def poly_reduce(f, g, *symbols):
"""Removes common content from a pair of polynomials.
>>> from sympy import *
>>> x = Symbol('x')
>>> f = Poly(2930944*x**6 + 2198208*x**4 + 549552*x**2 + 45796, x)
>>> g = Poly(17585664*x**5 + 8792832*x**3 + 1099104*x, x)
>>> F, G = poly_reduce(f, g)
>>> F
Poly(64*x**6 + 48*x**4 + 12*x**2 + 1, x)
>>> G
Poly(384*x**5 + 192*x**3 + 24*x, x)
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
fc = int(f.content)
gc = int(g.content)
cont = igcd(fc, gc)
if cont != 1:
f = f.div_term(cont)
g = g.div_term(cont)
return f, g
def poly_reduce(f, g, *symbols):
"""Removes common content from a pair of polynomials.
>>> from sympy import *
>>> x = Symbol('x')
>>> f = Poly(2930944*x**6 + 2198208*x**4 + 549552*x**2 + 45796, x)
>>> g = Poly(17585664*x**5 + 8792832*x**3 + 1099104*x, x)
>>> F, G = poly_reduce(f, g)
>>> F
Poly(64*x**6 + 48*x**4 + 12*x**2 + 1, x)
>>> G
Poly(384*x**5 + 192*x**3 + 24*x, x)
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
fc = int(f.content)
gc = int(g.content)
cont = igcd(fc, gc)
if cont != 1:
f = f.div_term(cont)
g = g.div_term(cont)
return f, g
def poly_pdiv(f, g, *symbols):
"""Univariate polynomial pseudo-division with remainder.
Given univariate polynomials f and g over an integral domain D[x]
applying classical division algorithm to LC(g)**(d + 1) * f and g
where d = max(-1, deg(f) - deg(g)), compute polynomials q and r
such that LC(g)**(d + 1)*f = g*q + r and r = 0 or deg(r) < deg(g).
Polynomials q and r are called the pseudo-quotient of f by g and
the pseudo-remainder of f modulo g respectively.
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
q, r = Poly((), *symbols, **flags), f
coeff, N = g.LC, f.degree - g.degree + 1
while not r.is_zero:
M = r.degree - g.degree
if M < 0:
break
else:
T, N = (r.LC, (M, )), N - 1
q = q.mul_term(coeff).add_term(*T)
r = r.mul_term(coeff) - g.mul_term(*T)
return (q.mul_term(coeff**N), r.mul_term(coeff**N))
def poly_pdiv(f, g, *symbols):
"""Univariate polynomial pseudo-division with remainder.
Given univariate polynomials f and g over an integral domain D[x]
applying classical division algorithm to LC(g)**(d + 1) * f and g
where d = max(-1, deg(f) - deg(g)), compute polynomials q and r
such that LC(g)**(d + 1)*f = g*q + r and r = 0 or deg(r) < deg(g).
Polynomials q and r are called the pseudo-quotient of f by g and
the pseudo-remainder of f modulo g respectively.
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
q, r = Poly((), *symbols, **flags), f
coeff, N = g.LC, f.degree - g.degree + 1
while not r.is_zero:
M = r.degree - g.degree
if M < 0:
break
else:
T, N = (r.LC, (M,)), N - 1
q = q.mul_term(coeff).add_term(*T)
r = r.mul_term(coeff)-g.mul_term(*T)
return (q.mul_term(coeff**N), r.mul_term(coeff**N))
def poly_half_gcdex(f, g, *symbols):
"""Half extended Euclidean algorithm.
Efficiently computes gcd(f, g) and one of the coefficients
in extended Euclidean algorithm. Formally, given univariate
polynomials f and g over an Euclidean domain, computes s
and h, such that h = gcd(f, g) and s*f = h (mod g).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
a = Poly(S.One, *symbols, **flags)
b = Poly((), *symbols, **flags)
while not g.is_zero:
q, r = poly_div(f, g)
f, g = g, r
c = a - q*b
a, b = b, c
return a.div_term(f.LC), f.as_monic()
def poly_half_gcdex(f, g, *symbols):
"""Half extended Euclidean algorithm.
