def _a(n, j, prec):
"""Compute the inner sum in the HRR formula."""
if j == 1:
return fone
s = fzero
pi = pi_fixed(prec)
for h in range(1, j):
if igcd(h, j) != 1:
continue
# & with mask to compute fractional part of fixed-point number
one = 1 << prec
onemask = one - 1
half = one >> 1
g = 0
if j >= 3:
for k in range(1, j):
t = h*k*one//j
if t > 0:
frac = t & onemask
else:
frac = -((-t) & onemask)
g += k*(frac - half)
g = ((g - 2*h*n*one)*pi//j) >> prec
s = mpf_add(s, mpf_cos(from_man_exp(g, -prec), prec), prec)
return s
def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())
for n in range(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)
if f == g:
break
else: # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")
roots = []
if not factor:
# get the indices in the right order so the computed
# roots will be sorted
h = n//2
ks = [i for i in range(1, n + 1) if igcd(i, n) == 1]
ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1))
d = 2*I*pi/n
for k in reversed(ks):
roots.append(exp(k*d).expand(complex=True))
else:
g = Poly(f, extension=root(-1, n))
for h, _ in ordered(g.factor_list()[1]):
roots.append(-h.TC())
return roots
def positive_roots(self):
"""
This method generates all the positive roots of
A_n. This is half of all of the roots of A_n;
by multiplying all the positive roots by -1 we
get the negative roots.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("A3")
>>> c.positive_roots()
{1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0],
5: [0, 1, 0, -1], 6: [0, 0, 1, -1]}
"""
n = self.n
posroots = {}
k = 0
for i in range(0, n):
for j in range(i+1, n+1):
k += 1
posroots[k] = self.basic_root(i, j)
return posroots
def find_simple_recurrence_vector(l):
"""
This function is used internally by other functions from the
sympy.concrete.guess module. While most users may want to rather use the
function find_simple_recurrence when looking for recurrence relations
among rational numbers, the current function may still be useful when
some post-processing has to be done.
The function returns a vector of length n when a recurrence relation of
order n is detected in the sequence of rational numbers v.
If the returned vector has a length 1, then the returned value is always
the list [0], which means that no relation has been found.
While the functions is intended to be used with rational numbers, it should
work for other kinds of real numbers except for some cases involving
quadratic numbers; for that reason it should be used with some caution when
the argument is not a list of rational numbers.
Examples
========
>>> from sympy.concrete.guess import find_simple_recurrence_vector
>>> from sympy import fibonacci
>>> find_simple_recurrence_vector([fibonacci(k) for k in range(12)])
[1, -1, -1]
See also
========
See the function sympy.concrete.guess.find_simple_recurrence which is more
user-friendly.
"""
q1 = [0]
q2 = [Integer(1)]
b, z = 0, len(l) >> 1
while len(q2) <= z:
while l[b]==0:
b += 1
if b == len(l):
c = 1
for x in q2:
c = lcm(c, denom(x))
if q2[0]*c < 0: c = -c
for k in range(len(q2)):
q2[k] = int(q2[k]*c)
return q2
a = Integer(1)/l[b]
m = [a]
for k in range(b+1, len(l)):
m.append(-sum(l[j+1]*m[b-j-1] for j in range(b, k))*a)
l, m = m, [0] * max(len(q2), b+len(q1))
for k in range(len(q2)):
m[k] = a*q2[k]
for k in range(b, b+len(q1)):
m[k] += q1[k-b]
while m[-1]==0: m.pop() # because trailing zeros can occur
q1, q2, b = q2, m, 1
return [0]
def RGS_generalized(m):
"""
Computes the m + 1 generalized unrestricted growth strings
and returns them as rows in matrix.
Examples
========
>>> from sympy.combinatorics.partitions import RGS_generalized
>>> RGS_generalized(6)
Matrix([
[ 1, 1, 1, 1, 1, 1, 1],
[ 1, 2, 3, 4, 5, 6, 0],
[ 2, 5, 10, 17, 26, 0, 0],
[ 5, 15, 37, 77, 0, 0, 0],
[ 15, 52, 151, 0, 0, 0, 0],
[ 52, 203, 0, 0, 0, 0, 0],
[203, 0, 0, 0, 0, 0, 0]])
"""
d = zeros(m + 1)
for i in range(0, m + 1):
d[0, i] = 1
for i in range(1, m + 1):
for j in range(m):
if j <= m - i:
d[i, j] = j * d[i - 1, j] + d[i - 1, j + 1]
else:
d[i, j] = 0
return d
def as_explicit(self):
"""
Returns a dense Matrix with elements represented explicitly
Returns an object of type ImmutableDenseMatrix.
Examples
========
>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
as_mutable: returns mutable Matrix type
"""
from sympy.matrices.immutable import ImmutableDenseMatrix
return ImmutableDenseMatrix([[ self[i, j]
for j in range(self.cols)]
for i in range(self.rows)])
def __array__(self):
from numpy import empty
a = empty(self.shape, dtype=object)
for i in range(self.rows):
for j in range(self.cols):
a[i, j] = self[i, j]
return a
def copyin_matrix(self, key, value):
# include this here because it's not part of BaseMatrix
rlo, rhi, clo, chi = self.key2bounds(key)
shape = value.shape
dr, dc = rhi - rlo, chi - clo
if shape != (dr, dc):
raise ShapeError(
"The Matrix `value` doesn't have the same dimensions "
"as the in sub-Matrix given by `key`.")
if not isinstance(value, SparseMatrix):
for i in range(value.rows):
for j in range(value.cols):
self[i + rlo, j + clo] = value[i, j]
else:
if (rhi - rlo)*(chi - clo) < len(self):
for i in range(rlo, rhi):
for j in range(clo, chi):
self._smat.pop((i, j), None)
else:
for i, j, v in self.row_list():
if rlo <= i < rhi and clo <= j < chi:
self._smat.pop((i, j), None)
for k, v in value._smat.items():
i, j = k
self[i + rlo, j + clo] = value[i, j]
def fill(self, value):
"""Fill self with the given value.
Notes
=====
Unless many values are going to be deleted (i.e. set to zero)
this will create a matrix that is slower than a dense matrix in
operations.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix.zeros(3); M
Matrix([
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]])
>>> M.fill(1); M
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
"""
if not value:
self._smat = {}
else:
v = self._sympify(value)
self._smat = {(i, j): v
for i in range(self.rows) for j in range(self.cols)}
def _dup_right_decompose(f, s, K):
"""Helper function for :func:`_dup_decompose`."""
n = len(f) - 1
lc = dup_LC(f, K)
f = dup_to_raw_dict(f)
g = { s: K.one }
r = n // s
for i in range(1, s):
coeff = K.zero
for j in range(0, i):
if not n + j - i in f:
continue
if not s - j in g:
continue
fc, gc = f[n + j - i], g[s - j]
coeff += (i - r*j)*fc*gc
g[s - i] = K.quo(coeff, i*r*lc)
return dup_from_raw_dict(g, K)
def test_basic_degree_0():
d = 0
knots = range(5)
splines = bspline_basis_set(d, knots, x)
for i in range(len(splines)):
assert splines[i] == Piecewise((1, Interval(i, i + 1).contains(x)),
(0, True))
def tolist(self):
"""Return the Matrix as a nested Python list.
Examples
========
>>> from sympy import Matrix, ones
>>> m = Matrix(3, 3, range(9))
>>> m
Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> m.tolist()
[[0, 1, 2], [3, 4, 5], [6, 7, 8]]
>>> ones(3, 0).tolist()
[[], [], []]
When there are no rows then it will not be possible to tell how
many columns were in the original matrix:
>>> ones(0, 3).tolist()
[]
"""
if not self.rows:
return []
if not self.cols:
return [[] for i in range(self.rows)]
return [self._mat[i: i + self.cols]
for i in range(0, len(self), self.cols)]
def _extend_y0(Holonomic, n):
"""
Tries to find more initial conditions by substituting the initial
value point in the differential equation.
"""
annihilator = Holonomic.annihilator
a = annihilator.order
x = Holonomic.x
listofpoly = []
y0 = Holonomic.y0
R = annihilator.parent.base
K = R.get_field()
for i, j in enumerate(annihilator.listofpoly):
if isinstance(j, annihilator.parent.base.dtype):
listofpoly.append(K.new(j.rep))
if len(y0) < a or n <= len(y0):
return y0
else:
list_red = [-listofpoly[i] / listofpoly[a]
for i in range(a)]
y1 = [i for i in y0]
for i in range(n - a):
sol = 0
for a, b in zip(y1, list_red):
r = DMFsubs(b, Holonomic.x0)
if not r.is_finite:
return y0
sol += a * r
y1.append(sol)
list_red = _derivate_diff_eq(list_red)
return y0 + y1[len(y0):]
def dup_from_dict(f, K):
"""
Create a ``K[x]`` polynomial from a ``dict``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dup_from_dict
>>> dup_from_dict({(0,): ZZ(7), (2,): ZZ(5), (4,): ZZ(1)}, ZZ)
[1, 0, 5, 0, 7]
>>> dup_from_dict({}, ZZ)
[]
"""
if not f:
return []
n, h = max(f.keys()), []
if type(n) is int:
for k in range(n, -1, -1):
h.append(f.get(k, K.zero))
else:
(n,) = n
for k in range(n, -1, -1):
h.append(f.get((k,), K.zero))
return dup_strip(h)
def random_circuit(ngates, nqubits, gate_space=(X, Y, Z, S, T, H, CNOT, SWAP)):
"""Return a random circuit of ngates and nqubits.
This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP)
gates.
Parameters
----------
ngates : int
The number of gates in the circuit.
nqubits : int
The number of qubits in the circuit.
gate_space : tuple
A tuple of the gate classes that will be used in the circuit.
Repeating gate classes multiple times in this tuple will increase
the frequency they appear in the random circuit.
"""
qubit_space = range(nqubits)
result = []
for i in range(ngates):
g = random.choice(gate_space)
if g == CNotGate or g == SwapGate:
qubits = random.sample(qubit_space, 2)
g = g(*qubits)
else:
qubit = random.choice(qubit_space)
g = g(qubit)
result.append(g)
return Mul(*result)
def _identity_matrix(n, domain):
M = [[domain.zero]*n for _ in range(n)]
for i in range(n):
M[i][i] = domain.one
return M
def _eval_expand_func(self, **hints):
from sympy import exp, I, floor, Add, Poly, Dummy, exp_polar, unpolarify
z, s, a = self.args
if z == 1:
return zeta(s, a)
if s.is_Integer and s <= 0:
t = Dummy('t')
p = Poly((t + a)**(-s), t)
start = 1/(1 - t)
res = S(0)
for c in reversed(p.all_coeffs()):
res += c*start
start = t*start.diff(t)
return res.subs(t, z)
if a.is_Rational:
# See section 18 of
# Kelly B. Roach. Hypergeometric Function Representations.
# In: Proceedings of the 1997 International Symposium on Symbolic and
# Algebraic Computation, pages 205-211, New York, 1997. ACM.
# TODO should something be polarified here?
add = S(0)
mul = S(1)
# First reduce a to the interaval (0, 1]
if a > 1:
n = floor(a)
if n == a:
n -= 1
a -= n
mul = z**(-n)
add = Add(*[-z**(k - n)/(a + k)**s for k in range(n)])
elif a <= 0:
n = floor(-a) + 1
a += n
mul = z**n
add = Add(*[z**(n - 1 - k)/(a - k - 1)**s for k in range(n)])
m, n = S([a.p, a.q])
zet = exp_polar(2*pi*I/n)
root = z**(1/n)
return add + mul*n**(s - 1)*Add(
*[polylog(s, zet**k*root)._eval_expand_func(**hints)
/ (unpolarify(zet)**k*root)**m for k in range(n)])
# TODO use minpoly instead of ad-hoc methods when issue 5888 is fixed
if z.func is exp and (z.args[0]/(pi*I)).is_Rational or z in [-1, I, -I]:
# TODO reference?
if z == -1:
p, q = S([1, 2])
elif z == I:
p, q = S([1, 4])
elif z == -I:
p, q = S([-1, 4])
else:
arg = z.args[0]/(2*pi*I)
p, q = S([arg.p, arg.q])
return Add(*[exp(2*pi*I*k*p/q)/q**s*zeta(s, (k + a)/q)
for k in range(q)])
return lerchphi(z, s, a)
def blocks(self):
from sympy.matrices.immutable import ImmutableDenseMatrix
mats = self.args
data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols)
for j in range(len(mats))]
for i in range(len(mats))]
return ImmutableDenseMatrix(data)
def test_issue_12533():
d = IndexedBase('d')
assert IndexedBase(range(5)) == Range(0, 5, 1)
assert d[0].subs(Symbol("d"), range(5)) == 0
assert d[0].subs(d, range(5)) == 0
assert d[1].subs(d, range(5)) == 1
assert Indexed(Range(5), 2) == 2
def guess_generating_function_rational(v, X=Symbol('x'), maxcoeff=1024):
"""
Tries to "guess" a rational generating function for a sequence of rational
numbers v.
Examples
========
>>> from sympy.concrete.guess import guess_generating_function_rational
>>> from sympy import fibonacci
>>> l = [fibonacci(k) for k in range(5,15)]
>>> guess_generating_function_rational(l)
(3*x + 5)/(-x**2 - x + 1)
See also
========
See function sympy.series.approximants and mpmath.pade
"""
# a) compute the denominator as q
q = find_simple_recurrence_vector(v, maxcoeff=maxcoeff)
n = len(q)
if n <= 1: return None
# b) compute the numerator as p
p = [sum(v[i-k]*q[k] for k in range(min(i+1, n)))
for i in range(len(v))] # TODO: maybe better with: len(v)>>1
return (sum(p[k]*X**k for k in range(len(p)))
/ sum(q[k]*X**k for k in range(n)))
def eval_levicivita(*args):
"""Evaluate Levi-Civita symbol."""
from sympy import factorial
n = len(args)
return prod(
prod(args[j] - args[i] for j in range(i + 1, n))
/ factorial(i) for i in range(n))
def _complexes_sorted(cls, complexes):
"""Make complex isolating intervals disjoint and sort roots. """
complexes = cls._refine_complexes(complexes)
# XXX don't sort until you are sure that it is compatible
# with the indexing method but assert that the desired state
# is not broken
C, F = 0, 1 # location of ComplexInterval and factor
fs = set([i[F] for i in complexes])
for i in range(1, len(complexes)):
if complexes[i][F] != complexes[i - 1][F]:
# if this fails the factors of a root were not
# contiguous because a discontinuity should only
# happen once
fs.remove(complexes[i - 1][F])
for i in range(len(complexes)):
# negative im part (conj=True) comes before
# positive im part (conj=False)
assert complexes[i][C].conj is (i % 2 == 0)
# update cache
cache = {}
# -- collate
for root, factor, _ in complexes:
cache.setdefault(factor, []).append(root)
# -- store
for factor, roots in cache.items():
_complexes_cache[factor] = roots
return complexes
def decrement_part(self, part):
"""Decrements part (a subrange of pstack), if possible, returning
True iff the part was successfully decremented.
If you think of the v values in the part as a multi-digit
integer (least significant digit on the right) this is
basically decrementing that integer, but with the extra
constraint that the leftmost digit cannot be decremented to 0.
Parameters
==========
part
The part, represented as a list of PartComponent objects,
which is to be decremented.
"""
plen = len(part)
for j in range(plen - 1, -1, -1):
if (j == 0 and part[j].v > 1) or (j > 0 and part[j].v > 0):
# found val to decrement
part[j].v -= 1
# Reset trailing parts back to maximum
for k in range(j + 1, plen):
part[k].v = part[k].u
return True
return False
def _eval_expand_func(self, **hints):
from sympy import Sum
n = self.args[0]
m = self.args[1] if len(self.args) == 2 else 1
if m == S.One:
if n.is_Add:
off = n.args[0]
nnew = n - off
if off.is_Integer and off.is_positive:
result = [S.One/(nnew + i) for i in range(off, 0, -1)] + [harmonic(nnew)]
return Add(*result)
elif off.is_Integer and off.is_negative:
result = [-S.One/(nnew + i) for i in range(0, off, -1)] + [harmonic(nnew)]
return Add(*result)
if n.is_Rational:
# Expansions for harmonic numbers at general rational arguments (u + p/q)
# Split n as u + p/q with p < q
p, q = n.as_numer_denom()
u = p // q
p = p - u * q
if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
k = Dummy("k")
t1 = q * Sum(1 / (q * k + p), (k, 0, u))
t2 = 2 * Sum(cos((2 * pi * p * k) / S(q)) *
log(sin((pi * k) / S(q))),
(k, 1, floor((q - 1) / S(2))))
t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
return t1 + t2 - t3
return self
def _initialize_enumeration(self, multiplicities):
"""Allocates and initializes the partition stack.
This is called from the enumeration/counting routines, so
there is no need to call it separately."""
num_components = len(multiplicities)
# cardinality is the total number of elements, whether or not distinct
cardinality = sum(multiplicities)
# pstack is the partition stack, which is segmented by
# f into parts.
self.pstack = [PartComponent() for i in
range(num_components * cardinality + 1)]
self.f = [0] * (cardinality + 1)
# Initial state - entire multiset in one part.
for j in range(num_components):
ps = self.pstack[j]
ps.c = j
ps.u = multiplicities[j]
ps.v = multiplicities[j]
self.f[0] = 0
self.f[1] = num_components
self.lpart = 0
def _walsh_hadamard_transform(seq, inverse=False):
"""Utility function for the Walsh Hadamard Transform"""
if not iterable(seq):
raise TypeError("Expected a sequence of coefficients "
"for Walsh Hadamard Transform")
a = [sympify(arg) for arg in seq]
n = len(a)
if n < 2:
return a
if n&(n - 1):
n = 2**n.bit_length()
a += [S.Zero]*(n - len(a))
h = 2
while h <= n:
hf, ut = h // 2, n // h
for i in range(0, n, h):
for j in range(hf):
u, v = a[i + j], a[i + j + hf]
a[i + j], a[i + j + hf] = u + v, u - v
h *= 2
if inverse:
a = [x/n for x in a]
return a
def _multiset_histogram(n):
"""Return tuple used in permutation and combination counting. Input
is a dictionary giving items with counts as values or a sequence of
items (which need not be sorted).
The data is stored in a class deriving from tuple so it is easily
recognized and so it can be converted easily to a list.
