6 - 5*x**4 + 5*x**2 + 4, x)
f, g = D, A - D.diff(x).mul_term(t)
res, R = poly_subresultants(f, g)
S = poly_sqf(Poly(res, t))
assert S == [Poly(45796, t), Poly(1, t), Poly(4*t**2 + 1, t)]
def test_diff():
f = Poly(a*x**2 + b*x + 2, x)
assert f.diff(x) == Poly(2*a*x + b, x)
assert f.diff(y) == Poly(0, x)
assert f.diff(a) == Poly(x**2, x)
assert f.diff(b) == Poly(x, x)
assert f.diff() == Poly(2*a*x + b, x)
g = Poly(a*x**2 + b*x*y + 2, x, y)
assert g.diff(x) == Poly(2*a*x + b*y, x, y)
assert g.diff(y) == Poly(b*x, x, y)
assert g.diff(a) == Poly(x**2, x, y)
assert g.diff(b) == Poly(x*y, x, y)
assert g.diff() == g
def test_diff():
f = Poly(a*x**2 + b*x + 2, x)
assert f.diff(x) == Poly(2*a*x + b, x)
assert f.diff(y) == Poly(0, x)
assert f.diff(a) == Poly(x**2, x)
assert f.diff(b) == Poly(x, x)
assert f.diff() == Poly(2*a*x + b, x)
g = Poly(a*x**2 + b*x*y + 2, x, y)
assert g.diff(x) == Poly(2*a*x + b*y, x, y)
assert g.diff(y) == Poly(b*x, x, y)
assert g.diff(a) == Poly(x**2, x, y)
assert g.diff(b) == Poly(x*y, x, y)
assert g.diff() == g
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def evtupoly(n,d):
"""
Generates a random polynomial in n
variables with largest degree d.
"""
from scipy.misc import comb
from sympy import var, Poly, Matrix
from numpy import zeros, empty
m = comb(d-1+n,n,exact=1)
print 'number of terms: ', m, '+', n, 'leading terms'
print ''
(C,E) = randpoly(n,d-1)
for i in range(0,n):
C.append(10)
t = [0 for j in range(0,n)]
t[i] = d
E.append(tuple(t))
print 'coefficients:'
print C
print ''
print 'exponents:'
print E
print ''
# Set up variables
S = ['x' + str(i) for i in range(1,n+1)]
# Construct symbolic polynomial
p = Poly(strpoly(S,C,E),var(S))
print 'sympy expression:'
print p
print ''
# Construct polynomial function
def pf(x):
return float(p.eval(zip(S,x)))
# Calculate symbolic gradient
grad = Matrix(n,1, lambda i,j: p.diff(S[i]))
print 'sympy gradient:'
print grad
print ''
# Construct gradient function
def pg(x):
gval = zeros(n)
for i in range(0,n):
gval[i] = float(grad[i].eval(zip(S,x)))
return gval
# Calculate symbolic Hessian
H = Matrix(n,n, lambda i,j: (p.diff(S[i])).diff(S[j]))
print 'sympy Hessian:'
print H
print ''
# Construct Hessian function
def pH(x):
Hval = zeros((n,n))
for i in range(0,n):
for j in range(0,n):
Hval[i,j] = float(H[i,j].eval(zip(S,x)))
return Hval
# Calculate symbolic derivative Tensor
T = empty((n,n,n),dtype='object')
for i in range(0,n):
for j in range(0,n):
for k in range(0,n):
T[i,j,k] = ((p.diff(S[i])).diff(S[j])).diff(S[k])
print 'sympy Tensor:'
print T
print ''
# Construct Hessian bounding function
def bndH(l,u):
"""
Bound Hessian over l_i <= x_i <= u_i
"""
LH = zeros((n,n))
UH = zeros((n,n))
# Lower bound
# See which entries have even exponent
for i in range(0,n):
for j in range(0,n):
# Get exponents and coefficients
E = H[i,j].monoms()
C = H[i,j].coeffs()
# For each term
for t in range(0,len(E)):
# Containers
L = 1
U = 1
# For each variable
for p in range(0,n):
# Get interval approx.
Lp,Up = intpow(l[p],u[p],E[t][p])
# Multiply onto previous
L,U = intmult(L,U,Lp,Up)
LH[i,j] = LH[i,j] + C[t]*L
UH[i,j] = UH[i,j] + C[t]*U
return LH, UH
# Construct derivative tensor bounding function
def bndT(l,u):
"""
Bound derivative Tensor over l_i <= x_i <= u_i
"""
LT = zeros((n,n,n))
UT = zeros((n,n,n))
# Lower bound
# See which entries have even exponent
for i in range(0,n):
for j in range(0,n):
for k in range(0,n):
# Get exponents and coefficients
E = T[i,j,k].monoms()
C = T[i,j,k].coeffs()
# For each term
for t in range(0,len(E)):
# Containers
L = 1
U = 1
# For each variable
for p in range(0,n):
# Get interval approx.
Lp,Up = intpow(l[p],u[p],E[t][p])
# Multiply onto previous
L,U = intmult(L,U,Lp,Up)
LT[i,j,k] = LT[i,j,k] + C[t]*L
UT[i,j,k] = UT[i,j,k] + C[t]*U
return LT, UT
# Return f(x), g(x), H(x) and H, T bounding functions
return pf, pg, pH, bndH, bndT
def evtupoly(n, d):
"""
Generates a random polynomial in n
variables with largest degree d.
