def dom_eigenvects(A, l=Dummy('lambda')):
charpoly = A.charpoly()
rows, cols = A.shape
domain = A.domain
_, factors = dup_factor_list(charpoly, domain)
rational_eigenvects = []
algebraic_eigenvects = []
for base, exp in factors:
if len(base) == 2:
field = domain
eigenval = -base[1] / base[0]
EE_items = [
[eigenval if i == j else field.zero for j in range(cols)]
for i in range(rows)]
EE = DomainMatrix(EE_items, (rows, cols), field)
basis = (A - EE).nullspace()
rational_eigenvects.append((field, eigenval, exp, basis))
else:
minpoly = Poly.from_list(base, l, domain=domain)
field = FiniteExtension(minpoly)
eigenval = field(l)
AA_items = [
[Poly.from_list([item], l, domain=domain).rep for item in row]
for row in A.rep]
AA_items = [[field(item) for item in row] for row in AA_items]
AA = DomainMatrix(AA_items, (rows, cols), field)
EE_items = [
[eigenval if i == j else field.zero for j in range(cols)]
for i in range(rows)]
EE = DomainMatrix(EE_items, (rows, cols), field)
basis = (AA - EE).nullspace()
algebraic_eigenvects.append((field, minpoly, exp, basis))
return rational_eigenvects, algebraic_eigenvects
def test_dup_factor_list():
assert dup_factor_list([], ZZ) == (ZZ(0), [])
assert dup_factor_list([], QQ) == (QQ(0), [])
assert dup_factor_list([], ZZ['y']) == (DMP([], ZZ), [])
assert dup_factor_list([], QQ['y']) == (DMP([], QQ), [])
assert dup_factor_list([], ZZ, include=True) == [([], 1)]
assert dup_factor_list([ZZ(7)], ZZ) == (ZZ(7), [])
assert dup_factor_list([QQ(1, 7)], QQ) == (QQ(1, 7), [])
assert dup_factor_list([DMP([ZZ(7)], ZZ)], ZZ['y']) == (DMP([ZZ(7)],
ZZ), [])
assert dup_factor_list([DMP([QQ(1, 7)], QQ)], QQ['y']) == (DMP([QQ(1, 7)],
QQ), [])
assert dup_factor_list([ZZ(7)], ZZ, include=True) == [([ZZ(7)], 1)]
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dup_factor_list([QQ(1,2),QQ(1),QQ(1,2)], QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ, include=True) == \
[([ZZ(1), ZZ(1)], 2)]
assert dup_factor_list([RR(1.0),RR(2.0),RR(1.0)], RR) == \
(RR(1.0), [([RR(1.0),RR(1.0)], 2)])
assert dup_factor_list([RR(2.0),RR(4.0),RR(2.0)], RR) == \
(RR(2.0), [([RR(1.0),RR(1.0)], 2)])
f = [DMP([ZZ(4), ZZ(0)], ZZ), DMP([ZZ(4), ZZ(0), ZZ(0)], ZZ), DMP([], ZZ)]
assert dup_factor_list(f, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([DMP([ZZ(1)],ZZ),DMP([],ZZ)], 1),
([DMP([ZZ(1),ZZ(0)],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([ZZ(1),ZZ(0)],ZZ)], 1)])
f = [
DMP([QQ(1, 2), QQ(0)], ZZ),
DMP([QQ(1, 2), QQ(0), QQ(0)], ZZ),
DMP([], ZZ)
]
assert dup_factor_list(f, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([DMP([QQ(1)],QQ),DMP([],QQ)], 1),
([DMP([QQ(1),QQ(0)],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([QQ(1),QQ(0)],QQ)], 1)])
raises(DomainError, "dup_factor_list([EX(sin(1))], EX)")
def test_dup_factor_list():
assert dup_factor_list([], ZZ) == (ZZ(0), [])
assert dup_factor_list([], QQ) == (QQ(0), [])
assert dup_factor_list([], ZZ['y']) == (DMP([],ZZ), [])
assert dup_factor_list([], QQ['y']) == (DMP([],QQ), [])
assert dup_factor_list_include([], ZZ) == [([], 1)]
assert dup_factor_list([ZZ(7)], ZZ) == (ZZ(7), [])
assert dup_factor_list([QQ(1,7)], QQ) == (QQ(1,7), [])
assert dup_factor_list([DMP([ZZ(7)],ZZ)], ZZ['y']) == (DMP([ZZ(7)],ZZ), [])
assert dup_factor_list([DMP([QQ(1,7)],QQ)], QQ['y']) == (DMP([QQ(1,7)],QQ), [])
assert dup_factor_list_include([ZZ(7)], ZZ) == [([ZZ(7)], 1)]
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dup_factor_list([QQ(1,2),QQ(1),QQ(1,2)], QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dup_factor_list_include([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
[([ZZ(1), ZZ(1)], 2)]
K = FF(2)
assert dup_factor_list([K(1),K(0),K(1)], K) == \
(K(1), [([K(1), K(1)], 2)])
assert dup_factor_list([RR(1.0),RR(2.0),RR(1.0)], RR) == \
(RR(1.0), [([RR(1.0),RR(1.0)], 2)])
assert dup_factor_list([RR(2.0),RR(4.0),RR(2.0)], RR) == \
(RR(2.0), [([RR(1.0),RR(1.0)], 2)])
f = [DMP([ZZ(4),ZZ(0)],ZZ),DMP([ZZ(4),ZZ(0),ZZ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([DMP([ZZ(1),ZZ(0)],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([ZZ(1),ZZ(0)],ZZ)], 1)])