Efficiently computes gcd(f, g) and one of the coefficients
in extended Euclidean algorithm. Formally, given univariate
polynomials f and g over an Euclidean domain, computes s
and h, such that h = gcd(f, g) and s*f = h (mod g).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
a = Poly(S.One, *symbols, **flags)
b = Poly((), *symbols, **flags)
while not g.is_zero:
q, r = poly_div(f, g)
f, g = g, r
c = a - q * b
a, b = b, c
return a.div_term(f.LC), f.as_monic()
def poly_subresultants(f, g, *symbols, **flags):
"""Computes subresultant PRS of two univariate polynomials.
Polynomial remainder sequence (PRS) is a fundamental tool in
computer algebra as it gives as a sub-product the polynomial
greatest common divisor (GCD), provided that the coefficient
domain is an unique factorization domain.
There are several methods for computing PRS, eg.: Euclidean
PRS, where the most famous algorithm is used, primitive PRS
and, finally, subresultants which are implemented here.
The Euclidean approach is reasonably efficient but suffers
severely from coefficient growth. The primitive algorithm
avoids this but requires a lot of coefficient computations.
Subresultants solve both problems and so it is efficient and
have moderate coefficient growth. The current implementation
uses pseudo-divisions which is well suited for coefficients
in integral domains or number fields.
Formally, given univariate polynomials f and g over an UFD,
then a sequence (R_0, R_1, ..., R_k, 0, ...) is a polynomial
remainder sequence where R_0 = f, R_1 = g, R_k != 0 and R_k
is similar to gcd(f, g).
The result is returned as tuple (res, R) where R is the PRS
sequence and res is the resultant of the input polynomials.
If only polynomial remainder sequence is important, then by
setting res=False in keyword arguments expensive computation
of the resultant can be avoided (only PRS is returned).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] M. Keber, Division-Free computation of subresultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006,
Saarbrucken, 2006
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
else:
symbols = f.symbols
n, m = f.degree, g.degree
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = S(-1)**(d + 1)
c = S(-1)
B, D = [b], [d]
h = poly_prem(f, g)
h = h.mul_term(b)
while not h.is_zero:
k = h.degree
R.append(h)
lc = g.LC
C = (-lc)**d / c**(d-1)
c = Poly.cancel(C)
b = -lc * c**(m-k)
f, g, m, d = g, h, k, m-k
B.append(b)
D.append(d)
h = poly_prem(f, g)
h = h.div_term(b)
if not flags.get('res', True):
return R
if R[-1].degree > 0:
return (Poly((), *symbols), R)
if R[-2].is_one:
return (R[-1], R)
s, c, i = 1, S(1), 1
for b, d in zip(B, D)[:-1]:
u = R[i-1].degree
v = R[i ].degree
w = R[i+1].degree
if u % 2 and v % 2:
s = -s
lc = R[i].LC
C = c*(b/lc**(1 + d))**v * lc**(u - w)
c = Poly.cancel(C)
i += 1
j = R[-2].degree
return (R[-1]**j*s*c, R)
def poly_subresultants(f, g, *symbols, **flags):
"""Computes subresultant PRS of two univariate polynomials.
Polynomial remainder sequence (PRS) is a fundamental tool in
computer algebra as it gives as a sub-product the polynomial
greatest common divisor (GCD), provided that the coefficient
domain is an unique factorization domain.
There are several methods for computing PRS, eg.: Euclidean
PRS, where the most famous algorithm is used, primitive PRS
and, finally, subresultants which are implemented here.
The Euclidean approach is reasonably efficient but suffers
severely from coefficient growth. The primitive algorithm
avoids this but requires a lot of coefficient computations.
Subresultants solve both problems and so it is efficient and
have moderate coefficient growth. The current implementation
uses pseudo-divisions which is well suited for coefficients
in integral domains or number fields.
Formally, given univariate polynomials f and g over an UFD,
then a sequence (R_0, R_1, ..., R_k, 0, ...) is a polynomial
remainder sequence where R_0 = f, R_1 = g, R_k != 0 and R_k
is similar to gcd(f, g).
The result is returned as tuple (res, R) where R is the PRS
sequence and res is the resultant of the input polynomials.