"""
if type(n) is dict: # item: count
if not all(isinstance(v, int) and v >= 0 for v in n.values()):
raise ValueError
tot = sum(n.values())
items = sum(1 for k in n if n[k] > 0)
return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
else:
n = list(n)
s = set(n)
if len(s) == len(n):
n = [1]*len(n)
n.extend([len(n), len(n)])
return _MultisetHistogram(n)
m = dict(zip(s, range(len(s))))
d = dict(zip(range(len(s)), [0]*len(s)))
for i in n:
d[m[i]] += 1
return _multiset_histogram(d)
def eval(cls, x, k):
x = sympify(x)
k = sympify(k)
if x is S.NaN:
return S.NaN
elif k.is_Integer:
if k is S.NaN:
return S.NaN
elif k is S.Zero:
return S.One
else:
if k.is_positive:
if x is S.Infinity:
return S.Infinity
elif x is S.NegativeInfinity:
if k.is_odd:
return S.NegativeInfinity
else:
return S.Infinity
else:
return reduce(lambda r, i: r * (x - i), range(0, int(k)), 1)
else:
if x is S.Infinity:
return S.Infinity
elif x is S.NegativeInfinity:
return S.Infinity
else:
return 1 / reduce(lambda r, i: r * (x + i), range(1, abs(int(k)) + 1), 1)
def f():
for u in range(1, len(self.u_set)):
glBegin(GL_QUAD_STRIP)
for v in range(len(self.v_set)):
pa = self.verts[u - 1][v]
pb = self.verts[u][v]
if pa is None or pb is None:
glEnd()
glBegin(GL_QUAD_STRIP)
continue
if use_cverts:
ca = self.cverts[u - 1][v]
cb = self.cverts[u][v]
if ca is None:
ca = (0, 0, 0)
if cb is None:
cb = (0, 0, 0)
else:
if use_solid_color:
ca = cb = self.default_solid_color
else:
ca = cb = self.default_wireframe_color
glColor3f(*ca)
glVertex3f(*pa)
glColor3f(*cb)
glVertex3f(*pb)
glEnd()
def _form_fr(self, fl):
"""Form the generalized active force."""
if fl != None and (len(fl) == 0 or not iterable(fl)):
raise ValueError('Force pairs must be supplied in an '
'non-empty iterable or None.')
N = self._inertial
# pull out relevant velocities for constructing partial velocities
vel_list, f_list = _f_list_parser(fl, N)
vel_list = [msubs(i, self._qdot_u_map) for i in vel_list]
# Fill Fr with dot product of partial velocities and forces
o = len(self.u)
b = len(f_list)
FR = zeros(o, 1)
partials = partial_velocity(vel_list, self.u, N)
for i in range(o):
FR[i] = sum(partials[j][i] & f_list[j] for j in range(b))
# In case there are dependent speeds
if self._udep:
p = o - len(self._udep)
FRtilde = FR[:p, 0]
FRold = FR[p:o, 0]
FRtilde += self._Ars.T * FRold
FR = FRtilde
self._forcelist = fl
self._fr = FR
return FR
def test_bicycle():
if ON_TRAVIS:
skip("Too slow for travis.")
# Code to get equations of motion for a bicycle modeled as in:
# J.P Meijaard, Jim M Papadopoulos, Andy Ruina and A.L Schwab. Linearized
# dynamics equations for the balance and steer of a bicycle: a benchmark
# and review. Proceedings of The Royal Society (2007) 463, 1955-1982
# doi: 10.1098/rspa.2007.1857
# Note that this code has been crudely ported from Autolev, which is the
# reason for some of the unusual naming conventions. It was purposefully as
# similar as possible in order to aide debugging.
# Declare Coordinates & Speeds
# Simple definitions for qdots - qd = u
# Speeds are: yaw frame ang. rate, roll frame ang. rate, rear wheel frame
# ang. rate (spinning motion), frame ang. rate (pitching motion), steering
# frame ang. rate, and front wheel ang. rate (spinning motion).
# Wheel positions are ignorable coordinates, so they are not introduced.
q1, q2, q4, q5 = dynamicsymbols('q1 q2 q4 q5')
q1d, q2d, q4d, q5d = dynamicsymbols('q1 q2 q4 q5', 1)
u1, u2, u3, u4, u5, u6 = dynamicsymbols('u1 u2 u3 u4 u5 u6')
u1d, u2d, u3d, u4d, u5d, u6d = dynamicsymbols('u1 u2 u3 u4 u5 u6', 1)
# Declare System's Parameters
WFrad, WRrad, htangle, forkoffset = symbols('WFrad WRrad htangle forkoffset')
forklength, framelength, forkcg1 = symbols('forklength framelength forkcg1')
forkcg3, framecg1, framecg3, Iwr11 = symbols('forkcg3 framecg1 framecg3 Iwr11')
Iwr22, Iwf11, Iwf22, Iframe11 = symbols('Iwr22 Iwf11 Iwf22 Iframe11')
Iframe22, Iframe33, Iframe31, Ifork11 = symbols('Iframe22 Iframe33 Iframe31 Ifork11')
Ifork22, Ifork33, Ifork31, g = symbols('Ifork22 Ifork33 Ifork31 g')
mframe, mfork, mwf, mwr = symbols('mframe mfork mwf mwr')
# Set up reference frames for the system
# N - inertial
# Y - yaw
# R - roll
# WR - rear wheel, rotation angle is ignorable coordinate so not oriented
# Frame - bicycle frame
# TempFrame - statically rotated frame for easier reference inertia definition
# Fork - bicycle fork
# TempFork - statically rotated frame for easier reference inertia definition
# WF - front wheel, again posses a ignorable coordinate
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
R = Y.orientnew('R', 'Axis', [q2, Y.x])
Frame = R.orientnew('Frame', 'Axis', [q4 + htangle, R.y])
WR = ReferenceFrame('WR')
TempFrame = Frame.orientnew('TempFrame', 'Axis', [-htangle, Frame.y])
Fork = Frame.orientnew('Fork', 'Axis', [q5, Frame.x])
TempFork = Fork.orientnew('TempFork', 'Axis', [-htangle, Fork.y])
WF = ReferenceFrame('WF')
# Kinematics of the Bicycle First block of code is forming the positions of
# the relevant points
# rear wheel contact -> rear wheel mass center -> frame mass center +
# frame/fork connection -> fork mass center + front wheel mass center ->
# front wheel contact point
WR_cont = Point('WR_cont')
WR_mc = WR_cont.locatenew('WR_mc', WRrad * R.z)
Steer = WR_mc.locatenew('Steer', framelength * Frame.z)
Frame_mc = WR_mc.locatenew('Frame_mc', - framecg1 * Frame.x
+ framecg3 * Frame.z)
Fork_mc = Steer.locatenew('Fork_mc', - forkcg1 * Fork.x
+ forkcg3 * Fork.z)
WF_mc = Steer.locatenew('WF_mc', forklength * Fork.x + forkoffset * Fork.z)
WF_cont = WF_mc.locatenew('WF_cont', WFrad * (dot(Fork.y, Y.z) * Fork.y -
Y.z).normalize())
# Set the angular velocity of each frame.
# Angular accelerations end up being calculated automatically by
# differentiating the angular velocities when first needed.
# u1 is yaw rate
# u2 is roll rate
# u3 is rear wheel rate
# u4 is frame pitch rate
# u5 is fork steer rate
# u6 is front wheel rate
Y.set_ang_vel(N, u1 * Y.z)
R.set_ang_vel(Y, u2 * R.x)
WR.set_ang_vel(Frame, u3 * Frame.y)
Frame.set_ang_vel(R, u4 * Frame.y)
Fork.set_ang_vel(Frame, u5 * Fork.x)
WF.set_ang_vel(Fork, u6 * Fork.y)
# Form the velocities of the previously defined points, using the 2 - point
# theorem (written out by hand here). Accelerations again are calculated
# automatically when first needed.
WR_cont.set_vel(N, 0)
WR_mc.v2pt_theory(WR_cont, N, WR)
Steer.v2pt_theory(WR_mc, N, Frame)
Frame_mc.v2pt_theory(WR_mc, N, Frame)
Fork_mc.v2pt_theory(Steer, N, Fork)
WF_mc.v2pt_theory(Steer, N, Fork)
WF_cont.v2pt_theory(WF_mc, N, WF)
# Sets the inertias of each body. Uses the inertia frame to construct the
# inertia dyadics. Wheel inertias are only defined by principle moments of
# inertia, and are in fact constant in the frame and fork reference frames;
# it is for this reason that the orientations of the wheels does not need
# to be defined. The frame and fork inertias are defined in the 'Temp'
# frames which are fixed to the appropriate body frames; this is to allow
# easier input of the reference values of the benchmark paper. Note that
# due to slightly different orientations, the products of inertia need to
# have their signs flipped; this is done later when entering the numerical
# value.
Frame_I = (inertia(TempFrame, Iframe11, Iframe22, Iframe33, 0, 0, Iframe31), Frame_mc)
Fork_I = (inertia(TempFork, Ifork11, Ifork22, Ifork33, 0, 0, Ifork31), Fork_mc)
WR_I = (inertia(Frame, Iwr11, Iwr22, Iwr11), WR_mc)
WF_I = (inertia(Fork, Iwf11, Iwf22, Iwf11), WF_mc)
# Declaration of the RigidBody containers. ::
BodyFrame = RigidBody('BodyFrame', Frame_mc, Frame, mframe, Frame_I)
BodyFork = RigidBody('BodyFork', Fork_mc, Fork, mfork, Fork_I)
BodyWR = RigidBody('BodyWR', WR_mc, WR, mwr, WR_I)
BodyWF = RigidBody('BodyWF', WF_mc, WF, mwf, WF_I)
# The kinematic differential equations; they are defined quite simply. Each
# entry in this list is equal to zero.
kd = [q1d - u1, q2d - u2, q4d - u4, q5d - u5]
# The nonholonomic constraints are the velocity of the front wheel contact
# point dotted into the X, Y, and Z directions; the yaw frame is used as it
# is "closer" to the front wheel (1 less DCM connecting them). These
# constraints force the velocity of the front wheel contact point to be 0
# in the inertial frame; the X and Y direction constraints enforce a
# "no-slip" condition, and the Z direction constraint forces the front
# wheel contact point to not move away from the ground frame, essentially
# replicating the holonomic constraint which does not allow the frame pitch
# to change in an invalid fashion.
conlist_speed = [WF_cont.vel(N) & Y.x, WF_cont.vel(N) & Y.y, WF_cont.vel(N) & Y.z]
# The holonomic constraint is that the position from the rear wheel contact
# point to the front wheel contact point when dotted into the
# normal-to-ground plane direction must be zero; effectively that the front
# and rear wheel contact points are always touching the ground plane. This
# is actually not part of the dynamic equations, but instead is necessary
# for the lineraization process.
conlist_coord = [WF_cont.pos_from(WR_cont) & Y.z]
# The force list; each body has the appropriate gravitational force applied
# at its mass center.
FL = [(Frame_mc, -mframe * g * Y.z),
(Fork_mc, -mfork * g * Y.z),
(WF_mc, -mwf * g * Y.z),
(WR_mc, -mwr * g * Y.z)]
BL = [BodyFrame, BodyFork, BodyWR, BodyWF]
# The N frame is the inertial frame, coordinates are supplied in the order
# of independent, dependent coordinates, as are the speeds. The kinematic
# differential equation are also entered here. Here the dependent speeds
# are specified, in the same order they were provided in earlier, along
# with the non-holonomic constraints. The dependent coordinate is also
# provided, with the holonomic constraint. Again, this is only provided
# for the linearization process.
KM = KanesMethod(N, q_ind=[q1, q2, q5],
q_dependent=[q4], configuration_constraints=conlist_coord,
u_ind=[u2, u3, u5],
u_dependent=[u1, u4, u6], velocity_constraints=conlist_speed,
kd_eqs=kd)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
(fr, frstar) = KM.kanes_equations(FL, BL)
# This is the start of entering in the numerical values from the benchmark
# paper to validate the eigen values of the linearized equations from this
# model to the reference eigen values. Look at the aforementioned paper for
# more information. Some of these are intermediate values, used to
# transform values from the paper into the coordinate systems used in this
# model.
PaperRadRear = 0.3
PaperRadFront = 0.35
HTA = evalf.N(pi / 2 - pi / 10)
TrailPaper = 0.08
rake = evalf.N(-(TrailPaper*sin(HTA)-(PaperRadFront*cos(HTA))))
PaperWb = 1.02
PaperFrameCgX = 0.3
PaperFrameCgZ = 0.9
PaperForkCgX = 0.9
PaperForkCgZ = 0.7
FrameLength = evalf.N(PaperWb*sin(HTA)-(rake-(PaperRadFront-PaperRadRear)*cos(HTA)))
FrameCGNorm = evalf.N((PaperFrameCgZ - PaperRadRear-(PaperFrameCgX/sin(HTA))*cos(HTA))*sin(HTA))
FrameCGPar = evalf.N((PaperFrameCgX / sin(HTA) + (PaperFrameCgZ - PaperRadRear - PaperFrameCgX / sin(HTA) * cos(HTA)) * cos(HTA)))
tempa = evalf.N((PaperForkCgZ - PaperRadFront))
tempb = evalf.N((PaperWb-PaperForkCgX))
tempc = evalf.N(sqrt(tempa**2+tempb**2))
PaperForkL = evalf.N((PaperWb*cos(HTA)-(PaperRadFront-PaperRadRear)*sin(HTA)))
ForkCGNorm = evalf.N(rake+(tempc * sin(pi/2-HTA-acos(tempa/tempc))))
ForkCGPar = evalf.N(tempc * cos((pi/2-HTA)-acos(tempa/tempc))-PaperForkL)
# Here is the final assembly of the numerical values. The symbol 'v' is the
# forward speed of the bicycle (a concept which only makes sense in the
# upright, static equilibrium case?). These are in a dictionary which will
# later be substituted in. Again the sign on the *product* of inertia
# values is flipped here, due to different orientations of coordinate
# systems.
v = symbols('v')
val_dict = {WFrad: PaperRadFront,
WRrad: PaperRadRear,
htangle: HTA,
forkoffset: rake,
forklength: PaperForkL,
framelength: FrameLength,
forkcg1: ForkCGPar,
forkcg3: ForkCGNorm,
framecg1: FrameCGNorm,
framecg3: FrameCGPar,
Iwr11: 0.0603,
Iwr22: 0.12,
Iwf11: 0.1405,
Iwf22: 0.28,
Ifork11: 0.05892,
Ifork22: 0.06,
Ifork33: 0.00708,
Ifork31: 0.00756,
Iframe11: 9.2,
Iframe22: 11,
Iframe33: 2.8,
Iframe31: -2.4,
mfork: 4,
mframe: 85,
mwf: 3,
mwr: 2,
g: 9.81,
q1: 0,
q2: 0,
q4: 0,
q5: 0,
u1: 0,
u2: 0,
u3: v / PaperRadRear,
u4: 0,
u5: 0,
u6: v / PaperRadFront}
# Linearizes the forcing vector; the equations are set up as MM udot =
# forcing, where MM is the mass matrix, udot is the vector representing the
# time derivatives of the generalized speeds, and forcing is a vector which
# contains both external forcing terms and internal forcing terms, such as
# centripital or coriolis forces. This actually returns a matrix with as
# many rows as *total* coordinates and speeds, but only as many columns as
# independent coordinates and speeds.
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
forcing_lin = KM.linearize()[0]
# As mentioned above, the size of the linearized forcing terms is expanded
# to include both q's and u's, so the mass matrix must have this done as
# well. This will likely be changed to be part of the linearized process,
# for future reference.
MM_full = KM.mass_matrix_full
MM_full_s = msubs(MM_full, val_dict)
forcing_lin_s = msubs(forcing_lin, KM.kindiffdict(), val_dict)
MM_full_s = MM_full_s.evalf()
forcing_lin_s = forcing_lin_s.evalf()