"""
from scipy.misc import comb
from sympy import var, Poly, Matrix
from numpy import zeros, empty
m = comb(d - 1 + n, n, exact=1)
print 'number of terms: ', m, '+', n, 'leading terms'
print ''
(C, E) = randpoly(n, d - 1)
for i in range(0, n):
C.append(10)
t = [0 for j in range(0, n)]
t[i] = d
E.append(tuple(t))
print 'coefficients:'
print C
print ''
print 'exponents:'
print E
print ''
# Set up variables
S = ['x' + str(i) for i in range(1, n + 1)]
# Construct symbolic polynomial
p = Poly(strpoly(S, C, E), var(S))
print 'sympy expression:'
print p
print ''
# Construct polynomial function
def pf(x):
return float(p.eval(zip(S, x)))
# Calculate symbolic gradient
grad = Matrix(n, 1, lambda i, j: p.diff(S[i]))
print 'sympy gradient:'
print grad
print ''
# Construct gradient function
def pg(x):
gval = zeros(n)
for i in range(0, n):
gval[i] = float(grad[i].eval(zip(S, x)))
return gval
# Calculate symbolic Hessian
H = Matrix(n, n, lambda i, j: (p.diff(S[i])).diff(S[j]))
print 'sympy Hessian:'
print H
print ''
# Construct Hessian function
def pH(x):
Hval = zeros((n, n))
for i in range(0, n):
for j in range(0, n):
Hval[i, j] = float(H[i, j].eval(zip(S, x)))
return Hval
# Calculate symbolic derivative Tensor
T = empty((n, n, n), dtype='object')
for i in range(0, n):
for j in range(0, n):
for k in range(0, n):
T[i, j, k] = ((p.diff(S[i])).diff(S[j])).diff(S[k])
print 'sympy Tensor:'
print T
print ''
# Construct Hessian bounding function
def bndH(l, u):
"""
Bound Hessian over l_i <= x_i <= u_i
"""
LH = zeros((n, n))
UH = zeros((n, n))
# Lower bound
# See which entries have even exponent
for i in range(0, n):
for j in range(0, n):
# Get exponents and coefficients
E = H[i, j].monoms()
C = H[i, j].coeffs()
# For each term
for t in range(0, len(E)):
# Containers
L = 1
U = 1
# For each variable
for p in range(0, n):
# Get interval approx.
Lp, Up = intpow(l[p], u[p], E[t][p])
# Multiply onto previous
L, U = intmult(L, U, Lp, Up)
LH[i, j] = LH[i, j] + C[t] * L
UH[i, j] = UH[i, j] + C[t] * U
return LH, UH
# Construct derivative tensor bounding function
def bndT(l, u):
"""
Bound derivative Tensor over l_i <= x_i <= u_i
"""
LT = zeros((n, n, n))
UT = zeros((n, n, n))
# Lower bound
# See which entries have even exponent
for i in range(0, n):
for j in range(0, n):
for k in range(0, n):
# Get exponents and coefficients
E = T[i, j, k].monoms()
C = T[i, j, k].coeffs()
# For each term
for t in range(0, len(E)):
# Containers
L = 1
U = 1
# For each variable
for p in range(0, n):
# Get interval approx.
Lp, Up = intpow(l[p], u[p], E[t][p])
# Multiply onto previous
L, U = intmult(L, U, Lp, Up)
LT[i, j, k] = LT[i, j, k] + C[t] * L
UT[i, j, k] = UT[i, j, k] + C[t] * U
return LT, UT
# Return f(x), g(x), H(x) and H, T bounding functions
return pf, pg, pH, bndH, bndT
els.append(LC(r, v))
return els
if __name__ == '__main__':
from sympy import Poly
P = Poly('x*(x^3 - x)')
Q = Poly('x^3 + x')
print(sRes(P, Q))
P = Poly('x^2 + y^2 + z^2 -1')
x, y, z = P.gens
P1 = P.diff(z)
print(SyHa(P, P1, 0, z)) # from [SPR, example 5.17]
for j in range(2):
print(j, sRes(P, P1, j).simplify())
Q = Poly('x^2 + y^2 -1')
x, y = Q.gens
Q1 = Q.diff(y)
print(SyHa(Q, Q1, 0, y))
P = Poly('x^2 + y^2 + z^2 -1')
x, y, z = P.gens
for j in range(2):
print(j, sRes(Q, Q1, j).simplify())