If only polynomial remainder sequence is important, then by
setting res=False in keyword arguments expensive computation
of the resultant can be avoided (only PRS is returned).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] M. Keber, Division-Free computation of subresultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006,
Saarbrucken, 2006
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
else:
symbols = f.symbols
n, m = f.degree, g.degree
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = S(-1)**(d + 1)
c = S(-1)
B, D = [b], [d]
h = poly_prem(f, g)
h = h.mul_term(b)
while not h.is_zero:
k = h.degree
R.append(h)
lc = g.LC
C = (-lc)**d / c**(d - 1)
c = Poly.cancel(C)
b = -lc * c**(m - k)
f, g, m, d = g, h, k, m - k
B.append(b)
D.append(d)
h = poly_prem(f, g)
h = h.div_term(b)
if not flags.get('res', True):
return R
if R[-1].degree > 0:
return (Poly((), *symbols), R)
if R[-2].is_one:
return (R[-1], R)
s, c, i = 1, S(1), 1
for b, d in zip(B, D)[:-1]:
u = R[i - 1].degree
v = R[i].degree
w = R[i + 1].degree
if u % 2 and v % 2:
s = -s
lc = R[i].LC
C = c * (b / lc**(1 + d))**v * lc**(u - w)
c = Poly.cancel(C)
i += 1
j = R[-2].degree
return (R[-1]**j * s * c, R)
def poly_resultant(f, g, *symbols):
"""Computes resultant of two univariate polynomials.
Resultants are a classical algebraic tool for determining if
a system of n polynomials in n-1 variables have common root
without explicitly solving for the roots.
They are efficiently represented as determinants of Bezout
matrices whose entries are computed using O(n**2) additions
and multiplications where n = max(deg(f), deg(g)).
>>> from sympy import *
>>> x,y = symbols('xy')
Polynomials x**2-1 and (x-1)**2 have common root:
>>> poly_resultant(x**2-1, (x-1)**2, x)
0
For more information on the implemented algorithm refer to:
[1] Eng-Wee Chionh, Fast Computation of the Bezout and Dixon
Resultant Matrices, Journal of Symbolic Computation, ACM,
Volume 33, Issue 1, January 2002, Pages 13-29
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
n, m = f.degree, g.degree
N = max(n, m)
if n < m:
p = f.as_uv_dict()
q = g.as_uv_dict()
else:
q = f.as_uv_dict()
p = g.as_uv_dict()
import sympy.matrices
B = sympy.matrices.zeros(N)
for i in xrange(N):
for j in xrange(i, N):
if i in p and j + 1 in q:
B[i, j] += p[i] * q[j + 1]
if j + 1 in p and i in q:
B[i, j] -= p[j + 1] * q[i]
for i in xrange(1, N - 1):
for j in xrange(i, N - 1):
B[i, j] += B[i - 1, j + 1]
for i in xrange(N):
for j in xrange(i + 1, N):
B[j, i] = B[i, j]
det = B.det()
if not det:
return det
else:
if n >= m:
det /= f.LC**(n - m)
else:
det /= g.LC**(m - n)
sign = (-1)**(n * (n - 1) // 2)
if det.is_Atom:
return sign * det
else:
return sign * Poly.cancel(det)
def poly_gcd(f, g, *symbols):
"""Compute greatest common divisor of two polynomials.
Given two univariate polynomials, subresultants are used
to compute the GCD. In multivariate case Groebner basis
approach is used together with f*g = gcd(f, g)*lcm(f, g)
well known formula.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_zero and g.is_zero:
return f
if f.is_constant:
if f.is_zero:
cont, g = g.as_primitive()
return g.mul_term(cont / g.LC)
if f.is_one:
return f
if g.is_constant:
if g.is_zero:
cont, f = f.as_primitive()
return f.mul_term(cont / f.LC)
if g.is_one:
return g
if f.is_monomial and g.is_monomial:
monom = monomial_gcd(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(igcd(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
cf, f = f.as_primitive()
cg, g = g.as_primitive()
gcd = igcd(int(cf), int(cg))
if f.is_multivariate:
h = poly_div(f * g, poly_lcm(f, g))[0]
else:
h = poly_subresultants(f, g, res=False)[-1]
if gcd != 1:
return h.mul_term(gcd / h.LC)
else:
return h.as_monic()
def poly_lcm(f, g, *symbols):
"""Computes least common multiple of two polynomials.