# Finally, we construct an "A" matrix for the form xdot = A x (x being the
# state vector, although in this case, the sizes are a little off). The
# following line extracts only the minimum entries required for eigenvalue
# analysis, which correspond to rows and columns for lean, steer, lean
# rate, and steer rate.
Amat = MM_full_s.inv() * forcing_lin_s
A = Amat.extract([1, 2, 4, 6], [1, 2, 3, 5])
# Precomputed for comparison
Res = Matrix([[ 0, 0, 1.0, 0],
[ 0, 0, 0, 1.0],
[9.48977444677355, -0.891197738059089*v**2 - 0.571523173729245, -0.105522449805691*v, -0.330515398992311*v],
[11.7194768719633, -1.97171508499972*v**2 + 30.9087533932407, 3.67680523332152*v, -3.08486552743311*v]])
# Actual eigenvalue comparison
eps = 1.e-12
for i in range(6):
error = Res.subs(v, i) - A.subs(v, i)
assert all(abs(x) < eps for x in error)
def test_factorint():
assert primefactors(123456) == [2, 3, 643]
assert factorint(0) == {0: 1}
assert factorint(1) == {}
assert factorint(-1) == {-1: 1}
assert factorint(-2) == {-1: 1, 2: 1}
assert factorint(-16) == {-1: 1, 2: 4}
assert factorint(2) == {2: 1}
assert factorint(126) == {2: 1, 3: 2, 7: 1}
assert factorint(123456) == {2: 6, 3: 1, 643: 1}
assert factorint(5951757) == {3: 1, 7: 1, 29: 2, 337: 1}
assert factorint(64015937) == {7993: 1, 8009: 1}
assert factorint(2**(2**6) + 1) == {274177: 1, 67280421310721: 1}
assert multiproduct(factorint(fac(200))) == fac(200)
for b, e in factorint(fac(150)).items():
assert e == fac_multiplicity(150, b)
assert factorint(103005006059**7) == {103005006059: 7}
assert factorint(31337**191) == {31337: 191}
assert factorint(2**1000 * 3**500 * 257**127 * 383**60) == \
{2: 1000, 3: 500, 257: 127, 383: 60}
assert len(factorint(fac(10000))) == 1229
assert factorint(12932983746293756928584532764589230) == \
{2: 1, 5: 1, 73: 1, 727719592270351: 1, 63564265087747: 1, 383: 1}
assert factorint(727719592270351) == {727719592270351: 1}
assert factorint(2**64 + 1, use_trial=False) == factorint(2**64 + 1)
for n in range(60000):
assert multiproduct(factorint(n)) == n
assert pollard_rho(2**64 + 1, seed=1) == 274177
assert pollard_rho(19, seed=1) is None
assert factorint(3, limit=2) == {3: 1}
assert factorint(12345) == {3: 1, 5: 1, 823: 1}
assert factorint(12345, limit=3) == {
4115: 1,
3: 1
} # the 5 is greater than the limit
assert factorint(1, limit=1) == {}
assert factorint(0, 3) == {0: 1}
assert factorint(12, limit=1) == {12: 1}
assert factorint(30, limit=2) == {2: 1, 15: 1}
assert factorint(16, limit=2) == {2: 4}
assert factorint(124, limit=3) == {2: 2, 31: 1}
assert factorint(4 * 31**2, limit=3) == {2: 2, 31: 2}
p1 = nextprime(2**32)
p2 = nextprime(2**16)
p3 = nextprime(p2)
assert factorint(p1 * p2 * p3) == {p1: 1, p2: 1, p3: 1}
assert factorint(13 * 17 * 19, limit=15) == {13: 1, 17 * 19: 1}
assert factorint(1951 * 15013 * 15053, limit=2000) == {
225990689: 1,
1951: 1
}
assert factorint(primorial(17) + 1, use_pm1=0) == \
{long(19026377261): 1, 3467: 1, 277: 1, 105229: 1}
# when prime b is closer than approx sqrt(8*p) to prime p then they are
# "close" and have a trivial factorization
a = nextprime(2**2**8) # 78 digits
b = nextprime(a + 2**2**4)
assert 'Fermat' in capture(lambda: factorint(a * b, verbose=1))
raises(ValueError, lambda: pollard_rho(4))
raises(ValueError, lambda: pollard_pm1(3))
raises(ValueError, lambda: pollard_pm1(10, B=2))
# verbose coverage
n = nextprime(2**16) * nextprime(2**17) * nextprime(1901)
assert 'with primes' in capture(lambda: factorint(n, verbose=1))
capture(lambda: factorint(nextprime(2**16) * 1012, verbose=1))
n = nextprime(2**17)
capture(lambda: factorint(n**3, verbose=1)) # perfect power termination
capture(lambda: factorint(2 * n, verbose=1)) # factoring complete msg
# exceed 1st
n = nextprime(2**17)
n *= nextprime(n)
assert '1000' in capture(lambda: factorint(n, limit=1000, verbose=1))
n *= nextprime(n)
assert len(factorint(n)) == 3
assert len(factorint(n, limit=p1)) == 3
n *= nextprime(2 * n)
# exceed 2nd
assert '2001' in capture(lambda: factorint(n, limit=2000, verbose=1))
assert capture(lambda: factorint(n, limit=4000, verbose=1)).count(
'Pollard') == 2
# non-prime pm1 result
n = nextprime(8069)
n *= nextprime(2 * n) * nextprime(2 * n, 2)
capture(lambda: factorint(n, verbose=1)) # non-prime pm1 result
# factor fermat composite
p1 = nextprime(2**17)
p2 = nextprime(2 * p1)
assert factorint((p1 * p2**2)**3) == {p1: 3, p2: 6}
# Test for non integer input
raises(ValueError, lambda: factorint(4.5))
def test_sort_variable():
vsort = Derivative._sort_variable_count
def vsort0(*v, **kw):
reverse = kw.get('reverse', False)
return [i[0] for i in vsort([(i, 0) for i in (
reversed(v) if reverse else v)])]
for R in range(2):
assert vsort0(y, x, reverse=R) == [x, y]
assert vsort0(f(x), x, reverse=R) == [x, f(x)]
assert vsort0(f(y), f(x), reverse=R) == [f(x), f(y)]
assert vsort0(g(x), f(y), reverse=R) == [f(y), g(x)]
assert vsort0(f(x, y), f(x), reverse=R) == [f(x), f(x, y)]
fx = f(x).diff(x)
assert vsort0(fx, y, reverse=R) == [y, fx]
fy = f(y).diff(y)
assert vsort0(fy, fx, reverse=R) == [fx, fy]
fxx = fx.diff(x)
assert vsort0(fxx, fx, reverse=R) == [fx, fxx]
assert vsort0(Basic(x), f(x), reverse=R) == [f(x), Basic(x)]
assert vsort0(Basic(y), Basic(x), reverse=R) == [Basic(x), Basic(y)]
assert vsort0(Basic(y, z), Basic(x), reverse=R) == [
Basic(x), Basic(y, z)]
assert vsort0(fx, x, reverse=R) == [
x, fx] if R else [fx, x]
assert vsort0(Basic(x), x, reverse=R) == [
x, Basic(x)] if R else [Basic(x), x]
assert vsort0(Basic(f(x)), f(x), reverse=R) == [
f(x), Basic(f(x))] if R else [Basic(f(x)), f(x)]
assert vsort0(Basic(x, z), Basic(x), reverse=R) == [
Basic(x), Basic(x, z)] if R else [Basic(x, z), Basic(x)]
assert vsort([]) == []
assert _aresame(vsort([(x, 1)]), [Tuple(x, 1)])
assert vsort([(x, y), (x, z)]) == [(x, y + z)]
assert vsort([(y, 1), (x, 1 + y)]) == [(x, 1 + y), (y, 1)]
# coverage complete; legacy tests below
assert vsort([(x, 3), (y, 2), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(x), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 2), (x, 3), (h(x), 1), (g(x), 1),
(f(x), 1)]) == [(x, 3), (y, 2), (z, 1), (f(x), 1), (g(x), 1),
(h(x), 1)]
assert vsort([(x, 1), (f(x), 1), (y, 1), (f(y), 1)]) == [(x, 1),
(y, 1), (f(x), 1), (f(y), 1)]
assert vsort([(y, 1), (x, 2), (g(x), 1), (f(x), 1), (z, 1),
(h(x), 1), (y, 2), (x, 1)]) == [(x, 3), (y, 3), (z, 1),
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 1), (f(x), 1), (x, 1), (f(x), 1),
(g(x), 1)]) == [(x, 1), (y, 1), (z, 1), (f(x), 2), (g(x), 1)]
assert vsort([(z, 1), (y, 2), (f(x), 1), (x, 2), (f(x), 2),
(g(x), 1), (z, 2), (z, 1), (y, 1), (x, 1)]) == [(x, 3), (y, 3),
(z, 4), (f(x), 3), (g(x), 1)]
assert vsort(((y, 2), (x, 1), (y, 1), (x, 1))) == [(x, 2), (y, 3)]
assert isinstance(vsort([(x, 3), (y, 2), (z, 1)])[0], Tuple)
assert vsort([(x, 1), (f(x), 1), (x, 1)]) == [(x, 2), (f(x), 1)]
assert vsort([(y, 2), (x, 3), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(y), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(y), 1)]
assert vsort([(x, 1), (y, 1), (x, 1)]) == [(x, 2), (y, 1)]
assert vsort([(f(x), 1), (f(y), 1), (f(x), 1)]) == [
(f(x), 2), (f(y), 1)]
dfx = f(x).diff(x)
self = [(dfx, 1), (x, 1)]
assert vsort(self) == self
assert vsort([
(dfx, 1), (y, 1), (f(x), 1), (x, 1), (f(y), 1), (x, 1)]) == [
(y, 1), (f(x), 1), (f(y), 1), (dfx, 1), (x, 2)]
dfy = f(y).diff(y)
assert vsort([(dfy, 1), (dfx, 1)]) == [(dfx, 1), (dfy, 1)]
d2fx = dfx.diff(x)
assert vsort([(d2fx, 1), (dfx, 1)]) == [(dfx, 1), (d2fx, 1)]
Matrix, Basic, Dict, oo, zoo, nan)
from sympy.utilities.pytest import XFAIL, raises
from sympy.core.basic import _aresame
from sympy.core.cache import clear_cache
from sympy.core.compatibility import range
from sympy.core.expr import unchanged
from sympy.core.function import PoleError, _mexpand, arity
from sympy.core.sympify import sympify
from sympy.sets.sets import FiniteSet
from sympy.solvers.solveset import solveset
from sympy.tensor.array import NDimArray
from sympy.utilities.iterables import subsets, variations
from sympy.abc import t, w, x, y, z
f, g, h = symbols('f g h', cls=Function)
_xi_1, _xi_2, _xi_3 = [Dummy() for i in range(3)]
def test_f_expand_complex():
x = Symbol('x', real=True)
assert f(x).expand(complex=True) == I*im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I*x).expand(complex=True) == cos(x) + I*sin(x)
assert exp(z).expand(complex=True) == cos(im(z))*exp(re(z)) + \
I*sin(im(z))*exp(re(z))
def test_bug1():
e = sqrt(-log(w))
assert e.subs(log(w), -x) == sqrt(x)
def test_ranking():
assert Permutation.unrank_lex(5, 10).rank() == 10
p = Permutation.unrank_lex(15, 225)
assert p.rank() == 225
p1 = p.next_lex()
assert p1.rank() == 226
assert Permutation.unrank_lex(15, 225).rank() == 225
assert Permutation.unrank_lex(10, 0).is_Identity
p = Permutation.unrank_lex(4, 23)
assert p.rank() == 23
assert p.array_form == [3, 2, 1, 0]
assert p.next_lex() is None
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
assert a == [[0, 1, 2, 3], [0, 1, 3, 2], [0, 3, 1, 2], [3, 0, 1,
2], [3, 0, 2, 1] ]
assert [Permutation(pa).rank_trotterjohnson() for pa in a] == list(range(5))
assert Permutation([0, 1, 2, 3]).next_trotterjohnson() == \
Permutation([0, 1, 3, 2])
assert q.rank_trotterjohnson() == 2283
assert p.rank_trotterjohnson() == 3389
assert Permutation([1, 0]).rank_trotterjohnson() == 1
a = Permutation(list(range(3)))
b = a
l = []
tj = []
for i in range(6):
l.append(a)
tj.append(b)
a = a.next_lex()
b = b.next_trotterjohnson()
assert a == b is None
assert {tuple(a) for a in l} == {tuple(a) for a in tj}
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert p.rank() == 1964
assert q.rank() == 870
assert Permutation([]).rank_nonlex() == 0
prank = p.rank_nonlex()
assert prank == 1600
assert Permutation.unrank_nonlex(7, 1600) == p
qrank = q.rank_nonlex()
assert qrank == 41
assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)
a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
assert a == [
[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0],
[2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1],
[1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2],
[2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3],
[3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]
N = 10
p1 = Permutation(a[0])
for i in range(1, N+1):
p1 = p1*Permutation(a[i])
p2 = Permutation.rmul_with_af(*[Permutation(h) for h in a[N::-1]])
assert p1 == p2
ok = []
p = Permutation([1, 0])
for i in range(3):
ok.append(p.array_form)
p = p.next_nonlex()
if p is None:
ok.append(None)
break
assert ok == [[1, 0], [0, 1], None]
assert Permutation([3, 2, 0, 1]).next_nonlex() == Permutation([1, 3, 0, 2])
assert [Permutation(pa).rank_nonlex() for pa in a] == list(range(24))
def test_Permutation():
# don't auto fill 0
raises(ValueError, lambda: Permutation([1]))
p = Permutation([0, 1, 2, 3])
# call as bijective
assert [p(i) for i in range(p.size)] == list(p)
# call as operator
assert p(list(range(p.size))) == list(p)
# call as function
assert list(p(1, 2)) == [0, 2, 1, 3]
# conversion to list
assert list(p) == list(range(4))
assert Permutation(size=4) == Permutation(3)
assert Permutation(Permutation(3), size=5) == Permutation(4)
# cycle form with size
assert Permutation([[1, 2]], size=4) == Permutation([[1, 2], [0], [3]])
# random generation
assert Permutation.random(2) in (Permutation([1, 0]), Permutation([0, 1]))
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[1], [0, 3, 5, 6, 2, 4]])
assert len({p, p}) == 1
r = Permutation([1, 3, 2, 0, 4, 6, 5])
ans = Permutation(_af_rmuln(*[w.array_form for w in (p, q, r)])).array_form
assert rmul(p, q, r).array_form == ans
# make sure no other permutation of p, q, r could have given
# that answer
for a, b, c in permutations((p, q, r)):
if (a, b, c) == (p, q, r):
continue
assert rmul(a, b, c).array_form != ans
assert p.support() == list(range(7))
assert q.support() == [0, 2, 3, 4, 5, 6]
assert Permutation(p.cyclic_form).array_form == p.array_form
assert p.cardinality == 5040
assert q.cardinality == 5040
assert q.cycles == 2
assert rmul(q, p) == Permutation([4, 6, 1, 2, 5, 3, 0])
assert rmul(p, q) == Permutation([6, 5, 3, 0, 2, 4, 1])
assert _af_rmul(p.array_form, q.array_form) == \
[6, 5, 3, 0, 2, 4, 1]
assert rmul(Permutation([[1, 2, 3], [0, 4]]),
Permutation([[1, 2, 4], [0], [3]])).cyclic_form == \
[[0, 4, 2], [1, 3]]
assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
assert q.cyclic_form == [[0, 3, 5, 6, 2, 4]]
assert q.full_cyclic_form == [[0, 3, 5, 6, 2, 4], [1]]
assert p.cyclic_form == [[0, 2, 1, 5], [3, 6, 4]]
t = p.transpositions()
assert t == [(0, 5), (0, 1), (0, 2), (3, 4), (3, 6)]
assert Permutation.rmul(*[Permutation(Cycle(*ti)) for ti in (t)])
assert Permutation([1, 0]).transpositions() == [(0, 1)]
assert p**13 == p
assert q**0 == Permutation(list(range(q.size)))
assert q**-2 == ~q**2
assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])
assert q**3 == q**2*q
assert q**4 == q**2*q**2
a = Permutation(1, 3)
b = Permutation(2, 0, 3)
I = Permutation(3)
assert ~a == a**-1
assert a*~a == I
assert a*b**-1 == a*~b
ans = Permutation(0, 5, 3, 1, 6)(2, 4)
assert (p + q.rank()).rank() == ans.rank()
assert (p + q.rank())._rank == ans.rank()
assert (q + p.rank()).rank() == ans.rank()
raises(TypeError, lambda: p + Permutation(list(range(10))))
assert (p - q.rank()).rank() == Permutation(0, 6, 3, 1, 2, 5, 4).rank()
assert p.rank() - q.rank() < 0 # for coverage: make sure mod is used
assert (q - p.rank()).rank() == Permutation(1, 4, 6, 2)(3, 5).rank()
assert p*q == Permutation(_af_rmuln(*[list(w) for w in (q, p)]))
assert p*Permutation([]) == p
assert Permutation([])*p == p
assert p*Permutation([[0, 1]]) == Permutation([2, 5, 0, 6, 3, 1, 4])
assert Permutation([[0, 1]])*p == Permutation([5, 2, 1, 6, 3, 0, 4])
pq = p ^ q
assert pq == Permutation([5, 6, 0, 4, 1, 2, 3])
assert pq == rmul(q, p, ~q)
qp = q ^ p
assert qp == Permutation([4, 3, 6, 2, 1, 5, 0])
assert qp == rmul(p, q, ~p)
raises(ValueError, lambda: p ^ Permutation([]))
assert p.commutator(q) == Permutation(0, 1, 3, 4, 6, 5, 2)
assert q.commutator(p) == Permutation(0, 2, 5, 6, 4, 3, 1)
assert p.commutator(q) == ~q.commutator(p)
raises(ValueError, lambda: p.commutator(Permutation([])))
assert len(p.atoms()) == 7
assert q.atoms() == {0, 1, 2, 3, 4, 5, 6}
assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
assert q.inversion_vector() == [3, 1, 2, 2, 0, 1]
assert Permutation.from_inversion_vector(p.inversion_vector()) == p
assert Permutation.from_inversion_vector(q.inversion_vector()).array_form\
== q.array_form
raises(ValueError, lambda: Permutation.from_inversion_vector([0, 2]))
assert Permutation([i for i in range(500, -1, -1)]).inversions() == 125250
s = Permutation([0, 4, 1, 3, 2])
assert s.parity() == 0
_ = s.cyclic_form # needed to create a value for _cyclic_form
assert len(s._cyclic_form) != s.size and s.parity() == 0
assert not s.is_odd
assert s.is_even
assert Permutation([0, 1, 4, 3, 2]).parity() == 1
assert _af_parity([0, 4, 1, 3, 2]) == 0
assert _af_parity([0, 1, 4, 3, 2]) == 1
s = Permutation([0])
assert s.is_Singleton
assert Permutation([]).is_Empty
r = Permutation([3, 2, 1, 0])
assert (r**2).is_Identity
assert rmul(~p, p).is_Identity
assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
assert ~(r**2).is_Identity
assert p.max() == 6
assert p.min() == 0
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert q.max() == 4
assert q.min() == 0
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
assert p.ascents() == [0, 3, 4]
assert q.ascents() == [1, 2, 4]
assert r.ascents() == []
assert p.descents() == [1, 2, 5]
assert q.descents() == [0, 3, 5]
assert Permutation(r.descents()).is_Identity
assert p.inversions() == 7
# test the merge-sort with a longer permutation
big = list(p) + list(range(p.max() + 1, p.max() + 130))
assert Permutation(big).inversions() == 7
assert p.signature() == -1
assert q.inversions() == 11
assert q.signature() == -1
assert rmul(p, ~p).inversions() == 0
assert rmul(p, ~p).signature() == 1
assert p.order() == 6
assert q.order() == 10
assert (p**(p.order())).is_Identity
assert p.length() == 6
assert q.length() == 7
assert r.length() == 4
assert p.runs() == [[1, 5], [2], [0, 3, 6], [4]]
assert q.runs() == [[4], [2, 3, 5], [0, 6], [1]]
assert r.runs() == [[3], [2], [1], [0]]
assert p.index() == 8
assert q.index() == 8
assert r.index() == 3
assert p.get_precedence_distance(q) == q.get_precedence_distance(p)
assert p.get_adjacency_distance(q) == p.get_adjacency_distance(q)
assert p.get_positional_distance(q) == p.get_positional_distance(q)
p = Permutation([0, 1, 2, 3])
q = Permutation([3, 2, 1, 0])
assert p.get_precedence_distance(q) == 6
assert p.get_adjacency_distance(q) == 3
assert p.get_positional_distance(q) == 8
p = Permutation([0, 3, 1, 2, 4])
q = Permutation.josephus(4, 5, 2)
assert p.get_adjacency_distance(q) == 3
raises(ValueError, lambda: p.get_adjacency_distance(Permutation([])))
raises(ValueError, lambda: p.get_positional_distance(Permutation([])))
raises(ValueError, lambda: p.get_precedence_distance(Permutation([])))
a = [Permutation.unrank_nonlex(4, i) for i in range(5)]
iden = Permutation([0, 1, 2, 3])
for i in range(5):
for j in range(i + 1, 5):
assert a[i].commutes_with(a[j]) == \
(rmul(a[i], a[j]) == rmul(a[j], a[i]))
if a[i].commutes_with(a[j]):
assert a[i].commutator(a[j]) == iden
assert a[j].commutator(a[i]) == iden
a = Permutation(3)
b = Permutation(0, 6, 3)(1, 2)
assert a.cycle_structure == {1: 4}
assert b.cycle_structure == {2: 1, 3: 1, 1: 2}
def kahane_simplify(expression):
r"""
This function cancels contracted elements in a product of four
dimensional gamma matrices, resulting in an expression equal to the given
one, without the contracted gamma matrices.
