def dup_primitive_prs(f, g, K):
"""
Primitive polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs = R.dup_primitive_prs(f, g)
>>> prs[0]
x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> prs[1]
3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs[2]
-5*x**4 + x**2 - 3
>>> prs[3]
13*x**2 + 25*x - 49
>>> prs[4]
4663*x - 6150
>>> prs[5]
1
"""
prs = [f, g]
_, h = dup_primitive(dup_prem(f, g, K), K)
while h:
prs.append(h)
f, g = g, h
_, h = dup_primitive(dup_prem(f, g, K), K)
return prs
def dup_primitive_prs(f, g, K):
"""
Primitive polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs = R.dup_primitive_prs(f, g)
>>> prs[0]
x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> prs[1]
3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs[2]
-5*x**4 + x**2 - 3
>>> prs[3]
13*x**2 + 25*x - 49
>>> prs[4]
4663*x - 6150
>>> prs[5]
1
"""
prs = [f, g]
_, h = dup_primitive(dup_prem(f, g, K), K)
while h:
prs.append(h)
f, g = g, h
_, h = dup_primitive(dup_prem(f, g, K), K)
return prs
def dup_primitive_prs(f, g, K):
"""
Primitive polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_primitive_prs
>>> f = ZZ.map([1, 0, 1, 0, -3, -3, 8, 2, -5])
>>> g = ZZ.map([3, 0, 5, 0, -4, -9, 21])
>>> prs = dup_primitive_prs(f, g, ZZ)
>>> prs[0]
[1, 0, 1, 0, -3, -3, 8, 2, -5]
>>> prs[1]
[3, 0, 5, 0, -4, -9, 21]
>>> prs[2]
[-5, 0, 1, 0, -3]
>>> prs[3]
[13, 25, -49]
>>> prs[4]
[4663, -6150]
>>> prs[5]
[1]
"""
prs = [f, g]
_, h = dup_primitive(dup_prem(f, g, K), K)
while h:
prs.append(h)
f, g = g, h
_, h = dup_primitive(dup_prem(f, g, K), K)
return prs
def dup_primitive_prs(f, g, K):
"""
Primitive polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_primitive_prs
>>> f = ZZ.map([1, 0, 1, 0, -3, -3, 8, 2, -5])
>>> g = ZZ.map([3, 0, 5, 0, -4, -9, 21])
>>> prs = dup_primitive_prs(f, g, ZZ)
>>> prs[0]
[1, 0, 1, 0, -3, -3, 8, 2, -5]
>>> prs[1]
[3, 0, 5, 0, -4, -9, 21]
>>> prs[2]
[-5, 0, 1, 0, -3]
>>> prs[3]
[13, 25, -49]
>>> prs[4]
[4663, -6150]
>>> prs[5]
[1]
"""
prs = [f, g]
_, h = dup_primitive(dup_prem(f, g, K), K)
while h:
prs.append(h)
f, g = g, h
_, h = dup_primitive(dup_prem(f, g, K), K)
return prs
def test_dup_pdiv():
f = dup_normal([3,1,1,5], ZZ)
g = dup_normal([5,-3,1], ZZ)
q = dup_normal([15, 14], ZZ)
r = dup_normal([52, 111], ZZ)
assert dup_pdiv(f, g, ZZ) == (q, r)
assert dup_pquo(f, g, ZZ) == q
assert dup_prem(f, g, ZZ) == r
raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, ZZ))
f = dup_normal([3,1,1,5], QQ)
g = dup_normal([5,-3,1], QQ)
q = dup_normal([15, 14], QQ)
r = dup_normal([52, 111], QQ)
assert dup_pdiv(f, g, QQ) == (q, r)
assert dup_pquo(f, g, QQ) == q
assert dup_prem(f, g, QQ) == r
raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, QQ))
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in ``K[x]``.
Computes the subresultant polynomial remainder sequence (PRS) of ``f``
and ``g``, and the values for $\\beta_i$ and $\\delta_i$. The last two
sequences of values are necessary for computing the resultant in
:func:`dup_prs_resultant`.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_inner_subresultants
>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([1, 0, -1])
>>> dup_inner_subresultants(f, g, ZZ)
([[1, 0, 1], [1, 0, -1], [-2]], [-1, -1], [0, 2])
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = (-K.one)**(d+1)
c = -K.one
B, D = [b], [d]
if not f or not g:
return R, B, D
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
while h:
k = dup_degree(h)
R.append(h)
lc = dup_LC(g, K)
if not d:
q = c
else:
q = c**(d-1)
c = K.exquo((-lc)**d, q)
b = -lc * c**(m-k)
f, g, m, d = g, h, k, m-k
B.append(b)
D.append(d)
h = dup_prem(f, g, K)
h = dup_exquo_ground(h, b, K)
return R, B, D
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS)
and the non-zero scalar subresultants of `f` and `g`.