Given two univariate polynomials, the LCM is computed via
f*g = gcd(f, g)*lcm(f, g) formula. In multivariate case, we
compute the unique generator of the intersection of the two
ideals, generated by f and g. This is done by computing a
Groebner basis, with respect to any lexicographic ordering,
of t*f and (1 - t)*g, where t is an unrelated symbol and
filtering out solution that does not contain t.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_monomial and g.is_monomial:
monom = monomial_lcm(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(ilcm(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
fc, f = f.as_primitive()
gc, g = g.as_primitive()
lcm = ilcm(int(fc), int(gc))
if f.is_multivariate:
t = Symbol('t', dummy=True)
lex = {'order': 'lex'}
f_monoms = [(1, ) + monom for monom in f.monoms]
F = Poly((f.coeffs, f_monoms), t, *symbols, **lex)
g_monoms = [ (0,) + monom for monom in g.monoms ] + \
[ (1,) + monom for monom in g.monoms ]
g_coeffs = list(g.coeffs) + [-coeff for coeff in g.coeffs]
G = Poly(dict(zip(g_monoms, g_coeffs)), t, *symbols, **lex)
def independent(h):
return all(not monom[0] for monom in h.monoms)
H = [h for h in poly_groebner((F, G)) if independent(h)]
if lcm != 1:
h_coeffs = [coeff * lcm for coeff in H[0].coeffs]
else:
h_coeffs = H[0].coeffs
h_monoms = [monom[1:] for monom in H[0].monoms]
return Poly(dict(zip(h_monoms, h_coeffs)), *symbols, **flags)
else:
h = poly_div(f * g, poly_gcd(f, g))[0]
if lcm != 1:
return h.mul_term(lcm / h.LC)
else:
return h.as_monic()
def poly_div(f, g, *symbols):
"""Generalized polynomial division with remainder.
Given polynomial f and a set of polynomials g = (g_1, ..., g_n)
compute a set of quotients q = (q_1, ..., q_n) and remainder r
such that f = q_1*f_1 + ... + q_n*f_n + r, where r = 0 or r is
a completely reduced polynomial with respect to g.
In particular g can be a tuple, list or a singleton. All g_i
and f can be given as Poly class instances or as expressions.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 62
[2] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
r = Poly((), *symbols, **flags)
if isinstance(g, Basic):
if g.is_constant:
if g.is_zero:
raise ZeroDivisionError
elif g.is_one:
return f, r
else:
return f.div_term(g.LC), r
if g.is_monomial:
LC, LM = g.lead_term
q_coeffs, q_monoms = [], []
r_coeffs, r_monoms = [], []
for coeff, monom in f.iter_terms():
quotient = monomial_div(monom, LM)
if quotient is not None:
coeff /= LC
q_coeffs.append(Poly.cancel(coeff))
q_monoms.append(quotient)
else:
r_coeffs.append(coeff)
r_monoms.append(monom)
return (Poly((q_coeffs, q_monoms), *symbols,
**flags), Poly((r_coeffs, r_monoms), *symbols,
**flags))
g, q = [g], [r]
else:
q = [r] * len(g)
while not f.is_zero:
for i, h in enumerate(g):
monom = monomial_div(f.LM, h.LM)
if monom is not None:
coeff = Poly.cancel(f.LC / h.LC)
q[i] = q[i].add_term(coeff, monom)
f -= h.mul_term(coeff, monom)
break
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
if len(q) != 1:
return q, r
else:
return q[0], r
def poly_groebner(f, *symbols, **flags):
"""Computes reduced Groebner basis for a set of polynomials.
Given a set of multivariate polynomials F, find another set G,
such that Ideal F = Ideal G and G is a reduced Groebner basis.
The resulting basis is unique and has monic generators.
Groebner bases can be used to choose specific generators for a
polynomial ideal. Because these bases are unique you can check
for ideal equality by comparing the Groebner bases. To see if
one polynomial lies in an ideal, divide by the elements in the
base and see if the remainder vanishes.