Parameters
==========
`expression` the tensor expression containing the gamma matrices to simplify.
Notes
=====
If spinor indices are given, the matrices must be given in
the order given in the product.
Algorithm
=========
The idea behind the algorithm is to use some well-known identities,
i.e., for contractions enclosing an even number of `\gamma` matrices
`\gamma^\mu \gamma_{a_1} \cdots \gamma_{a_{2N}} \gamma_\mu = 2 (\gamma_{a_{2N}} \gamma_{a_1} \cdots \gamma_{a_{2N-1}} + \gamma_{a_{2N-1}} \cdots \gamma_{a_1} \gamma_{a_{2N}} )`
for an odd number of `\gamma` matrices
`\gamma^\mu \gamma_{a_1} \cdots \gamma_{a_{2N+1}} \gamma_\mu = -2 \gamma_{a_{2N+1}} \gamma_{a_{2N}} \cdots \gamma_{a_{1}}`
Instead of repeatedly applying these identities to cancel out all contracted indices,
it is possible to recognize the links that would result from such an operation,
the problem is thus reduced to a simple rearrangement of free gamma matrices.
Examples
========
When using, always remember that the original expression coefficient
has to be handled separately
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex
>>> from sympy.physics.hep.gamma_matrices import kahane_simplify
>>> from sympy.tensor.tensor import tensor_indices
>>> i0, i1, i2 = tensor_indices('i0:3', LorentzIndex)
>>> ta = G(i0)*G(-i0)
>>> kahane_simplify(ta)
Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])
>>> tb = G(i0)*G(i1)*G(-i0)
>>> kahane_simplify(tb)
-2*GammaMatrix(i1)
>>> t = G(i0)*G(-i0)
>>> kahane_simplify(t)
Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])
>>> t = G(i0)*G(-i0)
>>> kahane_simplify(t)
Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])
If there are no contractions, the same expression is returned
>>> tc = G(i0)*G(i1)
>>> kahane_simplify(tc)
GammaMatrix(i0)*GammaMatrix(i1)
References
==========
[1] Algorithm for Reducing Contracted Products of gamma Matrices,
Joseph Kahane, Journal of Mathematical Physics, Vol. 9, No. 10, October 1968.
"""
if isinstance(expression, Mul):
return expression
if isinstance(expression, TensAdd):
return TensAdd(*[kahane_simplify(arg) for arg in expression.args])
if isinstance(expression, Tensor):
return expression
assert isinstance(expression, TensMul)
gammas = expression.args
for gamma in gammas:
assert gamma.component == GammaMatrix
free = expression.free
# spinor_free = [_ for _ in expression.free_in_args if _[1] != 0]
# if len(spinor_free) == 2:
# spinor_free.sort(key=lambda x: x[2])
# assert spinor_free[0][1] == 1 and spinor_free[-1][1] == 2
# assert spinor_free[0][2] == 0
# elif spinor_free:
# raise ValueError('spinor indices do not match')
dum = []
for dum_pair in expression.dum:
if expression.index_types[dum_pair[0]] == LorentzIndex:
dum.append((dum_pair[0], dum_pair[1]))
dum = sorted(dum)
if len(dum) == 0: # or GammaMatrixHead:
# no contractions in `expression`, just return it.
return expression
# find the `first_dum_pos`, i.e. the position of the first contracted
# gamma matrix, Kahane's algorithm as described in his paper requires the
# gamma matrix expression to start with a contracted gamma matrix, this is
# a workaround which ignores possible initial free indices, and re-adds
# them later.
first_dum_pos = min(map(min, dum))
# for p1, p2, a1, a2 in expression.dum_in_args:
# if p1 != 0 or p2 != 0:
# # only Lorentz indices, skip Dirac indices:
# continue
# first_dum_pos = min(p1, p2)
# break
total_number = len(free) + len(dum) * 2
number_of_contractions = len(dum)
free_pos = [None] * total_number
for i in free:
free_pos[i[1]] = i[0]
# `index_is_free` is a list of booleans, to identify index position
# and whether that index is free or dummy.
index_is_free = [False] * total_number
for i, indx in enumerate(free):
index_is_free[indx[1]] = True
# `links` is a dictionary containing the graph described in Kahane's paper,
# to every key correspond one or two values, representing the linked indices.
# All values in `links` are integers, negative numbers are used in the case
# where it is necessary to insert gamma matrices between free indices, in
# order to make Kahane's algorithm work (see paper).
links = dict()
for i in range(first_dum_pos, total_number):
links[i] = []
# `cum_sign` is a step variable to mark the sign of every index, see paper.
cum_sign = -1
# `cum_sign_list` keeps storage for all `cum_sign` (every index).
cum_sign_list = [None] * total_number
block_free_count = 0
# multiply `resulting_coeff` by the coefficient parameter, the rest
# of the algorithm ignores a scalar coefficient.
resulting_coeff = S.One
# initialize a list of lists of indices. The outer list will contain all
# additive tensor expressions, while the inner list will contain the
# free indices (rearranged according to the algorithm).
resulting_indices = [[]]
# start to count the `connected_components`, which together with the number
# of contractions, determines a -1 or +1 factor to be multiplied.
connected_components = 1
# First loop: here we fill `cum_sign_list`, and draw the links
# among consecutive indices (they are stored in `links`). Links among
# non-consecutive indices will be drawn later.
for i, is_free in enumerate(index_is_free):
# if `expression` starts with free indices, they are ignored here;
# they are later added as they are to the beginning of all
# `resulting_indices` list of lists of indices.
if i < first_dum_pos:
continue
if is_free:
block_free_count += 1
# if previous index was free as well, draw an arch in `links`.
if block_free_count > 1:
links[i - 1].append(i)
links[i].append(i - 1)
else:
# Change the sign of the index (`cum_sign`) if the number of free
# indices preceding it is even.
cum_sign *= 1 if (block_free_count % 2) else -1
if block_free_count == 0 and i != first_dum_pos:
# check if there are two consecutive dummy indices:
# in this case create virtual indices with negative position,
# these "virtual" indices represent the insertion of two
# gamma^0 matrices to separate consecutive dummy indices, as
# Kahane's algorithm requires dummy indices to be separated by
# free indices. The product of two gamma^0 matrices is unity,
# so the new expression being examined is the same as the
# original one.
if cum_sign == -1:
links[-1 - i] = [-1 - i + 1]
links[-1 - i + 1] = [-1 - i]
if (i - cum_sign) in links:
if i != first_dum_pos:
links[i].append(i - cum_sign)
if block_free_count != 0:
if i - cum_sign < len(index_is_free):
if index_is_free[i - cum_sign]:
links[i - cum_sign].append(i)
block_free_count = 0
cum_sign_list[i] = cum_sign
# The previous loop has only created links between consecutive free indices,
# it is necessary to properly create links among dummy (contracted) indices,
# according to the rules described in Kahane's paper. There is only one exception
# to Kahane's rules: the negative indices, which handle the case of some
# consecutive free indices (Kahane's paper just describes dummy indices
# separated by free indices, hinting that free indices can be added without
# altering the expression result).
for i in dum:
# get the positions of the two contracted indices:
pos1 = i[0]
pos2 = i[1]
# create Kahane's upper links, i.e. the upper arcs between dummy
# (i.e. contracted) indices:
links[pos1].append(pos2)
links[pos2].append(pos1)
# create Kahane's lower links, this corresponds to the arcs below
# the line described in the paper:
# first we move `pos1` and `pos2` according to the sign of the indices:
linkpos1 = pos1 + cum_sign_list[pos1]
linkpos2 = pos2 + cum_sign_list[pos2]
# otherwise, perform some checks before creating the lower arcs:
# make sure we are not exceeding the total number of indices:
if linkpos1 >= total_number:
continue
if linkpos2 >= total_number:
continue
# make sure we are not below the first dummy index in `expression`:
if linkpos1 < first_dum_pos:
continue
if linkpos2 < first_dum_pos:
continue
# check if the previous loop created "virtual" indices between dummy
# indices, in such a case relink `linkpos1` and `linkpos2`:
if (-1 - linkpos1) in links:
linkpos1 = -1 - linkpos1
if (-1 - linkpos2) in links:
linkpos2 = -1 - linkpos2
# move only if not next to free index:
if linkpos1 >= 0 and not index_is_free[linkpos1]:
linkpos1 = pos1
if linkpos2 >= 0 and not index_is_free[linkpos2]:
linkpos2 = pos2
# create the lower arcs:
if linkpos2 not in links[linkpos1]:
links[linkpos1].append(linkpos2)
if linkpos1 not in links[linkpos2]:
links[linkpos2].append(linkpos1)
# This loop starts from the `first_dum_pos` index (first dummy index)
# walks through the graph deleting the visited indices from `links`,
# it adds a gamma matrix for every free index in encounters, while it
# completely ignores dummy indices and virtual indices.
pointer = first_dum_pos
previous_pointer = 0
while True:
if pointer in links:
next_ones = links.pop(pointer)
else:
break
if previous_pointer in next_ones:
next_ones.remove(previous_pointer)
previous_pointer = pointer
if next_ones:
pointer = next_ones[0]
else:
break
if pointer == previous_pointer:
break
if pointer >= 0 and free_pos[pointer] is not None:
for ri in resulting_indices:
ri.append(free_pos[pointer])
# The following loop removes the remaining connected components in `links`.
# If there are free indices inside a connected component, it gives a
# contribution to the resulting expression given by the factor
# `gamma_a gamma_b ... gamma_z + gamma_z ... gamma_b gamma_a`, in Kahanes's
# paper represented as {gamma_a, gamma_b, ... , gamma_z},
# virtual indices are ignored. The variable `connected_components` is
# increased by one for every connected component this loop encounters.
# If the connected component has virtual and dummy indices only
# (no free indices), it contributes to `resulting_indices` by a factor of two.
# The multiplication by two is a result of the
# factor {gamma^0, gamma^0} = 2 I, as it appears in Kahane's paper.
# Note: curly brackets are meant as in the paper, as a generalized
# multi-element anticommutator!
while links:
connected_components += 1
pointer = min(links.keys())
previous_pointer = pointer
# the inner loop erases the visited indices from `links`, and it adds
# all free indices to `prepend_indices` list, virtual indices are
# ignored.
prepend_indices = []
while True:
if pointer in links:
next_ones = links.pop(pointer)
else:
break
if previous_pointer in next_ones:
if len(next_ones) > 1:
next_ones.remove(previous_pointer)
previous_pointer = pointer
if next_ones:
pointer = next_ones[0]
if pointer >= first_dum_pos and free_pos[pointer] is not None:
prepend_indices.insert(0, free_pos[pointer])
# if `prepend_indices` is void, it means there are no free indices
# in the loop (and it can be shown that there must be a virtual index),
# loops of virtual indices only contribute by a factor of two:
if len(prepend_indices) == 0:
resulting_coeff *= 2
# otherwise, add the free indices in `prepend_indices` to
# the `resulting_indices`:
else:
expr1 = prepend_indices
expr2 = list(reversed(prepend_indices))
resulting_indices = [
expri + ri for ri in resulting_indices
for expri in (expr1, expr2)
]
# sign correction, as described in Kahane's paper:
resulting_coeff *= -1 if (number_of_contractions - connected_components +
1) % 2 else 1
# power of two factor, as described in Kahane's paper:
resulting_coeff *= 2**(number_of_contractions)
# If `first_dum_pos` is not zero, it means that there are trailing free gamma
# matrices in front of `expression`, so multiply by them:
for i in range(0, first_dum_pos):
[ri.insert(0, free_pos[i]) for ri in resulting_indices]
resulting_expr = S.Zero
for i in resulting_indices:
temp_expr = S.One
for j in i:
temp_expr *= GammaMatrix(j)
resulting_expr += temp_expr
t = resulting_coeff * resulting_expr
t1 = None
if isinstance(t, TensAdd):
t1 = t.args[0]
elif isinstance(t, TensMul):
t1 = t
if t1:
pass
else:
t = eye(4) * t
return t
def __iter__(self):
for i in range(self.length):
pt = self._ith_point(i)
yield self.coeff(pt)
def _sympystr(self, p):
"""
Returns the string representation of 'self'.
Examples
========
>>> from sympy import TableForm
>>> t = TableForm([[5, 7], [4, 2], [10, 3]])
>>> s = t.as_str()
"""
column_widths = [0] * self._w
lines = []
for line in self._lines:
new_line = []
for i in range(self._w):
# Format the item somehow if needed:
s = str(line[i])
if self._wipe_zeros and (s == "0"):
s = " "
w = len(s)
if w > column_widths[i]:
column_widths[i] = w
new_line.append(s)
lines.append(new_line)
# Check heading:
if self._headings[0]:
self._headings[0] = [str(x) for x in self._headings[0]]
_head_width = max([len(x) for x in self._headings[0]])
if self._headings[1]:
new_line = []
for i in range(self._w):
# Format the item somehow if needed:
s = str(self._headings[1][i])
w = len(s)
if w > column_widths[i]:
column_widths[i] = w
new_line.append(s)
self._headings[1] = new_line
format_str = []
def _align(align, w):
return '%%%s%ss' % (("-" if align == "l" else ""), str(w))
format_str = [
_align(align, w)
for align, w in zip(self._alignments, column_widths)
]
if self._headings[0]:
format_str.insert(0, _align(self._head_align, _head_width))
format_str.insert(1, '|')
format_str = ' '.join(format_str) + '\n'
s = []
if self._headings[1]:
d = self._headings[1]
if self._headings[0]:
d = [""] + d
first_line = format_str % tuple(d)
s.append(first_line)
s.append("-" * (len(first_line) - 1) + "\n")
for i, line in enumerate(lines):
d = [
l if self._alignments[j] != 'c' else l.center(column_widths[j])
for j, l in enumerate(line)
]
if self._headings[0]:
l = self._headings[0][i]
l = (l if self._head_align != 'c' else l.center(_head_width))
d = [l] + d
s.append(format_str % tuple(d))
return ''.join(s)[:-1] # don't include trailing newline
def find_linear_recurrence(self,n,d=None,gfvar=None):
r"""
Finds the shortest linear recurrence that satisfies the first n
terms of sequence of order `\leq` n/2 if possible.
If d is specified, find shortest linear recurrence of order
`\leq` min(d, n/2) if possible.
Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the
recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...``
Returns ``[]`` if no recurrence is found.
If gfvar is specified, also returns ordinary generating function as a
function of gfvar.
Examples
========
>>> from sympy import sequence, sqrt, oo, lucas
>>> from sympy.abc import n, x, y
>>> sequence(n**2).find_linear_recurrence(10, 2)
[]
>>> sequence(n**2).find_linear_recurrence(10)
[3, -3, 1]
>>> sequence(2**n).find_linear_recurrence(10)
[2]
>>> sequence(23*n**4+91*n**2).find_linear_recurrence(10)
[5, -10, 10, -5, 1]
>>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10)
[1, 1]
>>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30)
[1/2, 1/2]
>>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x)
([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1)))
>>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x)
([1, 1], (x - 2)/(x**2 + x - 1))
"""
from sympy.matrices import Matrix
x = [simplify(expand(t)) for t in self[:n]]
lx = len(x)
if d is None:
r = lx//2
else:
r = min(d,lx//2)
coeffs = []
for l in range(1, r+1):
l2 = 2*l
mlist = []
for k in range(l):
mlist.append(x[k:k+l])
m = Matrix(mlist)
if m.det() != 0:
y = simplify(m.LUsolve(Matrix(x[l:l2])))
if lx == l2:
coeffs = flatten(y[::-1])
break
mlist = []
for k in range(l,lx-l):
mlist.append(x[k:k+l])
m = Matrix(mlist)
if m*y == Matrix(x[l2:]):
coeffs = flatten(y[::-1])
break
if gfvar is None:
return coeffs
else:
l = len(coeffs)
if l == 0:
return [], None
else:
n, d = x[l-1]*gfvar**(l-1), 1 - coeffs[l-1]*gfvar**l
for i in range(l-1):
n += x[i]*gfvar**i
for j in range(l-i-1):
n -= coeffs[i]*x[j]*gfvar**(i+j+1)
d -= coeffs[i]*gfvar**(i+1)
return coeffs, simplify(factor(n)/factor(d))
def rsolve(f, y, init=None):
"""
Solve univariate recurrence with rational coefficients.
Given `k`-th order linear recurrence `\operatorname{L} y = f`,
or equivalently:
.. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) +
\cdots + a_{0}(n) y(n) = f(n)
where `a_{i}(n)`, for `i=0, \ldots, k`, are polynomials or rational
functions in `n`, and `f` is a hypergeometric function or a sum
of a fixed number of pairwise dissimilar hypergeometric terms in
`n`, finds all solutions or returns ``None``, if none were found.
Initial conditions can be given as a dictionary in two forms:
(1) ``{ n_0 : v_0, n_1 : v_1, ..., n_m : v_m }``
(2) ``{ y(n_0) : v_0, y(n_1) : v_1, ..., y(n_m) : v_m }``
or as a list ``L`` of values:
``L = [ v_0, v_1, ..., v_m ]``
where ``L[i] = v_i``, for `i=0, \ldots, m`, maps to `y(n_i)`.