f = [DMP([QQ(1,2),QQ(0)],ZZ),DMP([QQ(1,2),QQ(0),QQ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([DMP([QQ(1),QQ(0)],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([QQ(1),QQ(0)],QQ)], 1)])
K = QQ.algebraic_field(I)
h = [QQ(1,1), QQ(0,1), QQ(1,1)]
f = [ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ), ANP([], h, QQ), ANP([], h, QQ)]
assert dup_factor_list(f, K) == \
(ANP([QQ(1,1)], h, QQ), [([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ)], 2),
([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ)], 1)])
raises(DomainError, "dup_factor_list([EX(sin(1))], EX)")
def test_dup_factor_list():
assert dup_factor_list([], ZZ) == (ZZ(0), [])
assert dup_factor_list([], QQ) == (QQ(0), [])
assert dup_factor_list([], ZZ['y']) == (DMP([],ZZ), [])
assert dup_factor_list([], QQ['y']) == (DMP([],QQ), [])
assert dup_factor_list_include([], ZZ) == [([], 1)]
assert dup_factor_list([ZZ(7)], ZZ) == (ZZ(7), [])
assert dup_factor_list([QQ(1,7)], QQ) == (QQ(1,7), [])
assert dup_factor_list([DMP([ZZ(7)],ZZ)], ZZ['y']) == (DMP([ZZ(7)],ZZ), [])
assert dup_factor_list([DMP([QQ(1,7)],QQ)], QQ['y']) == (DMP([QQ(1,7)],QQ), [])
assert dup_factor_list_include([ZZ(7)], ZZ) == [([ZZ(7)], 1)]
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dup_factor_list([QQ(1,2),QQ(1),QQ(1,2)], QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dup_factor_list_include([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
[([ZZ(1), ZZ(1)], 2)]
K = FF(2)
assert dup_factor_list([K(1),K(0),K(1)], K) == \
(K(1), [([K(1), K(1)], 2)])
assert dup_factor_list([RR(1.0),RR(2.0),RR(1.0)], RR) == \
(RR(1.0), [([RR(1.0),RR(1.0)], 2)])
assert dup_factor_list([RR(2.0),RR(4.0),RR(2.0)], RR) == \
(RR(2.0), [([RR(1.0),RR(1.0)], 2)])
f = [DMP([ZZ(4),ZZ(0)],ZZ),DMP([ZZ(4),ZZ(0),ZZ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([DMP([ZZ(1),ZZ(0)],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([ZZ(1),ZZ(0)],ZZ)], 1)])
f = [DMP([QQ(1,2),QQ(0)],ZZ),DMP([QQ(1,2),QQ(0),QQ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([DMP([QQ(1),QQ(0)],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([QQ(1),QQ(0)],QQ)], 1)])
K = QQ.algebraic_field(I)
h = [QQ(1,1), QQ(0,1), QQ(1,1)]
f = [ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ), ANP([], h, QQ), ANP([], h, QQ)]
assert dup_factor_list(f, K) == \
(ANP([QQ(1,1)], h, QQ), [([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ)], 2),
([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ)], 1)])
raises(DomainError, "dup_factor_list([EX(sin(1))], EX)")
def test_dup_factor_list():
assert dup_factor_list([], ZZ) == (ZZ(0), [])
assert dup_factor_list([], QQ) == (QQ(0), [])
assert dup_factor_list([], ZZ['y']) == (DMP([],ZZ), [])
assert dup_factor_list([], QQ['y']) == (DMP([],QQ), [])
assert dup_factor_list([], ZZ, include=True) == [([], 1)]
assert dup_factor_list([ZZ(7)], ZZ) == (ZZ(7), [])
assert dup_factor_list([QQ(1,7)], QQ) == (QQ(1,7), [])
assert dup_factor_list([DMP([ZZ(7)],ZZ)], ZZ['y']) == (DMP([ZZ(7)],ZZ), [])
assert dup_factor_list([DMP([QQ(1,7)],QQ)], QQ['y']) == (DMP([QQ(1,7)],QQ), [])
assert dup_factor_list([ZZ(7)], ZZ, include=True) == [([ZZ(7)], 1)]
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dup_factor_list([QQ(1,2),QQ(1),QQ(1,2)], QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ, include=True) == \
[([ZZ(1), ZZ(1)], 2)]
assert dup_factor_list([RR(1.0),RR(2.0),RR(1.0)], RR) == \
(RR(1.0), [([RR(1.0),RR(1.0)], 2)])
assert dup_factor_list([RR(2.0),RR(4.0),RR(2.0)], RR) == \
(RR(2.0), [([RR(1.0),RR(1.0)], 2)])
f = [DMP([ZZ(4),ZZ(0)],ZZ),DMP([ZZ(4),ZZ(0),ZZ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([DMP([ZZ(1)],ZZ),DMP([],ZZ)], 1),
([DMP([ZZ(1),ZZ(0)],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([ZZ(1),ZZ(0)],ZZ)], 1)])
f = [DMP([QQ(1,2),QQ(0)],ZZ),DMP([QQ(1,2),QQ(0),QQ(0)],ZZ),DMP([],ZZ)]
assert dup_factor_list(f, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([DMP([QQ(1)],QQ),DMP([],QQ)], 1),
([DMP([QQ(1),QQ(0)],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([QQ(1),QQ(0)],QQ)], 1)])
raises(DomainError, "dup_factor_list([EX(sin(1))], EX)")