They can also be used to solve systems of polynomial equations
as, by choosing lexicographic ordering, you can eliminate one
variable at a time, provided that the ideal is zero-dimensional
(finite number of solutions).
>>> from sympy import *
>>> x,y = symbols('xy')
>>> G = poly_groebner([x**2 + y**3, y**2-x], x, y, order='lex')
>>> [ g.as_basic() for g in G ]
[x - y**2, y**3 + y**4]
For more information on the implemented algorithm refer to:
[1] N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional
Systems Theory and Applications, Springer, 2003, pp. 98+
[2] A. Giovini, T. Mora, "One sugar cube, please" or Selection
strategies in Buchberger algorithm, Proc. ISSAC '91, ACM
[3] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
[4] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 62
"""
if isinstance(f, (tuple, list, set)):
f, g = f[0], list(f[1:])
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
else:
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
return [f.as_monic()]
compare = monomial_cmp(flags.get('order'))
f = [h for h in [f] + g if h]
if not f:
return [Poly((), *symbols, **flags)]
R, P, G, B, F = set(), set(), set(), {}, {}
for i, h in enumerate(f):
F[h] = i
R.add(i)
def normal(g, H):
h = poly_div(g, [f[i] for i in H])[1]
if h.is_zero:
return None
else:
if not h in F:
F[h] = len(f)
f.append(h)
return F[h], h.LM
def generate(R, P, G, B):
while R:
h = normal(f[R.pop()], G | P)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
P0 = set(p for p in P if monomial_div(f[p].LM, LM))
G, P, R = G - G0, P - P0 | set([k]), R | G0 | P0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
G |= P
for i in G:
for j in P:
if i == j:
continue
if i < j:
k = (i, j)
else:
k = (j, i)
if k not in B:
B[k] = monomial_lcm(f[i].LM, f[j].LM)
G = set([normal(f[g], G - set([g]))[0] for g in G])
return R, P, G, B
R, P, G, B = generate(R, P, G, B)
while B:
k, M = B.items()[0]
for l, N in B.iteritems():
if compare(M, N) == 1:
k, M = l, N
del B[k]
i, j = k[0], k[1]
p, q = f[i], f[j]
p_LM, q_LM = p.LM, q.LM
if M == monomial_mul(p_LM, q_LM):
continue
criterion = False
for g in G:
if g == i or g == j:
continue
if (min(i, g), max(i, g)) not in B:
continue
if (min(j, g), max(j, g)) not in B:
continue
if not monomial_div(M, f[g].LM):
continue
criterion = True
break
if criterion:
continue
p = p.mul_term(1 / p.LC, monomial_div(M, p_LM))
q = q.mul_term(1 / q.LC, monomial_div(M, q_LM))
h = normal(p - q, G)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
R, P, G = G0, set([k]), G - G0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
R, P, G, B = generate(R, P, G, B)
G = [f[g].as_monic() for g in G]
G = sorted(G, compare, lambda p: p.LM)
return list(reversed(G))
def poly_groebner(f, *symbols, **flags):
"""Computes reduced Groebner basis for a set of polynomials.
Given a set of multivariate polynomials F, find another set G,
such that Ideal F = Ideal G and G is a reduced Groebner basis.
The resulting basis is unique and has monic generators.
Groebner bases can be used to choose specific generators for a
polynomial ideal. Because these bases are unique you can check
for ideal equality by comparing the Groebner bases. To see if
one polynomial lies in an ideal, divide by the elements in the
base and see if the remainder vanishes.
They can also be used to solve systems of polynomial equations
as, by choosing lexicographic ordering, you can eliminate one
variable at a time, provided that the ideal is zero-dimensional
(finite number of solutions).