By [1] Thm. 3, these are the constants '-c' (- to optimize
computation of sign).
The first subdeterminant is set to 1 by convention to match
the polynomial and the scalar subdeterminants.
If 'deg(f) < deg(g)', the subresultants of '(g,f)' are computed.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1)
([x**2 + 1, x**2 - 1, -2], [1, 1, 4])
References
==========
[1] W.S. Brown, The Subresultant PRS Algorithm.
ACM Transaction of Mathematical Software 4 (1978) 237-249
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
if not f:
return [], []
if not g:
return [f], [K.one]
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
lc = dup_LC(g, K)
c = lc**d
# Conventional first scalar subdeterminant is 1
S = [K.one, c]
c = -c
while h:
k = dup_degree(h)
R.append(h)
f, g, m, d = g, h, k, m - k
b = -lc * c**d
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
lc = dup_LC(g, K)
if d > 1: # abnormal case
q = c**(d - 1)
c = K.quo((-lc)**d, q)
else:
c = -lc
S.append(-c)
return R, S
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS) of `f`
and `g`, and the values for `\beta_i` and `\delta_i`. The last two
sequences of values are necessary for computing the resultant in
:func:`dup_prs_resultant`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1)
([x**2 + 1, x**2 - 1, -2], [-1, -1], [0, 2])
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
c = -K.one
B, D = [b], [d]
if not f or not g:
return R, B, D
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
while h:
k = dup_degree(h)
R.append(h)
lc = dup_LC(g, K)
if not d:
q = c
else:
q = c**(d - 1)
c = K.quo((-lc)**d, q)
b = -lc * c**(m - k)
f, g, m, d = g, h, k, m - k
B.append(b)
D.append(d)
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
return R, B, D
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS) of `f`
and `g`, and the values for `\beta_i` and `\delta_i`. The last two
sequences of values are necessary for computing the resultant in
:func:`dup_prs_resultant`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1)
([x**2 + 1, x**2 - 1, -2], [-1, -1], [0, 2])
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
c = -K.one
B, D = [b], [d]
if not f or not g:
return R, B, D
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
while h:
k = dup_degree(h)
R.append(h)
lc = dup_LC(g, K)
if not d:
q = c
else:
q = c**(d - 1)
c = K.quo((-lc)**d, q)
b = -lc * c**(m - k)
f, g, m, d = g, h, k, m - k
B.append(b)
D.append(d)
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
return R, B, D
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS)
and the non-zero scalar subresultants of `f` and `g`.
By [1] Thm. 3, these are the constants '-c' (- to optimize
computation of sign).
The first subdeterminant is set to 1 by convention to match
the polynomial and the scalar subdeterminants.
If 'deg(f) < deg(g)', the subresultants of '(g,f)' are computed.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1)
([x**2 + 1, x**2 - 1, -2], [1, 1, 4])
References
==========
[1] W.S. Brown, The Subresultant PRS Algorithm.
ACM Transaction of Mathematical Software 4 (1978) 237-249
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
if not f:
return [], []
if not g:
return [f], [K.one]
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
lc = dup_LC(g, K)
c = lc**d
# Conventional first scalar subdeterminant is 1
S = [K.one, c]
c = -c
while h:
k = dup_degree(h)
R.append(h)
f, g, m, d = g, h, k, m - k
b = -lc * c**d
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
lc = dup_LC(g, K)
if d > 1: # abnormal case
q = c**(d - 1)
c = K.quo((-lc)**d, q)
else:
c = -lc
S.append(-c)
return R, S
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in ``K[x]``.
Computes the subresultant polynomial remainder sequence (PRS) of ``f``
and ``g``, and the values for $\\beta_i$ and $\\delta_i$. The last two
sequences of values are necessary for computing the resultant in
:func:`dup_prs_resultant`.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_inner_subresultants
>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([1, 0, -1])
>>> dup_inner_subresultants(f, g, ZZ)
([[1, 0, 1], [1, 0, -1], [-2]], [-1, -1], [0, 2])
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
c = -K.one
B, D = [b], [d]
if not f or not g:
return R, B, D
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
while h:
k = dup_degree(h)
R.append(h)
lc = dup_LC(g, K)
if not d:
q = c
else:
q = c**(d - 1)
c = K.quo((-lc)**d, q)
b = -lc * c**(m - k)
f, g, m, d = g, h, k, m - k
B.append(b)
D.append(d)
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
return R, B, D