Examples
========
Lets consider the following recurrence:
.. math:: (n - 1) y(n + 2) - (n^2 + 3 n - 2) y(n + 1) +
2 n (n + 1) y(n) = 0
>>> from sympy import Function, rsolve
>>> from sympy.abc import n
>>> y = Function('y')
>>> f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
>>> rsolve(f, y(n))
2**n*C0 + C1*factorial(n)
>>> rsolve(f, y(n), { y(0):0, y(1):3 })
3*2**n - 3*factorial(n)
See Also
========
rsolve_poly, rsolve_ratio, rsolve_hyper
"""
if isinstance(f, Equality):
f = f.lhs - f.rhs
n = y.args[0]
k = Wild('k', exclude=(n,))
# Preprocess user input to allow things like
# y(n) + a*(y(n + 1) + y(n - 1))/2
f = f.expand().collect(y.func(Wild('m', integer=True)))
h_part = defaultdict(lambda: S.Zero)
i_part = S.Zero
for g in Add.make_args(f):
coeff = S.One
kspec = None
for h in Mul.make_args(g):
if h.is_Function:
if h.func == y.func:
result = h.args[0].match(n + k)
if result is not None:
kspec = int(result[k])
else:
raise ValueError(
"'%s(%s+k)' expected, got '%s'" % (y.func, n, h))
else:
raise ValueError(
"'%s' expected, got '%s'" % (y.func, h.func))
else:
coeff *= h
if kspec is not None:
h_part[kspec] += coeff
else:
i_part += coeff
for k, coeff in h_part.items():
h_part[k] = simplify(coeff)
common = S.One
for coeff in h_part.values():
if coeff.is_rational_function(n):
if not coeff.is_polynomial(n):
common = lcm(common, coeff.as_numer_denom()[1], n)
else:
raise ValueError(
"Polynomial or rational function expected, got '%s'" % coeff)
i_numer, i_denom = i_part.as_numer_denom()
if i_denom.is_polynomial(n):
common = lcm(common, i_denom, n)
if common is not S.One:
for k, coeff in h_part.items():
numer, denom = coeff.as_numer_denom()
h_part[k] = numer*quo(common, denom, n)
i_part = i_numer*quo(common, i_denom, n)
K_min = min(h_part.keys())
if K_min < 0:
K = abs(K_min)
H_part = defaultdict(lambda: S.Zero)
i_part = i_part.subs(n, n + K).expand()
common = common.subs(n, n + K).expand()
for k, coeff in h_part.items():
H_part[k + K] = coeff.subs(n, n + K).expand()
else:
H_part = h_part
K_max = max(H_part.keys())
coeffs = [H_part[i] for i in range(K_max + 1)]
result = rsolve_hyper(coeffs, -i_part, n, symbols=True)
if result is None:
return None
solution, symbols = result
if init == {} or init == []:
init = None
if symbols and init is not None:
if type(init) is list:
init = {i: init[i] for i in range(len(init))}
equations = []
for k, v in init.items():
try:
i = int(k)
except TypeError:
if k.is_Function and k.func == y.func:
i = int(k.args[0])
else:
raise ValueError("Integer or term expected, got '%s'" % k)
try:
eq = solution.limit(n, i) - v
except NotImplementedError:
eq = solution.subs(n, i) - v
equations.append(eq)
result = solve(equations, *symbols)
if not result:
return None
else:
solution = solution.subs(result)
return solution
def __init__(self, data, **kwarg):
"""
Creates a TableForm.
Parameters:
data ...
2D data to be put into the table; data can be
given as a Matrix
headings ...
gives the labels for rows and columns:
Can be a single argument that applies to both
dimensions:
- None ... no labels
- "automatic" ... labels are 1, 2, 3, ...
Can be a list of labels for rows and columns:
The lables for each dimension can be given
as None, "automatic", or [l1, l2, ...] e.g.
["automatic", None] will number the rows
[default: None]
alignments ...
alignment of the columns with:
- "left" or "<"
- "center" or "^"
- "right" or ">"
When given as a single value, the value is used for
all columns. The row headings (if given) will be
right justified unless an explicit alignment is
given for it and all other columns.
[default: "left"]
formats ...
a list of format strings or functions that accept
3 arguments (entry, row number, col number) and
return a string for the table entry. (If a function
returns None then the _print method will be used.)
wipe_zeros ...
Don't show zeros in the table.
[default: True]
pad ...
the string to use to indicate a missing value (e.g.
elements that are None or those that are missing
from the end of a row (i.e. any row that is shorter
than the rest is assumed to have missing values).
When None, nothing will be shown for values that
are missing from the end of a row; values that are
None, however, will be shown.
[default: None]
Examples
========
>>> from sympy import TableForm, Matrix
>>> TableForm([[5, 7], [4, 2], [10, 3]])
5 7
4 2
10 3
>>> TableForm([list('.'*i) for i in range(1, 4)], headings='automatic')
| 1 2 3
---------
1 | .
2 | . .
3 | . . .
>>> TableForm([['.'*(j if not i%2 else 1) for i in range(3)]
... for j in range(4)], alignments='rcl')
.
. . .
.. . ..
... . ...
"""
from sympy import Symbol, S, Matrix
from sympy.core.sympify import SympifyError
# We only support 2D data. Check the consistency:
if isinstance(data, Matrix):
data = data.tolist()
_w = len(data[0])
_h = len(data)
# fill out any short lines
pad = kwarg.get('pad', None)
ok_None = False
if pad is None:
pad = " "
ok_None = True
pad = Symbol(pad)
_w = max(len(line) for line in data)
for i, line in enumerate(data):
if len(line) != _w:
line.extend([pad] * (_w - len(line)))
for j, lj in enumerate(line):
if lj is None:
if not ok_None:
lj = pad
else:
try:
lj = S(lj)
except SympifyError:
lj = Symbol(str(lj))
line[j] = lj
data[i] = line
_lines = Tuple(*data)
headings = kwarg.get("headings", [None, None])
if headings == "automatic":
_headings = [range(1, _h + 1), range(1, _w + 1)]
else:
h1, h2 = headings
if h1 == "automatic":
h1 = range(1, _h + 1)
if h2 == "automatic":
h2 = range(1, _w + 1)
_headings = [h1, h2]
allow = ('l', 'r', 'c')
alignments = kwarg.get("alignments", "l")
def _std_align(a):
a = a.strip().lower()
if len(a) > 1:
return {'left': 'l', 'right': 'r', 'center': 'c'}.get(a, a)
else:
return {'<': 'l', '>': 'r', '^': 'c'}.get(a, a)
std_align = _std_align(alignments)
if std_align in allow:
_alignments = [std_align] * _w
else:
_alignments = []
for a in alignments:
std_align = _std_align(a)
_alignments.append(std_align)
if std_align not in ('l', 'r', 'c'):
raise ValueError('alignment "%s" unrecognized' %
alignments)
if _headings[0] and len(_alignments) == _w + 1:
_head_align = _alignments[0]
_alignments = _alignments[1:]
else:
_head_align = 'r'
if len(_alignments) != _w:
raise ValueError(
'wrong number of alignments: expected %s but got %s' %
(_w, len(_alignments)))
_column_formats = kwarg.get("formats", [None] * _w)
_wipe_zeros = kwarg.get("wipe_zeros", True)
self._w = _w
self._h = _h
self._lines = _lines
self._headings = _headings
self._head_align = _head_align
self._alignments = _alignments
self._column_formats = _column_formats
self._wipe_zeros = _wipe_zeros
def rsolve_ratio(coeffs, f, n, **hints):
"""
Given linear recurrence operator `\operatorname{L}` of order `k`
with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f`, where `f` is a polynomial, we seek
for all rational solutions over field `K` of characteristic zero.
This procedure accepts only polynomials, however if you are
interested in solving recurrence with rational coefficients
then use ``rsolve`` which will pre-process the given equation
and run this procedure with polynomial arguments.
The algorithm performs two basic steps:
(1) Compute polynomial `v(n)` which can be used as universal
denominator of any rational solution of equation
`\operatorname{L} y = f`.
(2) Construct new linear difference equation by substitution
`y(n) = u(n)/v(n)` and solve it for `u(n)` finding all its
polynomial solutions. Return ``None`` if none were found.
Algorithm implemented here is a revised version of the original
Abramov's algorithm, developed in 1989. The new approach is much
simpler to implement and has better overall efficiency. This
method can be easily adapted to q-difference equations case.
Besides finding rational solutions alone, this functions is
an important part of Hyper algorithm were it is used to find
particular solution of inhomogeneous part of a recurrence.
Examples
========
>>> from sympy.abc import x
>>> from sympy.solvers.recurr import rsolve_ratio
>>> rsolve_ratio([-2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x,
... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x)
C2*(2*x - 3)/(2*(x**2 - 1))
References
==========
.. [1] S. A. Abramov, Rational solutions of linear difference
and q-difference equations with polynomial coefficients,
in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York,
1995, 285-289
See Also
========
rsolve_hyper
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
coeffs = list(map(sympify, coeffs))
r = len(coeffs) - 1
A, B = coeffs[r], coeffs[0]
A = A.subs(n, n - r).expand()
h = Dummy('h')
res = resultant(A, B.subs(n, n + h), n)
if not res.is_polynomial(h):
p, q = res.as_numer_denom()
res = quo(p, q, h)
nni_roots = list(roots(res, h, filter='Z',
predicate=lambda r: r >= 0).keys())
if not nni_roots:
return rsolve_poly(coeffs, f, n, **hints)
else:
C, numers = S.One, [S.Zero]*(r + 1)
for i in range(int(max(nni_roots)), -1, -1):
d = gcd(A, B.subs(n, n + i), n)
A = quo(A, d, n)
B = quo(B, d.subs(n, n - i), n)
C *= Mul(*[ d.subs(n, n - j) for j in range(0, i + 1) ])
denoms = [ C.subs(n, n + i) for i in range(0, r + 1) ]
for i in range(0, r + 1):
g = gcd(coeffs[i], denoms[i], n)
numers[i] = quo(coeffs[i], g, n)
denoms[i] = quo(denoms[i], g, n)
for i in range(0, r + 1):
numers[i] *= Mul(*(denoms[:i] + denoms[i + 1:]))
result = rsolve_poly(numers, f * Mul(*denoms), n, **hints)
if result is not None:
if hints.get('symbols', False):
return (simplify(result[0] / C), result[1])
else:
return simplify(result / C)
else:
return None
def rsolve_poly(coeffs, f, n, **hints):
"""
Given linear recurrence operator `\operatorname{L}` of order
`k` with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f`, where `f` is a polynomial, we seek for
all polynomial solutions over field `K` of characteristic zero.
The algorithm performs two basic steps:
(1) Compute degree `N` of the general polynomial solution.
(2) Find all polynomials of degree `N` or less
of `\operatorname{L} y = f`.
There are two methods for computing the polynomial solutions.
If the degree bound is relatively small, i.e. it's smaller than
or equal to the order of the recurrence, then naive method of
undetermined coefficients is being used. This gives system
of algebraic equations with `N+1` unknowns.
In the other case, the algorithm performs transformation of the
initial equation to an equivalent one, for which the system of
algebraic equations has only `r` indeterminates. This method is
quite sophisticated (in comparison with the naive one) and was
invented together by Abramov, Bronstein and Petkovsek.
It is possible to generalize the algorithm implemented here to
the case of linear q-difference and differential equations.
Lets say that we would like to compute `m`-th Bernoulli polynomial
up to a constant. For this we can use `b(n+1) - b(n) = m n^{m-1}`
recurrence, which has solution `b(n) = B_m + C`. For example:
>>> from sympy import Symbol, rsolve_poly
>>> n = Symbol('n', integer=True)
>>> rsolve_poly([-1, 1], 4*n**3, n)
C0 + n**4 - 2*n**3 + n**2
References
==========
.. [1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial
solutions of linear operator equations, in: T. Levelt, ed.,
Proc. ISSAC '95, ACM Press, New York, 1995, 290-296.
.. [2] M. Petkovsek, Hypergeometric solutions of linear recurrences
with polynomial coefficients, J. Symbolic Computation,
14 (1992), 243-264.
.. [3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
homogeneous = f.is_zero
r = len(coeffs) - 1
coeffs = [ Poly(coeff, n) for coeff in coeffs ]
polys = [ Poly(0, n) ] * (r + 1)
terms = [ (S.Zero, S.NegativeInfinity) ] *(r + 1)
for i in range(0, r + 1):
for j in range(i, r + 1):
polys[i] += coeffs[j]*binomial(j, i)
if not polys[i].is_zero:
(exp,), coeff = polys[i].LT()
terms[i] = (coeff, exp)
d = b = terms[0][1]
for i in range(1, r + 1):
if terms[i][1] > d:
d = terms[i][1]
if terms[i][1] - i > b:
b = terms[i][1] - i
d, b = int(d), int(b)
x = Dummy('x')
degree_poly = S.Zero
for i in range(0, r + 1):
if terms[i][1] - i == b:
degree_poly += terms[i][0]*FallingFactorial(x, i)
nni_roots = list(roots(degree_poly, x, filter='Z',
predicate=lambda r: r >= 0).keys())
if nni_roots:
N = [max(nni_roots)]
else:
N = []
if homogeneous:
N += [-b - 1]
else:
N += [f.as_poly(n).degree() - b, -b - 1]
N = int(max(N))
if N < 0:
if homogeneous:
if hints.get('symbols', False):
return (S.Zero, [])
else:
return S.Zero
else:
return None
if N <= r:
C = []
y = E = S.Zero
for i in range(0, N + 1):
C.append(Symbol('C' + str(i)))
y += C[i] * n**i
for i in range(0, r + 1):
E += coeffs[i].as_expr()*y.subs(n, n + i)
solutions = solve_undetermined_coeffs(E - f, C, n)
if solutions is not None:
C = [ c for c in C if (c not in solutions) ]
result = y.subs(solutions)
else:
return None # TBD
else:
A = r
U = N + A + b + 1
nni_roots = list(roots(polys[r], filter='Z',
predicate=lambda r: r >= 0).keys())
if nni_roots != []:
a = max(nni_roots) + 1
else:
a = S.Zero
def _zero_vector(k):
return [S.Zero] * k
def _one_vector(k):
return [S.One] * k
def _delta(p, k):
B = S.One
D = p.subs(n, a + k)
for i in range(1, k + 1):
B *= -Rational(k - i + 1, i)
D += B * p.subs(n, a + k - i)
return D
alpha = {}
for i in range(-A, d + 1):
I = _one_vector(d + 1)
for k in range(1, d + 1):
I[k] = I[k - 1] * (x + i - k + 1)/k
alpha[i] = S.Zero
for j in range(0, A + 1):
for k in range(0, d + 1):
B = binomial(k, i + j)
D = _delta(polys[j].as_expr(), k)
alpha[i] += I[k]*B*D
V = Matrix(U, A, lambda i, j: int(i == j))
if homogeneous:
for i in range(A, U):
v = _zero_vector(A)
for k in range(1, A + b + 1):
if i - k < 0:
break
B = alpha[k - A].subs(x, i - k)
for j in range(0, A):
v[j] += B * V[i - k, j]
denom = alpha[-A].subs(x, i)
for j in range(0, A):
V[i, j] = -v[j] / denom
else:
G = _zero_vector(U)
for i in range(A, U):
v = _zero_vector(A)
g = S.Zero
for k in range(1, A + b + 1):
if i - k < 0:
break
B = alpha[k - A].subs(x, i - k)
for j in range(0, A):
v[j] += B * V[i - k, j]
g += B * G[i - k]
denom = alpha[-A].subs(x, i)
for j in range(0, A):
V[i, j] = -v[j] / denom
G[i] = (_delta(f, i - A) - g) / denom
P, Q = _one_vector(U), _zero_vector(A)
for i in range(1, U):
P[i] = (P[i - 1] * (n - a - i + 1)/i).expand()
for i in range(0, A):
Q[i] = Add(*[ (v*p).expand() for v, p in zip(V[:, i], P) ])
if not homogeneous:
h = Add(*[ (g*p).expand() for g, p in zip(G, P) ])
C = [ Symbol('C' + str(i)) for i in range(0, A) ]
g = lambda i: Add(*[ c*_delta(q, i) for c, q in zip(C, Q) ])
if homogeneous:
E = [ g(i) for i in range(N + 1, U) ]
else:
E = [ g(i) + _delta(h, i) for i in range(N + 1, U) ]
if E != []:
solutions = solve(E, *C)
if not solutions:
if homogeneous:
if hints.get('symbols', False):
return (S.Zero, [])
else:
return S.Zero
else:
return None
else:
solutions = {}
if homogeneous:
result = S.Zero
else:
result = h
for c, q in list(zip(C, Q)):
if c in solutions:
s = solutions[c]*q
C.remove(c)
else:
s = c*q
result += s.expand()
if hints.get('symbols', False):
return (result, C)
else:
return result
SSUP = lambda symb: U('SUPERSCRIPT %s' % symb_2txt[symb])
sub = {} # symb -> subscript symbol
sup = {} # symb -> superscript symbol
# latin subscripts
for l in 'aeioruvxhklmnpst':
sub[l] = LSUB(l)
for l in 'in':
sup[l] = LSUP(l)
for gl in ['beta', 'gamma', 'rho', 'phi', 'chi']:
sub[gl] = GSUB(gl)
for d in [str(i) for i in range(10)]:
sub[d] = DSUB(d)
sup[d] = DSUP(d)
for s in '+-=()':
sub[s] = SSUB(s)
sup[s] = SSUP(s)
# Variable modifiers
# TODO: Is it worth trying to handle faces with, e.g., 'MATHEMATICAL BOLD CAPITAL A'?
# TODO: Make brackets adjust to height of contents
modifier_dict = {
# Accents
'mathring':
lambda s: s + u('\N{COMBINING RING ABOVE}'),
'ddddot':
def rsolve_hyper(coeffs, f, n, **hints):
"""
Given linear recurrence operator `\operatorname{L}` of order `k`
with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f` we seek for all hypergeometric solutions
over field `K` of characteristic zero.
The inhomogeneous part can be either hypergeometric or a sum
of a fixed number of pairwise dissimilar hypergeometric terms.
The algorithm performs three basic steps:
(1) Group together similar hypergeometric terms in the
inhomogeneous part of `\operatorname{L} y = f`, and find
particular solution using Abramov's algorithm.
(2) Compute generating set of `\operatorname{L}` and find basis
in it, so that all solutions are linearly independent.
(3) Form final solution with the number of arbitrary
constants equal to dimension of basis of `\operatorname{L}`.
Term `a(n)` is hypergeometric if it is annihilated by first order
linear difference equations with polynomial coefficients or, in
simpler words, if consecutive term ratio is a rational function.
The output of this procedure is a linear combination of fixed
number of hypergeometric terms. However the underlying method
can generate larger class of solutions - D'Alembertian terms.
Note also that this method not only computes the kernel of the
inhomogeneous equation, but also reduces in to a basis so that
solutions generated by this procedure are linearly independent
Examples
========
>>> from sympy.solvers import rsolve_hyper
>>> from sympy.abc import x
>>> rsolve_hyper([-1, -1, 1], 0, x)
C0*(1/2 + sqrt(5)/2)**x + C1*(-sqrt(5)/2 + 1/2)**x
>>> rsolve_hyper([-1, 1], 1 + x, x)
C0 + x*(x + 1)/2
References
==========
.. [1] M. Petkovsek, Hypergeometric solutions of linear recurrences
with polynomial coefficients, J. Symbolic Computation,
14 (1992), 243-264.
.. [2] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.