>>> from sympy import *
>>> x,y = symbols('xy')
>>> G = poly_groebner([x**2 + y**3, y**2-x], x, y, order='lex')
>>> [ g.as_basic() for g in G ]
[x - y**2, y**3 + y**4]
For more information on the implemented algorithm refer to:
[1] N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional
Systems Theory and Applications, Springer, 2003, pp. 98+
[2] A. Giovini, T. Mora, "One sugar cube, please" or Selection
strategies in Buchberger algorithm, Proc. ISSAC '91, ACM
[3] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
[4] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 62
"""
if isinstance(f, (tuple, list, set)):
f, g = f[0], list(f[1:])
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
else:
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
return [f.as_monic()]
compare = monomial_cmp(flags.get('order'))
f = [ h for h in [f] + g if h ]
if not f:
return [Poly((), *symbols, **flags)]
R, P, G, B, F = set(), set(), set(), {}, {}
for i, h in enumerate(f):
F[h] = i; R.add(i)
def normal(g, H):
h = poly_div(g, [ f[i] for i in H ])[1]
if h.is_zero:
return None
else:
if not F.has_key(h):
F[h] = len(f)
f.append(h)
return F[h], h.LM
def generate(R, P, G, B):
while R:
h = normal(f[R.pop()], G | P)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
P0 = set(p for p in P if monomial_div(f[p].LM, LM))
G, P, R = G - G0, P - P0 | set([k]), R | G0 | P0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
G |= P
for i in G:
for j in P:
if i == j:
continue
if i < j:
k = (i, j)
else:
k = (j, i)
if not B.has_key(k):
B[k] = monomial_lcm(f[i].LM, f[j].LM)
G = set([ normal(f[g], G - set([g]))[0] for g in G ])
return R, P, G, B
R, P, G, B = generate(R, P, G, B)
while B:
k, M = B.items()[0]
for l, N in B.iteritems():
if compare(M, N) == 1:
k, M = l, N
del B[k]
i, j = k[0], k[1]
p, q = f[i], f[j]
p_LM, q_LM = p.LM, q.LM
if M == monomial_mul(p_LM, q_LM):
continue
criterion = False
for g in G:
if g == i or g == j:
continue
if not B.has_key((min(i, g), max(i, g))):
continue
if not B.has_key((min(j, g), max(j, g))):
continue
if not monomial_div(M, f[g].LM):
continue
criterion = True
break
if criterion:
continue
p = p.mul_term(1/p.LC, monomial_div(M, p_LM))
q = q.mul_term(1/q.LC, monomial_div(M, q_LM))
h = normal(p - q, G)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
R, P, G = G0, set([k]), G - G0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
R, P, G, B = generate(R, P, G, B)
G = [ f[g].as_monic() for g in G ]
G = sorted(G, compare, lambda p: p.LM)
return list(reversed(G))
def poly_subresultants(f, g, *symbols):
"""Computes subresultant PRS of two univariate polynomials.
Polynomial remainder sequence (PRS) is a fundamental tool in
computer algebra as it gives as a sub-product the polynomial
greatest common divisor (GCD), provided that the coefficient
domain is an unique factorization domain.
There are several methods for computing PRS, eg.: Euclidean
PRS, where the most famous algorithm is used, primitive PRS
and, finally, subresultants which are implemented here.
The Euclidean approach is reasonably efficient but suffers
severely from coefficient growth. The primitive algorithm
avoids this but requires a lot of coefficient computations.
Subresultants solve both problems and so it is efficient and
have moderate coefficient growth. The current implementation
uses pseudo-divisions which is well suited for coefficients
in integral domains or number fields.
Formally, given univariate polynomials f and g over an UFD,
then a sequence (R_0, R_1, ..., R_k, 0, ...) is a polynomial
remainder sequence where R_0 = f, R_1 = g, R_k != 0 and R_k
is similar to gcd(f, g).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] M. Keber, Division-Free computation of subresultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006,
Saarbrucken, 2006
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
n, m = f.degree, g.degree
if n < m:
f, g = g, f
n, m = m, n
prs = [f, g]
d = n - m
b = (-1)**(d + 1)
h = poly_pdiv(f, g)[1]
h = h.mul_term(b)
k = h.degree
c = S.NegativeOne
while not h.is_zero:
prs.append(h)
coeff = g.LC
c = (-coeff)**d / c**(d - 1)
b = -coeff * c**(m - k)
f, g, m, d = g, h, k, m - k
h = poly_pdiv(f, g)[1]
h = h.div_term(b)
k = h.degree
return prs
def poly_subresultants(f, g, *symbols):
"""Computes subresultant PRS of two univariate polynomials.
