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_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_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_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_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)