"""
coeffs = list(map(sympify, coeffs))
f = sympify(f)
r, kernel, symbols = len(coeffs) - 1, [], set()
if not f.is_zero:
if f.is_Add:
similar = {}
for g in f.expand().args:
if not g.is_hypergeometric(n):
return None
for h in similar.keys():
if hypersimilar(g, h, n):
similar[h] += g
break
else:
similar[g] = S.Zero
inhomogeneous = []
for g, h in similar.items():
inhomogeneous.append(g + h)
elif f.is_hypergeometric(n):
inhomogeneous = [f]
else:
return None
for i, g in enumerate(inhomogeneous):
coeff, polys = S.One, coeffs[:]
denoms = [ S.One ] * (r + 1)
s = hypersimp(g, n)
for j in range(1, r + 1):
coeff *= s.subs(n, n + j - 1)
p, q = coeff.as_numer_denom()
polys[j] *= p
denoms[j] = q
for j in range(0, r + 1):
polys[j] *= Mul(*(denoms[:j] + denoms[j + 1:]))
R = rsolve_poly(polys, Mul(*denoms), n)
if not (R is None or R is S.Zero):
inhomogeneous[i] *= R
else:
return None
result = Add(*inhomogeneous)
else:
result = S.Zero
Z = Dummy('Z')
p, q = coeffs[0], coeffs[r].subs(n, n - r + 1)
p_factors = [ z for z in roots(p, n).keys() ]
q_factors = [ z for z in roots(q, n).keys() ]
factors = [ (S.One, S.One) ]
for p in p_factors:
for q in q_factors:
if p.is_integer and q.is_integer and p <= q:
continue
else:
factors += [(n - p, n - q)]
p = [ (n - p, S.One) for p in p_factors ]
q = [ (S.One, n - q) for q in q_factors ]
factors = p + factors + q
for A, B in factors:
polys, degrees = [], []
D = A*B.subs(n, n + r - 1)
for i in range(0, r + 1):
a = Mul(*[ A.subs(n, n + j) for j in range(0, i) ])
b = Mul(*[ B.subs(n, n + j) for j in range(i, r) ])
poly = quo(coeffs[i]*a*b, D, n)
polys.append(poly.as_poly(n))
if not poly.is_zero:
degrees.append(polys[i].degree())
if degrees:
d, poly = max(degrees), S.Zero
else:
return None
for i in range(0, r + 1):
coeff = polys[i].nth(d)
if coeff is not S.Zero:
poly += coeff * Z**i
for z in roots(poly, Z).keys():
if z.is_zero:
continue
(C, s) = rsolve_poly([ polys[i]*z**i for i in range(r + 1) ], 0, n, symbols=True)
if C is not None and C is not S.Zero:
symbols |= set(s)
ratio = z * A * C.subs(n, n + 1) / B / C
ratio = simplify(ratio)
# If there is a nonnegative root in the denominator of the ratio,
# this indicates that the term y(n_root) is zero, and one should
# start the product with the term y(n_root + 1).
n0 = 0
for n_root in roots(ratio.as_numer_denom()[1], n).keys():
if n_root.has(I):
return None
elif (n0 < (n_root + 1)) == True:
n0 = n_root + 1
K = product(ratio, (n, n0, n - 1))
if K.has(factorial, FallingFactorial, RisingFactorial):
K = simplify(K)
if casoratian(kernel + [K], n, zero=False) != 0:
kernel.append(K)
kernel.sort(key=default_sort_key)
sk = list(zip(numbered_symbols('C'), kernel))
if sk:
for C, ker in sk:
result += C * ker
else:
return None
if hints.get('symbols', False):
symbols |= {s for s, k in sk}
return (result, list(symbols))
else:
return result
def canonicalize(g, dummies, msym, *v):
"""
canonicalize tensor formed by tensors
Parameters
==========
g : permutation representing the tensor
dummies : list representing the dummy indices
it can be a list of dummy indices of the same type
or a list of lists of dummy indices, one list for each
type of index;
the dummy indices must come after the free indices,
and put in order contravariant, covariant
[d0, -d0, d1,-d1,...]
msym : symmetry of the metric(s)
it can be an integer or a list;
in the first case it is the symmetry of the dummy index metric;
in the second case it is the list of the symmetries of the
index metric for each type
v : list, (base_i, gens_i, n_i, sym_i) for tensors of type `i`
base_i, gens_i : BSGS for tensors of this type.
The BSGS should have minimal base under lexicographic ordering;
if not, an attempt is made do get the minimal BSGS;
in case of failure,
canonicalize_naive is used, which is much slower.
n_i : number of tensors of type `i`.
sym_i : symmetry under exchange of component tensors of type `i`.
Both for msym and sym_i the cases are
* None no symmetry
* 0 commuting
* 1 anticommuting
Returns
=======
0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.
Algorithm
=========
First one uses canonical_free to get the minimum tensor under
lexicographic order, using only the slot symmetries.
If the component tensors have not minimal BSGS, it is attempted
to find it; if the attempt fails canonicalize_naive
is used instead.
Compute the residual slot symmetry keeping fixed the free indices
using tensor_gens(base, gens, list_free_indices, sym).
Reduce the problem eliminating the free indices.
Then use double_coset_can_rep and lift back the result reintroducing
the free indices.
Examples
========
one type of index with commuting metric;
`A_{a b}` and `B_{a b}` antisymmetric and commuting
`T = A_{d0 d1} * B^{d0}{}_{d2} * B^{d2 d1}`
`ord = [d0,-d0,d1,-d1,d2,-d2]` order of the indices
g = [1, 3, 0, 5, 4, 2, 6, 7]
`T_c = 0`
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, canonicalize, bsgs_direct_product
>>> from sympy.combinatorics import Permutation
>>> base2a, gens2a = get_symmetric_group_sgs(2, 1)
>>> t0 = (base2a, gens2a, 1, 0)
>>> t1 = (base2a, gens2a, 2, 0)
>>> g = Permutation([1, 3, 0, 5, 4, 2, 6, 7])
>>> canonicalize(g, range(6), 0, t0, t1)
0
same as above, but with `B_{a b}` anticommuting
`T_c = -A^{d0 d1} * B_{d0}{}^{d2} * B_{d1 d2}`
can = [0,2,1,4,3,5,7,6]
>>> t1 = (base2a, gens2a, 2, 1)
>>> canonicalize(g, range(6), 0, t0, t1)
[0, 2, 1, 4, 3, 5, 7, 6]
two types of indices `[a,b,c,d,e,f]` and `[m,n]`, in this order,
both with commuting metric
`f^{a b c}` antisymmetric, commuting
`A_{m a}` no symmetry, commuting
`T = f^c{}_{d a} * f^f{}_{e b} * A_m{}^d * A^{m b} * A_n{}^a * A^{n e}`
ord = [c,f,a,-a,b,-b,d,-d,e,-e,m,-m,n,-n]
g = [0,7,3, 1,9,5, 11,6, 10,4, 13,2, 12,8, 14,15]
The canonical tensor is
`T_c = -f^{c a b} * f^{f d e} * A^m{}_a * A_{m d} * A^n{}_b * A_{n e}`
can = [0,2,4, 1,6,8, 10,3, 11,7, 12,5, 13,9, 15,14]
>>> base_f, gens_f = get_symmetric_group_sgs(3, 1)
>>> base1, gens1 = get_symmetric_group_sgs(1)
>>> base_A, gens_A = bsgs_direct_product(base1, gens1, base1, gens1)
>>> t0 = (base_f, gens_f, 2, 0)
>>> t1 = (base_A, gens_A, 4, 0)
>>> dummies = [range(2, 10), range(10, 14)]
>>> g = Permutation([0, 7, 3, 1, 9, 5, 11, 6, 10, 4, 13, 2, 12, 8, 14, 15])
>>> canonicalize(g, dummies, [0, 0], t0, t1)
[0, 2, 4, 1, 6, 8, 10, 3, 11, 7, 12, 5, 13, 9, 15, 14]
"""
from sympy.combinatorics.testutil import canonicalize_naive
if not isinstance(msym, list):
if not msym in [0, 1, None]:
raise ValueError('msym must be 0, 1 or None')
num_types = 1
else:
num_types = len(msym)
if not all(msymx in [0, 1, None] for msymx in msym):
raise ValueError('msym entries must be 0, 1 or None')
if len(dummies) != num_types:
raise ValueError(
'dummies and msym must have the same number of elements')
size = g.size
num_tensors = 0
v1 = []
for i in range(len(v)):
base_i, gens_i, n_i, sym_i = v[i]
# check that the BSGS is minimal;
# this property is used in double_coset_can_rep;
# if it is not minimal use canonicalize_naive
if not _is_minimal_bsgs(base_i, gens_i):
mbsgs = get_minimal_bsgs(base_i, gens_i)
if not mbsgs:
can = canonicalize_naive(g, dummies, msym, *v)
return can
base_i, gens_i = mbsgs
v1.append((base_i, gens_i, [[]] * n_i, sym_i))
num_tensors += n_i
if num_types == 1 and not isinstance(msym, list):
dummies = [dummies]
msym = [msym]
flat_dummies = []
for dumx in dummies:
flat_dummies.extend(dumx)
if flat_dummies and flat_dummies != list(
range(flat_dummies[0], flat_dummies[-1] + 1)):
raise ValueError('dummies is not valid')
# slot symmetry of the tensor
size1, sbase, sgens = gens_products(*v1)
if size != size1:
raise ValueError('g has size %d, generators have size %d' %
(size, size1))
free = [i for i in range(size - 2) if i not in flat_dummies]
num_free = len(free)
# g1 minimal tensor under slot symmetry
g1 = canonical_free(sbase, sgens, g, num_free)
if not flat_dummies:
return g1
# save the sign of g1
sign = 0 if g1[-1] == size - 1 else 1
# the free indices are kept fixed.
# Determine free_i, the list of slots of tensors which are fixed
# since they are occupied by free indices, which are fixed.
start = 0
for i in range(len(v)):
free_i = []
base_i, gens_i, n_i, sym_i = v[i]
len_tens = gens_i[0].size - 2
# for each component tensor get a list od fixed islots
for j in range(n_i):
# get the elements corresponding to the component tensor
h = g1[start:(start + len_tens)]
fr = []
# get the positions of the fixed elements in h
for k in free:
if k in h:
fr.append(h.index(k))
free_i.append(fr)
start += len_tens
v1[i] = (base_i, gens_i, free_i, sym_i)
# BSGS of the tensor with fixed free indices
# if tensor_gens fails in gens_product, use canonicalize_naive
size, sbase, sgens = gens_products(*v1)
# reduce the permutations getting rid of the free indices
pos_dummies = [g1.index(x) for x in flat_dummies]
pos_free = [g1.index(x) for x in range(num_free)]
size_red = size - num_free
g1_red = [x - num_free for x in g1 if x in flat_dummies]
if sign:
g1_red.extend([size_red - 1, size_red - 2])
else:
g1_red.extend([size_red - 2, size_red - 1])
map_slots = _get_map_slots(size, pos_free)
sbase_red = [map_slots[i] for i in sbase if i not in pos_free]
sgens_red = [
_af_new([map_slots[i] for i in y._array_form if i not in pos_free])
for y in sgens
]
dummies_red = [[x - num_free for x in y] for y in dummies]
transv_red = get_transversals(sbase_red, sgens_red)
g1_red = _af_new(g1_red)
g2 = double_coset_can_rep(dummies_red, msym, sbase_red, sgens_red,
transv_red, g1_red)
if g2 == 0:
return 0
# lift to the case with the free indices
g3 = _lift_sgens(size, pos_free, free, g2)
return g3
def test_norm():
# Maximum "n" which is tested:
n_max = 2 # it works, but is slow, for n_max > 2
for n in range(n_max + 1):
for l in range(n):
assert integrate(R_nl(n, l, r)**2 * r**2, (r, 0, oo)) == 1
def double_coset_can_rep(dummies, sym, b_S, sgens, S_transversals, g):
"""
Butler-Portugal algorithm for tensor canonicalization with dummy indices
dummies
list of lists of dummy indices,
one list for each type of index;
the dummy indices are put in order contravariant, covariant
[d0, -d0, d1, -d1, ...].
sym
list of the symmetries of the index metric for each type.
possible symmetries of the metrics
* 0 symmetric
* 1 antisymmetric
* None no symmetry
b_S
base of a minimal slot symmetry BSGS.
sgens
generators of the slot symmetry BSGS.
S_transversals
transversals for the slot BSGS.
g
permutation representing the tensor.
Return 0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.
A tensor with dummy indices can be represented in a number
of equivalent ways which typically grows exponentially with
the number of indices. To be able to establish if two tensors
with many indices are equal becomes computationally very slow
in absence of an efficient algorithm.
The Butler-Portugal algorithm [3] is an efficient algorithm to
put tensors in canonical form, solving the above problem.
Portugal observed that a tensor can be represented by a permutation,
and that the class of tensors equivalent to it under slot and dummy
symmetries is equivalent to the double coset `D*g*S`
(Note: in this documentation we use the conventions for multiplication
of permutations p, q with (p*q)(i) = p[q[i]] which is opposite
to the one used in the Permutation class)
Using the algorithm by Butler to find a representative of the
double coset one can find a canonical form for the tensor.
To see this correspondence,
let `g` be a permutation in array form; a tensor with indices `ind`
(the indices including both the contravariant and the covariant ones)
can be written as
`t = T(ind[g[0],..., ind[g[n-1]])`,
where `n= len(ind)`;
`g` has size `n + 2`, the last two indices for the sign of the tensor
(trick introduced in [4]).
A slot symmetry transformation `s` is a permutation acting on the slots
`t -> T(ind[(g*s)[0]],..., ind[(g*s)[n-1]])`
A dummy symmetry transformation acts on `ind`
`t -> T(ind[(d*g)[0]],..., ind[(d*g)[n-1]])`
Being interested only in the transformations of the tensor under
these symmetries, one can represent the tensor by `g`, which transforms
as
`g -> d*g*s`, so it belongs to the coset `D*g*S`.
Let us explain the conventions by an example.
Given a tensor `T^{d3 d2 d1}{}_{d1 d2 d3}` with the slot symmetries
`T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`
`T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`
and symmetric metric, find the tensor equivalent to it which
is the lowest under the ordering of indices:
lexicographic ordering `d1, d2, d3` then and contravariant index
before covariant index; that is the canonical form of the tensor.
The canonical form is `-T^{d1 d2 d3}{}_{d1 d2 d3}`
obtained using `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`.
To convert this problem in the input for this function,
use the following labelling of the index names
(- for covariant for short) `d1, -d1, d2, -d2, d3, -d3`
`T^{d3 d2 d1}{}_{d1 d2 d3}` corresponds to `g = [4, 2, 0, 1, 3, 5, 6, 7]`
where the last two indices are for the sign
`sgens = [Permutation(0, 2)(6, 7), Permutation(0, 4)(6, 7)]`
sgens[0] is the slot symmetry `-(0, 2)`
`T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`
sgens[1] is the slot symmetry `-(0, 4)`
`T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`
The dummy symmetry group D is generated by the strong base generators
`[(0, 1), (2, 3), (4, 5), (0, 1)(2, 3),(2, 3)(4, 5)]`
The dummy symmetry acts from the left
`d = [1, 0, 2, 3, 4, 5, 6, 7]` exchange `d1 -> -d1`
`T^{d3 d2 d1}{}_{d1 d2 d3} == T^{d3 d2}{}_{d1}{}^{d1}{}_{d2 d3}`
`g=[4, 2, 0, 1, 3, 5, 6, 7] -> [4, 2, 1, 0, 3, 5, 6, 7] = _af_rmul(d, g)`
which differs from `_af_rmul(g, d)`.
The slot symmetry acts from the right
`s = [2, 1, 0, 3, 4, 5, 7, 6]` exchanges slots 0 and 2 and changes sign
`T^{d3 d2 d1}{}_{d1 d2 d3} == -T^{d1 d2 d3}{}_{d1 d2 d3}`
`g=[4,2,0,1,3,5,6,7] -> [0, 2, 4, 1, 3, 5, 7, 6] = _af_rmul(g, s)`
Example in which the tensor is zero, same slot symmetries as above:
`T^{d3}{}_{d1,d2}{}^{d1}{}_{d3}{}^{d2}`
`= -T^{d3}{}_{d1,d3}{}^{d1}{}_{d2}{}^{d2}` under slot symmetry `-(2,4)`;
`= T_{d3 d1}{}^{d3}{}^{d1}{}_{d2}{}^{d2}` under slot symmetry `-(0,2)`;
`= T^{d3}{}_{d1 d3}{}^{d1}{}_{d2}{}^{d2}` symmetric metric;
`= 0` since two of these lines have tensors differ only for the sign.
The double coset D*g*S consists of permutations `h = d*g*s` corresponding
to equivalent tensors; if there are two `h` which are the same apart
from the sign, return zero; otherwise
choose as representative the tensor with indices
ordered lexicographically according to `[d1, -d1, d2, -d2, d3, -d3]`
that is `rep = min(D*g*S) = min([d*g*s for d in D for s in S])`
The indices are fixed one by one; first choose the lowest index
for slot 0, then the lowest remaining index for slot 1, etc.
Doing this one obtains a chain of stabilizers
`S -> S_{b0} -> S_{b0,b1} -> ...` and
`D -> D_{p0} -> D_{p0,p1} -> ...`
where `[b0, b1, ...] = range(b)` is a base of the symmetric group;
the strong base `b_S` of S is an ordered sublist of it;
therefore it is sufficient to compute once the
strong base generators of S using the Schreier-Sims algorithm;
the stabilizers of the strong base generators are the
strong base generators of the stabilizer subgroup.
`dbase = [p0, p1, ...]` is not in general in lexicographic order,
so that one must recompute the strong base generators each time;
however this is trivial, there is no need to use the Schreier-Sims
algorithm for D.
The algorithm keeps a TAB of elements `(s_i, d_i, h_i)`
where `h_i = d_i*g*s_i` satisfying `h_i[j] = p_j` for `0 <= j < i`
starting from `s_0 = id, d_0 = id, h_0 = g`.
The equations `h_0[0] = p_0, h_1[1] = p_1,...` are solved in this order,
choosing each time the lowest possible value of p_i
For `j < i`
`d_i*g*s_i*S_{b_0,...,b_{i-1}}*b_j = D_{p_0,...,p_{i-1}}*p_j`
so that for dx in `D_{p_0,...,p_{i-1}}` and sx in
`S_{base[0],...,base[i-1]}` one has `dx*d_i*g*s_i*sx*b_j = p_j`
Search for dx, sx such that this equation holds for `j = i`;
it can be written as `s_i*sx*b_j = J, dx*d_i*g*J = p_j`
`sx*b_j = s_i**-1*J; sx = trace(s_i**-1, S_{b_0,...,b_{i-1}})`
`dx**-1*p_j = d_i*g*J; dx = trace(d_i*g*J, D_{p_0,...,p_{i-1}})`
`s_{i+1} = s_i*trace(s_i**-1*J, S_{b_0,...,b_{i-1}})`
`d_{i+1} = trace(d_i*g*J, D_{p_0,...,p_{i-1}})**-1*d_i`
`h_{i+1}*b_i = d_{i+1}*g*s_{i+1}*b_i = p_i`
`h_n*b_j = p_j` for all j, so that `h_n` is the solution.