Polynomial remainder sequence (PRS) is a fundamental tool in
computer algebra as it gives as a sub-product the polynomial
greatest common divisor (GCD), provided that the coefficient
domain is an unique factorization domain.
There are several methods for computing PRS, eg.: Euclidean
PRS, where the most famous algorithm is used, primitive PRS
and, finally, subresultants which are implemented here.
The Euclidean approach is reasonably efficient but suffers
severely from coefficient growth. The primitive algorithm
avoids this but requires a lot of coefficient computations.
Subresultants solve both problems and so it is efficient and
have moderate coefficient growth. The current implementation
uses pseudo-divisions which is well suited for coefficients
in integral domains or number fields.
Formally, given univariate polynomials f and g over an UFD,
then a sequence (R_0, R_1, ..., R_k, 0, ...) is a polynomial
remainder sequence where R_0 = f, R_1 = g, R_k != 0 and R_k
is similar to gcd(f, g).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] M. Keber, Division-Free computation of subresultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006,
Saarbrucken, 2006
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
n, m = f.degree, g.degree
if n < m:
f, g = g, f
n, m = m, n
prs = [f, g]
d = n - m
b = (-1)**(d + 1)
h = poly_pdiv(f, g)[1]
h = h.mul_term(b)
k = h.degree
c = S.NegativeOne
while not h.is_zero:
prs.append(h)
coeff = g.LC
c = (-coeff)**d / c**(d-1)
b = -coeff * c**(m-k)
f, g, m, d = g, h, k, m-k
h = poly_pdiv(f, g)[1]
h = h.div_term(b)
k = h.degree
return prs
def poly_resultant(f, g, *symbols):
"""Computes resultant of two univariate polynomials.
Resultants are a classical algebraic tool for determining if
a system of n polynomials in n-1 variables have common root
without explicitly solving for the roots.
They are efficiently represented as determinants of Bezout
matrices whose entries are computed using O(n**2) additions
and multiplications where n = max(deg(f), deg(g)).
>>> from sympy import *
>>> x,y = symbols('xy')
Polynomials x**2-1 and (x-1)**2 have common root:
>>> poly_resultant(x**2-1, (x-1)**2, x)
0
For more information on the implemented algorithm refer to:
[1] Eng-Wee Chionh, Fast Computation of the Bezout and Dixon
Resultant Matrices, Journal of Symbolic Computation, ACM,
Volume 33, Issue 1, January 2002, Pages 13-29
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
n, m = f.degree, g.degree
N = max(n, m)
if n < m:
p = f.as_uv_dict()
q = g.as_uv_dict()
else:
q = f.as_uv_dict()
p = g.as_uv_dict()
import sympy.matrices
B = sympy.matrices.zeros(N)
for i in xrange(N):
for j in xrange(i, N):
if p.has_key(i) and q.has_key(j+1):
B[i, j] += p[i] * q[j+1]
if p.has_key(j+1) and q.has_key(i):
B[i, j] -= p[j+1] * q[i]
for i in xrange(1, N-1):
for j in xrange(i, N-1):
B[i, j] += B[i-1, j+1]
for i in xrange(N):
for j in xrange(i+1, N):
B[j, i] = B[i, j]
det = B.det()
if not det:
return det
else:
if n >= m:
det /= f.LC**(n-m)
else:
det /= g.LC**(m-n)
sign = (-1)**(n*(n-1)//2)
if det.is_Atom:
return sign * det
else:
return sign * Poly.cancel(det)
def poly_gcd(f, g, *symbols):
"""Compute greatest common divisor of two polynomials.
Given two univariate polynomials, subresultants are used
to compute the GCD. In multivariate case Groebner basis
approach is used together with f*g = gcd(f, g)*lcm(f, g)
well known formula.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_zero and g.is_zero:
return f
if f.is_constant:
if f.is_zero:
cont, g = g.as_primitive()
return g.mul_term(cont / g.LC)
if f.is_one:
return f
if g.is_constant:
if g.is_zero:
cont, f = f.as_primitive()
return f.mul_term(cont / f.LC)
if g.is_one:
return g
if f.is_monomial and g.is_monomial:
monom = monomial_gcd(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(igcd(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
cf, f = f.as_primitive()
cg, g = g.as_primitive()
gcd = igcd(int(cf), int(cg))
if f.is_multivariate:
h = poly_div(f*g, poly_lcm(f, g))[0]
else:
h = poly_subresultants(f, g)[-1]
if gcd != 1:
return h.mul_term(gcd / h.LC)
else:
return h.as_monic()
def poly_lcm(f, g, *symbols):
"""Computes least common multiple of two polynomials.
Given two univariate polynomials, the LCM is computed via
f*g = gcd(f, g)*lcm(f, g) formula. In multivariate case, we
compute the unique generator of the intersection of the two
ideals, generated by f and g. This is done by computing a
Groebner basis, with respect to any lexicographic ordering,
of t*f and (1 - t)*g, where t is an unrelated symbol and
filtering out solution that does not contain t.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_monomial and g.is_monomial:
monom = monomial_lcm(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(ilcm(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
fc, f = f.as_primitive()
gc, g = g.as_primitive()
lcm = ilcm(int(fc), int(gc))
if f.is_multivariate:
t = Symbol('t', dummy=True)
lex = { 'order' : 'lex' }
f_monoms = [ (1,) + monom for monom in f.monoms ]
F = Poly((f.coeffs, f_monoms), t, *symbols, **lex)
g_monoms = [ (0,) + monom for monom in g.monoms ] + \
[ (1,) + monom for monom in g.monoms ]
g_coeffs = list(g.coeffs) + [ -coeff for coeff in g.coeffs ]
G = Poly(dict(zip(g_monoms, g_coeffs)), t, *symbols, **lex)
def independent(h):
return all(not monom[0] for monom in h.monoms)
H = [ h for h in poly_groebner((F, G)) if independent(h) ]
if lcm != 1:
h_coeffs = [ coeff*lcm for coeff in H[0].coeffs ]
else:
h_coeffs = H[0].coeffs
h_monoms = [ monom[1:] for monom in H[0].monoms ]
return Poly(dict(zip(h_monoms, h_coeffs)), *symbols, **flags)
else:
h = poly_div(f * g, poly_gcd(f, g))[0]
if lcm != 1:
return h.mul_term(lcm / h.LC)
else:
return h.as_monic()
def poly_div(f, g, *symbols):
"""Generalized polynomial division with remainder.
Given polynomial f and a set of polynomials g = (g_1, ..., g_n)
compute a set of quotients q = (q_1, ..., q_n) and remainder r
such that f = q_1*f_1 + ... + q_n*f_n + r, where r = 0 or r is
a completely reduced polynomial with respect to g.
In particular g can be a tuple, list or a singleton. All g_i
and f can be given as Poly class instances or as expressions.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 62
[2] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
r = Poly((), *symbols, **flags)
if isinstance(g, Basic):
if g.is_constant:
if g.is_zero:
raise ZeroDivisionError
elif g.is_one:
return f, r
else:
return f.div_term(g.LC), r
if g.is_monomial:
LC, LM = g.lead_term
q_coeffs, q_monoms = [], []
r_coeffs, r_monoms = [], []
for coeff, monom in f.iter_terms():
quotient = monomial_div(monom, LM)
if quotient is not None:
coeff /= LC
q_coeffs.append(Poly.cancel(coeff))
q_monoms.append(quotient)
else:
r_coeffs.append(coeff)
r_monoms.append(monom)
return (Poly((q_coeffs, q_monoms), *symbols, **flags),
Poly((r_coeffs, r_monoms), *symbols, **flags))
g, q = [g], [r]
else:
q = [r] * len(g)
while not f.is_zero:
for i, h in enumerate(g):
monom = monomial_div(f.LM, h.LM)
if monom is not None:
coeff = Poly.cancel(f.LC / h.LC)
q[i] = q[i].add_term(coeff, monom)
f -= h.mul_term(coeff, monom)
break
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
if len(q) != 1:
return q, r
else:
return q[0], r