Add the found `(s, d, h)` to TAB1.
At the end of the iteration sort TAB1 with respect to the `h`;
if there are two consecutive `h` in TAB1 which differ only for the
sign, the tensor is zero, so return 0;
if there are two consecutive `h` which are equal, keep only one.
Then stabilize the slot generators under `i` and the dummy generators
under `p_i`.
Assign `TAB = TAB1` at the end of the iteration step.
At the end `TAB` contains a unique `(s, d, h)`, since all the slots
of the tensor `h` have been fixed to have the minimum value according
to the symmetries. The algorithm returns `h`.
It is important that the slot BSGS has lexicographic minimal base,
otherwise there is an `i` which does not belong to the slot base
for which `p_i` is fixed by the dummy symmetry only, while `i`
is not invariant from the slot stabilizer, so `p_i` is not in
general the minimal value.
This algorithm differs slightly from the original algorithm [3]:
the canonical form is minimal lexicographically, and
the BSGS has minimal base under lexicographic order.
Equal tensors `h` are eliminated from TAB.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.tensor_can import double_coset_can_rep, get_transversals
>>> gens = [Permutation(x) for x in [[2, 1, 0, 3, 4, 5, 7, 6], [4, 1, 2, 3, 0, 5, 7, 6]]]
>>> base = [0, 2]
>>> g = Permutation([4, 2, 0, 1, 3, 5, 6, 7])
>>> transversals = get_transversals(base, gens)
>>> double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)
[0, 1, 2, 3, 4, 5, 7, 6]
>>> g = Permutation([4, 1, 3, 0, 5, 2, 6, 7])
>>> double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)
0
"""
size = g.size
g = g.array_form
num_dummies = size - 2
indices = list(range(num_dummies))
all_metrics_with_sym = all([_ is not None for _ in sym])
num_types = len(sym)
dumx = dummies[:]
dumx_flat = []
for dx in dumx:
dumx_flat.extend(dx)
b_S = b_S[:]
sgensx = [h._array_form for h in sgens]
if b_S:
S_transversals = transversal2coset(size, b_S, S_transversals)
# strong generating set for D
dsgsx = []
for i in range(num_types):
dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
ginv = _af_invert(g)
idn = list(range(size))
# TAB = list of entries (s, d, h) where h = _af_rmuln(d,g,s)
# for short, in the following d*g*s means _af_rmuln(d,g,s)
TAB = [(idn, idn, g)]
for i in range(size - 2):
b = i
testb = b in b_S and sgensx
if testb:
sgensx1 = [_af_new(_) for _ in sgensx]
deltab = _orbit(size, sgensx1, b)
else:
deltab = set([b])
# p1 = min(IMAGES) = min(Union D_p*h*deltab for h in TAB)
if all_metrics_with_sym:
md = _min_dummies(dumx, sym, indices)
else:
md = [
min(_orbit(size, [_af_new(ddx) for ddx in dsgsx], ii))
for ii in range(size - 2)
]
p_i = min([min([md[h[x]] for x in deltab]) for s, d, h in TAB])
dsgsx1 = [_af_new(_) for _ in dsgsx]
Dxtrav = _orbit_transversal(size, dsgsx1, p_i, False, af=True) \
if dsgsx else None
if Dxtrav:
Dxtrav = [_af_invert(x) for x in Dxtrav]
# compute the orbit of p_i
for ii in range(num_types):
if p_i in dumx[ii]:
# the orbit is made by all the indices in dum[ii]
if sym[ii] is not None:
deltap = dumx[ii]
else:
# the orbit is made by all the even indices if p_i
# is even, by all the odd indices if p_i is odd
p_i_index = dumx[ii].index(p_i) % 2
deltap = dumx[ii][p_i_index::2]
break
else:
deltap = [p_i]
TAB1 = []
nTAB = len(TAB)
while TAB:
s, d, h = TAB.pop()
if min([md[h[x]] for x in deltab]) != p_i:
continue
deltab1 = [x for x in deltab if md[h[x]] == p_i]
# NEXT = s*deltab1 intersection (d*g)**-1*deltap
dg = _af_rmul(d, g)
dginv = _af_invert(dg)
sdeltab = [s[x] for x in deltab1]
gdeltap = [dginv[x] for x in deltap]
NEXT = [x for x in sdeltab if x in gdeltap]
# d, s satisfy
# d*g*s*base[i-1] = p_{i-1}; using the stabilizers
# d*g*s*S_{base[0],...,base[i-1]}*base[i-1] =
# D_{p_0,...,p_{i-1}}*p_{i-1}
# so that to find d1, s1 satisfying d1*g*s1*b = p_i
# one can look for dx in D_{p_0,...,p_{i-1}} and
# sx in S_{base[0],...,base[i-1]}
# d1 = dx*d; s1 = s*sx
# d1*g*s1*b = dx*d*g*s*sx*b = p_i
for j in NEXT:
if testb:
# solve s1*b = j with s1 = s*sx for some element sx
# of the stabilizer of ..., base[i-1]
# sx*b = s**-1*j; sx = _trace_S(s, j,...)
# s1 = s*trace_S(s**-1*j,...)
s1 = _trace_S(s, j, b, S_transversals)
if not s1:
continue
else:
s1 = [s[ix] for ix in s1]
else:
s1 = s
# assert s1[b] == j # invariant
# solve d1*g*j = p_i with d1 = dx*d for some element dg
# of the stabilizer of ..., p_{i-1}
# dx**-1*p_i = d*g*j; dx**-1 = trace_D(d*g*j,...)
# d1 = trace_D(d*g*j,...)**-1*d
# to save an inversion in the inner loop; notice we did
# Dxtrav = [perm_af_invert(x) for x in Dxtrav] out of the loop
if Dxtrav:
d1 = _trace_D(dg[j], p_i, Dxtrav)
if not d1:
continue
else:
if p_i != dg[j]:
continue
d1 = idn
assert d1[dg[j]] == p_i # invariant
d1 = [d1[ix] for ix in d]
h1 = [d1[g[ix]] for ix in s1]
# assert h1[b] == p_i # invariant
TAB1.append((s1, d1, h1))
# if TAB contains equal permutations, keep only one of them;
# if TAB contains equal permutations up to the sign, return 0
TAB1.sort(key=lambda x: x[-1])
nTAB1 = len(TAB1)
prev = [0] * size
while TAB1:
s, d, h = TAB1.pop()
if h[:-2] == prev[:-2]:
if h[-1] != prev[-1]:
return 0
else:
TAB.append((s, d, h))
prev = h
# stabilize the SGS
sgensx = [h for h in sgensx if h[b] == b]
if b in b_S:
b_S.remove(b)
_dumx_remove(dumx, dumx_flat, p_i)
dsgsx = []
for i in range(num_types):
dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
return TAB[0][-1]
def test_nC_nP_nT():
from sympy.utilities.iterables import (
multiset_permutations, multiset_combinations, multiset_partitions,
partitions, subsets, permutations)
from sympy.functions.combinatorial.numbers import (
nP, nC, nT, stirling, _multiset_histogram, _AOP_product)
from sympy.combinatorics.permutations import Permutation
from sympy.core.numbers import oo
from random import choice
c = string.ascii_lowercase
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nP(s, i)
tot += check
assert len(list(multiset_permutations(s, i))) == check
if u:
assert nP(len(s), i) == check
assert nP(s) == tot
except AssertionError:
print(s, i, 'failed perm test')
raise ValueError()
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nC(s, i)
tot += check
assert len(list(multiset_combinations(s, i))) == check
if u:
assert nC(len(s), i) == check
assert nC(s) == tot
if u:
assert nC(len(s)) == tot
except AssertionError:
print(s, i, 'failed combo test')
raise ValueError()
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(i, j)
assert check.is_Integer
tot += check
assert sum(1 for p in partitions(i, j, size=True) if p[0] == j) == check
assert nT(i) == tot
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(range(i), j)
tot += check
assert len(list(multiset_partitions(list(range(i)), j))) == check
assert nT(range(i)) == tot
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(1, 8):
check = nT(s, i)
tot += check
assert len(list(multiset_partitions(s, i))) == check
if u:
assert nT(range(len(s)), i) == check
if u:
assert nT(range(len(s))) == tot
assert nT(s) == tot
except AssertionError:
print(s, i, 'failed partition test')
raise ValueError()
# tests for Stirling numbers of the first kind that are not tested in the
# above
assert [stirling(9, i, kind=1) for i in range(11)] == [
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
perms = list(permutations(range(4)))
assert [sum(1 for p in perms if Permutation(p).cycles == i)
for i in range(5)] == [0, 6, 11, 6, 1] == [
stirling(4, i, kind=1) for i in range(5)]
# http://oeis.org/A008275
assert [stirling(n, k, signed=1)
for n in range(10) for k in range(1, n + 1)] == [
1, -1,
1, 2, -3,
1, -6, 11, -6,
1, 24, -50, 35, -10,
1, -120, 274, -225, 85, -15,
1, 720, -1764, 1624, -735, 175, -21,
1, -5040, 13068, -13132, 6769, -1960, 322, -28,
1, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
assert [stirling(n, k, kind=1)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 2, 3, 1,
0, 6, 11, 6, 1,
0, 24, 50, 35, 10, 1,
0, 120, 274, 225, 85, 15, 1,
0, 720, 1764, 1624, 735, 175, 21, 1,
0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1,
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
assert [stirling(n, k, kind=2)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 1, 3, 1,
0, 1, 7, 6, 1,
0, 1, 15, 25, 10, 1,
0, 1, 31, 90, 65, 15, 1,
0, 1, 63, 301, 350, 140, 21, 1,
0, 1, 127, 966, 1701, 1050, 266, 28, 1,
0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1]
assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
raises(ValueError, lambda: stirling(-2, 2))
def delta(p):
if len(p) == 1:
return oo
return min(abs(i[0] - i[1]) for i in subsets(p, 2))
parts = multiset_partitions(range(5), 3)
d = 2
assert (sum(1 for p in parts if all(delta(i) >= d for i in p)) ==
stirling(5, 3, d=d) == 7)
# other coverage tests
assert nC('abb', 2) == nC('aab', 2) == 2
assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
assert nP(3, 4) == 0
assert nP('aabc', 5) == 0
assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
len(list(multiset_combinations('aabbccdd', 2))) == 10
assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
assert nC(list('abcdd'), 4) == 4
assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
assert nC('aabb'*3, 3) == 4 # aaa, bbb, abb, baa
assert dict(_AOP_product((4,1,1,1))) == {
0: 1, 1: 4, 2: 7, 3: 8, 4: 8, 5: 7, 6: 4, 7: 1}
# the following was the first t that showed a problem in a previous form of
# the function, so it's not as random as it may appear
t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
raises(ValueError, lambda: _multiset_histogram({1:'a'}))
class prettyForm(stringPict):
"""
Extension of the stringPict class that knows about basic math applications,
optimizing double minus signs.
"Binding" is interpreted as follows::
ATOM this is an atom: never needs to be parenthesized
FUNC this is a function application: parenthesize if added (?)
DIV this is a division: make wider division if divided
POW this is a power: only parenthesize if exponent
MUL this is a multiplication: parenthesize if powered
ADD this is an addition: parenthesize if multiplied or powered
NEG this is a negative number: optimize if added, parenthesize if
multiplied or powered
OPEN this is an open object: parenthesize if added, multiplied, or
powered (example: Piecewise)
"""
ATOM, FUNC, DIV, POW, MUL, ADD, NEG, OPEN = range(8)
def __init__(self, s, baseline=0, binding=0, unicode=None):
"""Initialize from stringPict and binding power."""
stringPict.__init__(self, s, baseline)
self.binding = binding
self.unicode = unicode or s
# Note: code to handle subtraction is in _print_Add
def __add__(self, *others):
"""Make a pretty addition.
Addition of negative numbers is simplified.
"""
arg = self
if arg.binding > prettyForm.NEG:
arg = stringPict(*arg.parens())
result = [arg]
for arg in others:
#add parentheses for weak binders
if arg.binding > prettyForm.NEG:
arg = stringPict(*arg.parens())
#use existing minus sign if available
if arg.binding != prettyForm.NEG:
result.append(' + ')
result.append(arg)
return prettyForm(binding=prettyForm.ADD, *stringPict.next(*result))
def __div__(self, den, slashed=False):
"""Make a pretty division; stacked or slashed.
"""
if slashed:
raise NotImplementedError("Can't do slashed fraction yet")
num = self
if num.binding == prettyForm.DIV:
num = stringPict(*num.parens())
if den.binding == prettyForm.DIV:
den = stringPict(*den.parens())
if num.binding == prettyForm.NEG:
num = num.right(" ")[0]
return prettyForm(binding=prettyForm.DIV,
*stringPict.stack(num, stringPict.LINE, den))
def __truediv__(self, o):
return self.__div__(o)
def __mul__(self, *others):
"""Make a pretty multiplication.
Parentheses are needed around +, - and neg.
"""
quantity = {'degree': u"\N{DEGREE SIGN}"}
if len(others) == 0:
return self # We aren't actually multiplying... So nothing to do here.
args = self
if args.binding > prettyForm.MUL:
arg = stringPict(*args.parens())
result = [args]
for arg in others:
if arg.picture[0] not in quantity.values():
result.append(xsym('*'))
#add parentheses for weak binders
if arg.binding > prettyForm.MUL:
arg = stringPict(*arg.parens())
result.append(arg)
len_res = len(result)
for i in range(len_res):
if i < len_res - 1 and result[i] == '-1' and result[i + 1] == xsym(
'*'):
# substitute -1 by -, like in -1*x -> -x
result.pop(i)
result.pop(i)
result.insert(i, '-')
if result[0][0] == '-':
# if there is a - sign in front of all
# This test was failing to catch a prettyForm.__mul__(prettyForm("-1", 0, 6)) being negative
bin = prettyForm.NEG
if result[0] == '-':
right = result[1]
if right.picture[right.baseline][0] == '-':
result[0] = '- '
else:
bin = prettyForm.MUL
return prettyForm(binding=bin, *stringPict.next(*result))
def __repr__(self):
return "prettyForm(%r,%d,%d)" % ('\n'.join(
self.picture), self.baseline, self.binding)
def __pow__(self, b):
"""Make a pretty power.
"""
a = self
use_inline_func_form = False
if b.binding == prettyForm.POW:
b = stringPict(*b.parens())
if a.binding > prettyForm.FUNC:
a = stringPict(*a.parens())
elif a.binding == prettyForm.FUNC:
# heuristic for when to use inline power
if b.height() > 1:
a = stringPict(*a.parens())
else:
use_inline_func_form = True
if use_inline_func_form:
# 2
# sin + + (x)
b.baseline = a.prettyFunc.baseline + b.height()
func = stringPict(*a.prettyFunc.right(b))
return prettyForm(*func.right(a.prettyArgs))
else:
# 2 <-- top
# (x+y) <-- bot
top = stringPict(*b.left(' ' * a.width()))
bot = stringPict(*a.right(' ' * b.width()))
return prettyForm(binding=prettyForm.POW, *bot.above(top))
simpleFunctions = ["sin", "cos", "tan"]
@staticmethod
def apply(function, *args):
"""Functions of one or more variables.
"""
if function in prettyForm.simpleFunctions:
#simple function: use only space if possible
assert len(
args
) == 1, "Simple function %s must have 1 argument" % function
arg = args[0].__pretty__()
if arg.binding <= prettyForm.DIV:
#optimization: no parentheses necessary
return prettyForm(binding=prettyForm.FUNC,
*arg.left(function + ' '))
argumentList = []
for arg in args:
argumentList.append(',')
argumentList.append(arg.__pretty__())
argumentList = stringPict(*stringPict.next(*argumentList[1:]))
argumentList = stringPict(*argumentList.parens())
return prettyForm(binding=prettyForm.ATOM,
*argumentList.left(function))
def canonical_free(base, gens, g, num_free):
"""
canonicalization of a tensor with respect to free indices
choosing the minimum with respect to lexicographical ordering
in the free indices
``base``, ``gens`` BSGS for slot permutation group
``g`` permutation representing the tensor
``num_free`` number of free indices
The indices must be ordered with first the free indices
see explanation in double_coset_can_rep
The algorithm is a variation of the one given in [2].
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import canonical_free
>>> gens = [[1, 0, 2, 3, 5, 4], [2, 3, 0, 1, 4, 5],[0, 1, 3, 2, 5, 4]]
>>> gens = [Permutation(h) for h in gens]
>>> base = [0, 2]
>>> g = Permutation([2, 1, 0, 3, 4, 5])
>>> canonical_free(base, gens, g, 4)
[0, 3, 1, 2, 5, 4]
Consider the product of Riemann tensors
``T = R^{a}_{d0}^{d1,d2}*R_{d2,d1}^{d0,b}``
The order of the indices is ``[a, b, d0, -d0, d1, -d1, d2, -d2]``
The permutation corresponding to the tensor is
``g = [0, 3, 4, 6, 7, 5, 2, 1, 8, 9]``
In particular ``a`` is position ``0``, ``b`` is in position ``9``.
Use the slot symmetries to get `T` is a form which is the minimal
in lexicographic order in the free indices ``a`` and ``b``, e.g.
``-R^{a}_{d0}^{d1,d2}*R^{b,d0}_{d2,d1}`` corresponding to
``[0, 3, 4, 6, 1, 2, 7, 5, 9, 8]``
>>> from sympy.combinatorics.tensor_can import riemann_bsgs, tensor_gens
>>> base, gens = riemann_bsgs
>>> size, sbase, sgens = tensor_gens(base, gens, [[], []], 0)
>>> g = Permutation([0, 3, 4, 6, 7, 5, 2, 1, 8, 9])
>>> canonical_free(sbase, [Permutation(h) for h in sgens], g, 2)
[0, 3, 4, 6, 1, 2, 7, 5, 9, 8]
"""
g = g.array_form
size = len(g)
if not base:
return g[:]
transversals = get_transversals(base, gens)
m = len(base)
for x in sorted(g[:-2]):
if x not in base:
base.append(x)
h = g
for i, transv in enumerate(transversals):
b = base[i]
h_i = [size] * num_free
# find the element s in transversals[i] such that
# _af_rmul(h, s) has its free elements with the lowest position in h
s = None
for sk in transv.values():
h1 = _af_rmul(h, sk)
hi = [h1.index(ix) for ix in range(num_free)]
if hi < h_i:
h_i = hi
s = sk
if s:
h = _af_rmul(h, s)
return h
def euler_equations(L, funcs=(), vars=()):
r"""
Find the Euler-Lagrange equations [1]_ for a given Lagrangian.
Parameters
==========
L : Expr
The Lagrangian that should be a function of the functions listed
in the second argument and their derivatives.
For example, in the case of two functions `f(x,y)`, `g(x,y)` and
two independent variables `x`, `y` the Lagrangian would have the form:
.. math:: L\left(f(x,y),g(x,y),\frac{\partial f(x,y)}{\partial x},
\frac{\partial f(x,y)}{\partial y},
\frac{\partial g(x,y)}{\partial x},
\frac{\partial g(x,y)}{\partial y},x,y\right)
In many cases it is not necessary to provide anything, except the
Lagrangian, it will be auto-detected (and an error raised if this
couldn't be done).
funcs : Function or an iterable of Functions
The functions that the Lagrangian depends on. The Euler equations
are differential equations for each of these functions.
vars : Symbol or an iterable of Symbols
The Symbols that are the independent variables of the functions.
Returns
=======
eqns : list of Eq
The list of differential equations, one for each function.
Examples
========
>>> from sympy import Symbol, Function
>>> from sympy.calculus.euler import euler_equations
>>> x = Function('x')
>>> t = Symbol('t')
>>> L = (x(t).diff(t))**2/2 - x(t)**2/2
>>> euler_equations(L, x(t), t)
[Eq(-x(t) - Derivative(x(t), t, t), 0)]
>>> u = Function('u')
>>> x = Symbol('x')
>>> L = (u(t, x).diff(t))**2/2 - (u(t, x).diff(x))**2/2
>>> euler_equations(L, u(t, x), [t, x])
[Eq(-Derivative(u(t, x), t, t) + Derivative(u(t, x), x, x), 0)]
References
==========
.. [1] http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation
"""
funcs = tuple(funcs) if iterable(funcs) else (funcs, )
if not funcs:
funcs = tuple(L.atoms(Function))
else:
for f in funcs:
if not isinstance(f, Function):
raise TypeError('Function expected, got: %s' % f)
vars = tuple(vars) if iterable(vars) else (vars, )
if not vars:
vars = funcs[0].args
else:
vars = tuple(sympify(var) for var in vars)
if not all(isinstance(v, Symbol) for v in vars):
raise TypeError('Variables are not symbols, got %s' % vars)
for f in funcs:
if not vars == f.args:
raise ValueError("Variables %s don't match args: %s" % (vars, f))
order = max(
len(d.variables) for d in L.atoms(Derivative) if d.expr in funcs)
eqns = []
for f in funcs:
eq = diff(L, f)
for i in range(1, order + 1):
for p in combinations_with_replacement(vars, i):
eq = eq + S.NegativeOne**i * diff(L, diff(f, *p), *p)
eqns.append(Eq(eq))
return eqns
def tensor_gens(base, gens, list_free_indices, sym=0):
"""
Returns size, res_base, res_gens BSGS for n tensors of the
same type
base, gens BSGS for tensors of this type
list_free_indices list of the slots occupied by fixed indices
for each of the tensors
sym symmetry under commutation of two tensors
sym None no symmetry
sym 0 commuting
sym 1 anticommuting
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import tensor_gens, get_symmetric_group_sgs
>>> Permutation.print_cyclic = True
two symmetric tensors with 3 indices without free indices
>>> base, gens = get_symmetric_group_sgs(3)
>>> tensor_gens(base, gens, [[], []])
(8, [0, 1, 3, 4], [Permutation(7)(0, 1), Permutation(7)(1, 2), Permutation(7)(3, 4), Permutation(7)(4, 5), Permutation(7)(0, 3)(1, 4)(2, 5)])
two symmetric tensors with 3 indices with free indices in slot 1 and 0
>>> tensor_gens(base, gens, [[1], [0]])
(8, [0, 4], [Permutation(7)(0, 2), Permutation(7)(4, 5)])
four symmetric tensors with 3 indices, two of which with free indices
"""
def _get_bsgs(G, base, gens, free_indices):
"""
return the BSGS for G.pointwise_stabilizer(free_indices)
"""
if not free_indices:
return base[:], gens[:]
else:
H = G.pointwise_stabilizer(free_indices)
base, sgs = H.schreier_sims_incremental()
return base, sgs
# if not base there is no slot symmetry for the component tensors
# if list_free_indices.count([]) < 2 there is no commutation symmetry
# so there is no resulting slot symmetry
if not base and list_free_indices.count([]) < 2:
n = len(list_free_indices)
size = gens[0].size
size = n * (gens[0].size - 2) + 2
return size, [], [_af_new(list(range(size)))]
# if any(list_free_indices) one needs to compute the pointwise
# stabilizer, so G is needed
if any(list_free_indices):
G = PermutationGroup(gens)
else:
G = None
# no_free list of lists of indices for component tensors without fixed
# indices
no_free = []
size = gens[0].size
id_af = list(range(size))
num_indices = size - 2
if not list_free_indices[0]:
no_free.append(list(range(num_indices)))
res_base, res_gens = _get_bsgs(G, base, gens, list_free_indices[0])
for i in range(1, len(list_free_indices)):
base1, gens1 = _get_bsgs(G, base, gens, list_free_indices[i])
res_base, res_gens = bsgs_direct_product(res_base, res_gens, base1,
gens1, 1)
if not list_free_indices[i]:
no_free.append(list(range(size - 2, size - 2 + num_indices)))
size += num_indices
nr = size - 2
res_gens = [h for h in res_gens if h._array_form != id_af]
# if sym there are no commuting tensors stop here
if sym is None or not no_free:
if not res_gens:
res_gens = [_af_new(id_af)]
return size, res_base, res_gens
# if the component tensors have moinimal BSGS, so is their direct
# product P; the slot symmetry group is S = P*C, where C is the group
# to (anti)commute the component tensors with no free indices
# a stabilizer has the property S_i = P_i*C_i;
# the BSGS of P*C has SGS_P + SGS_C and the base is
# the ordered union of the bases of P and C.
# If P has minimal BSGS, so has S with this base.
base_comm = []
for i in range(len(no_free) - 1):
ind1 = no_free[i]
ind2 = no_free[i + 1]
a = list(range(ind1[0]))
a.extend(ind2)
a.extend(ind1)
base_comm.append(ind1[0])
a.extend(list(range(ind2[-1] + 1, nr)))
if sym == 0:
a.extend([nr, nr + 1])
else:
a.extend([nr + 1, nr])
res_gens.append(_af_new(a))
res_base = list(res_base)
# each base is ordered; order the union of the two bases
for i in base_comm:
if i not in res_base:
res_base.append(i)
res_base.sort()
if not res_gens:
res_gens = [_af_new(id_af)]
return size, res_base, res_gens
def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True):
"""
Return an Euler-Maclaurin approximation of self, where m is the
number of leading terms to sum directly and n is the number of
terms in the tail.
With m = n = 0, this is simply the corresponding integral
plus a first-order endpoint correction.
Returns (s, e) where s is the Euler-Maclaurin approximation
and e is the estimated error (taken to be the magnitude of
the first omitted term in the tail):
>>> from sympy.abc import k, a, b
>>> from sympy import Sum
>>> Sum(1/k, (k, 2, 5)).doit().evalf()
1.28333333333333
>>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
>>> s
-log(2) + 7/20 + log(5)
>>> from sympy import sstr
>>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
(1.26629073187415, 0.0175000000000000)
The endpoints may be symbolic:
>>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
>>> s
-log(a) + log(b) + 1/(2*b) + 1/(2*a)
>>> e
Abs(1/(12*b**2) - 1/(12*a**2))
If the function is a polynomial of degree at most 2n+1, the
Euler-Maclaurin formula becomes exact (and e = 0 is returned):
>>> Sum(k, (k, 2, b)).euler_maclaurin()
(b**2/2 + b/2 - 1, 0)
>>> Sum(k, (k, 2, b)).doit()
b**2/2 + b/2 - 1
With a nonzero eps specified, the summation is ended
as soon as the remainder term is less than the epsilon.
"""
from sympy.functions import bernoulli, factorial
from sympy.integrals import Integral
m = int(m)
n = int(n)
f = self.function
if len(self.limits) != 1:
raise ValueError("More than 1 limit")
i, a, b = self.limits[0]
if (a > b) == True:
if a - b == 1:
return S.Zero, S.Zero
a, b = b + 1, a - 1
f = -f
s = S.Zero
if m:
if b.is_Integer and a.is_Integer:
m = min(m, b - a + 1)
if not eps or f.is_polynomial(i):
for k in range(m):
s += f.subs(i, a + k)
else:
term = f.subs(i, a)
if term:
test = abs(term.evalf(3)) < eps
if test == True:
return s, abs(term)
elif not (test == False):
# a symbolic Relational class, can't go further
return term, S.Zero
s += term
for k in range(1, m):
term = f.subs(i, a + k)
if abs(term.evalf(3)) < eps and term != 0:
return s, abs(term)
s += term
if b - a + 1 == m:
return s, S.Zero
a += m
x = Dummy('x')
I = Integral(f.subs(i, x), (x, a, b))
if eval_integral:
I = I.doit()
s += I
def fpoint(expr):
if b is S.Infinity:
return expr.subs(i, a), 0
return expr.subs(i, a), expr.subs(i, b)
fa, fb = fpoint(f)
iterm = (fa + fb) / 2
g = f.diff(i)
for k in range(1, n + 2):
ga, gb = fpoint(g)
term = bernoulli(2 * k) / factorial(2 * k) * (gb - ga)
if (eps and term and abs(term.evalf(3)) < eps) or (k > n):
break
s += term
g = g.diff(i, 2, simplify=False)
return s + iterm, abs(term)
def test_chop_value():
for i in range(-27, 28):
assert (Pow(10, i) *
2).n(chop=10**i) and not (Pow(10, i)).n(chop=10**i)
def _eval_is_Identity(self):
if not all(self[i, i] == 1 for i in range(self.rows)):
return False
return len(self._smat) == self.rows
def eval_sum_direct(expr, limits):
from sympy.core import Add
(i, a, b) = limits
dif = b - a
return Add(*[expr.subs(i, a + j) for j in range(dif + 1)])
def _eval_eye(cls, rows, cols):
entries = {(i, i): S.One for i in range(min(rows, cols))}
return cls._new(rows, cols, entries)
def is_convergent(self):
r"""Checks for the convergence of a Sum.
We divide the study of convergence of infinite sums and products in
two parts.
First Part:
One part is the question whether all the terms are well defined, i.e.,
they are finite in a sum and also non-zero in a product. Zero
is the analogy of (minus) infinity in products as
:math:`e^{-\infty} = 0`.
Second Part:
The second part is the question of convergence after infinities,
and zeros in products, have been omitted assuming that their number
is finite. This means that we only consider the tail of the sum or
product, starting from some point after which all terms are well
defined.
For example, in a sum of the form:
.. math::
\sum_{1 \leq i < \infty} \frac{1}{n^2 + an + b}
where a and b are numbers. The routine will return true, even if there
are infinities in the term sequence (at most two). An analogous
product would be:
.. math::
\prod_{1 \leq i < \infty} e^{\frac{1}{n^2 + an + b}}
This is how convergence is interpreted. It is concerned with what
happens at the limit. Finding the bad terms is another independent
matter.
Note: It is responsibility of user to see that the sum or product
is well defined.
There are various tests employed to check the convergence like
divergence test, root test, integral test, alternating series test,
comparison tests, Dirichlet tests. It returns true if Sum is convergent
and false if divergent and NotImplementedError if it can not be checked.
References
==========
.. [1] https://en.wikipedia.org/wiki/Convergence_tests
Examples
========
>>> from sympy import factorial, S, Sum, Symbol, oo
>>> n = Symbol('n', integer=True)
>>> Sum(n/(n - 1), (n, 4, 7)).is_convergent()
True
>>> Sum(n/(2*n + 1), (n, 1, oo)).is_convergent()
False
>>> Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent()
False
>>> Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent()
True
See Also
========
Sum.is_absolutely_convergent()
Product.is_convergent()
"""
from sympy import Interval, Integral, Limit, log, symbols, Ge, Gt, simplify
p, q, r = symbols('p q r', cls=Wild)
sym = self.limits[0][0]
lower_limit = self.limits[0][1]
upper_limit = self.limits[0][2]
sequence_term = self.function
if len(sequence_term.free_symbols) > 1:
raise NotImplementedError(
"convergence checking for more than one symbol "
"containing series is not handled")
if lower_limit.is_finite and upper_limit.is_finite:
return S.true
# transform sym -> -sym and swap the upper_limit = S.Infinity
# and lower_limit = - upper_limit
if lower_limit is S.NegativeInfinity:
if upper_limit is S.Infinity:
return Sum(sequence_term, (sym, 0, S.Infinity)).is_convergent() and \
Sum(sequence_term, (sym, S.NegativeInfinity, 0)).is_convergent()
sequence_term = simplify(sequence_term.xreplace({sym: -sym}))
lower_limit = -upper_limit
upper_limit = S.Infinity
sym_ = Dummy(sym.name, integer=True, positive=True)
sequence_term = sequence_term.xreplace({sym: sym_})
sym = sym_
interval = Interval(lower_limit, upper_limit)
# Piecewise function handle
if sequence_term.is_Piecewise:
for func, cond in sequence_term.args:
# see if it represents something going to oo
if cond == True or cond.as_set().sup is S.Infinity:
s = Sum(func, (sym, lower_limit, upper_limit))
return s.is_convergent()
return S.true
### -------- Divergence test ----------- ###
try:
lim_val = limit(sequence_term, sym, upper_limit)
if lim_val.is_number and lim_val is not S.Zero:
return S.false
except NotImplementedError:
pass
try:
lim_val_abs = limit(abs(sequence_term), sym, upper_limit)
if lim_val_abs.is_number and lim_val_abs is not S.Zero:
return S.false
except NotImplementedError:
pass
order = O(sequence_term, (sym, S.Infinity))
### --------- p-series test (1/n**p) ---------- ###
p1_series_test = order.expr.match(sym**p)
if p1_series_test is not None:
if p1_series_test[p] < -1:
return S.true
if p1_series_test[p] >= -1:
return S.false
p2_series_test = order.expr.match((1 / sym)**p)
if p2_series_test is not None:
if p2_series_test[p] > 1:
return S.true
if p2_series_test[p] <= 1:
return S.false
### ------------- comparison test ------------- ###
# 1/(n**p*log(n)**q*log(log(n))**r) comparison
n_log_test = order.expr.match(
1 / (sym**p * log(sym)**q * log(log(sym))**r))
if n_log_test is not None:
if (n_log_test[p] > 1 or (n_log_test[p] == 1 and n_log_test[q] > 1)
or
(n_log_test[p] == n_log_test[q] == 1 and n_log_test[r] > 1)):
return S.true
return S.false
### ------------- Limit comparison test -----------###
# (1/n) comparison
try:
lim_comp = limit(sym * sequence_term, sym, S.Infinity)
if lim_comp.is_number and lim_comp > 0:
return S.false
except NotImplementedError:
pass
### ----------- ratio test ---------------- ###
next_sequence_term = sequence_term.xreplace({sym: sym + 1})
ratio = combsimp(powsimp(next_sequence_term / sequence_term))
try:
lim_ratio = limit(ratio, sym, upper_limit)
if lim_ratio.is_number:
if abs(lim_ratio) > 1:
return S.false
if abs(lim_ratio) < 1:
return S.true
except NotImplementedError:
pass
### ----------- root test ---------------- ###
lim = Limit(abs(sequence_term)**(1 / sym), sym, S.Infinity)
try:
lim_evaluated = lim.doit()
if lim_evaluated.is_number:
if lim_evaluated < 1:
return S.true
if lim_evaluated > 1:
return S.false
except NotImplementedError:
pass
### ------------- alternating series test ----------- ###
dict_val = sequence_term.match((-1)**(sym + p) * q)
if not dict_val[p].has(sym) and is_decreasing(dict_val[q], interval):
return S.true
### ------------- integral test -------------- ###
check_interval = None
maxima = solveset(sequence_term.diff(sym), sym, interval)
if not maxima:
check_interval = interval
elif isinstance(maxima, FiniteSet) and maxima.sup.is_number:
check_interval = Interval(maxima.sup, interval.sup)
if (check_interval is not None
and (is_decreasing(sequence_term, check_interval)
or is_decreasing(-sequence_term, check_interval))):
integral_val = Integral(sequence_term,
(sym, lower_limit, upper_limit))
try:
integral_val_evaluated = integral_val.doit()
if integral_val_evaluated.is_number:
return S(integral_val_evaluated.is_finite)
except NotImplementedError:
pass
### ----- Dirichlet and bounded times convergent tests ----- ###
# TODO
#
# Dirichlet_test
# https://en.wikipedia.org/wiki/Dirichlet%27s_test
#
# Bounded times convergent test
# It is based on comparison theorems for series.
# In particular, if the general term of a series can
# be written as a product of two terms a_n and b_n
# and if a_n is bounded and if Sum(b_n) is absolutely
# convergent, then the original series Sum(a_n * b_n)
# is absolutely convergent and so convergent.
#
# The following code can grows like 2**n where n is the
# number of args in order.expr
# Possibly combined with the potentially slow checks
# inside the loop, could make this test extremely slow
# for larger summation expressions.
if order.expr.is_Mul:
args = order.expr.args
argset = set(args)
### -------------- Dirichlet tests -------------- ###
m = Dummy('m', integer=True)
def _dirichlet_test(g_n):
try:
ing_val = limit(
Sum(g_n, (sym, interval.inf, m)).doit(), m, S.Infinity)
if ing_val.is_finite:
return S.true
except NotImplementedError:
pass
### -------- bounded times convergent test ---------###
def _bounded_convergent_test(g1_n, g2_n):
try:
lim_val = limit(g1_n, sym, upper_limit)
if lim_val.is_finite or (
isinstance(lim_val, AccumulationBounds) and
(lim_val.max - lim_val.min).is_finite):
if Sum(g2_n, (sym, lower_limit,
upper_limit)).is_absolutely_convergent():
return S.true
except NotImplementedError:
pass
for n in range(1, len(argset)):
for a_tuple in itertools.combinations(args, n):
b_set = argset - set(a_tuple)
a_n = Mul(*a_tuple)
b_n = Mul(*b_set)
if is_decreasing(a_n, interval):
dirich = _dirichlet_test(b_n)
if dirich is not None:
return dirich
bc_test = _bounded_convergent_test(a_n, b_n)
if bc_test is not None:
return bc_test
_sym = self.limits[0][0]
sequence_term = sequence_term.xreplace({sym: _sym})
raise NotImplementedError(
"The algorithm to find the Sum convergence of %s "
"is not yet implemented" % (sequence_term))
