def test_trim():
f = Function("f")
assert trim((f(x) ** 2 + f(x)) / f(x)) == 1 + f(x)
assert trim((sin(x) ** 2 + sin(x)) / sin(x)) == 1 + sin(x)
assert trim((f(x) + y * f(x)) / f(x)) == 1 + y
expr = integrate(1 / (x ** 3 + 1), x)
assert trim(together(expr.diff(x))) == 1 / (x ** 3 + 1)
assert cancel(together(expr.diff(x))) == 1 / (x ** 3 + 1)
expr = together(expr.subs(x, sin(x)).diff(x))
assert trim(expr) == cos(x) / (1 + sin(x) ** 3)
assert trim((2 * (1 / n - cos(n * pi) / n)) / pi) == 1 / pi / n * (2 - 2 * cos(pi * n))
assert trim(sin((f(x) ** 2 + f(x)) / f(x))) == sin(1 + f(x))
assert trim(exp(x) * sin(x) / 2 + cos(x) * exp(x)) == exp(x) * (sin(x) + 2 * cos(x)) / 2
def test_trim():
f = Function('f')
assert trim((f(x)**2 + f(x)) / f(x)) == 1 + f(x)
assert trim((sin(x)**2 + sin(x)) / sin(x)) == 1 + sin(x)
assert trim((f(x) + y * f(x)) / f(x)) == 1 + y
expr = integrate(1 / (x**3 + 1), x)
assert trim(together(expr.diff(x))) == 1 / (x**3 + 1)
assert cancel(together(expr.diff(x))) == 1 / (x**3 + 1)
expr = together(expr.subs(x, sin(x)).diff(x))
assert trim(expr) == cos(x) / (1 + sin(x)**3)
assert trim((2 * (1/n - cos(n * pi)/n))/pi) == \
1/pi/n*(2 - 2*cos(pi*n))
assert trim(sin((f(x)**2 + f(x)) / f(x))) == sin(1 + f(x))
assert trim(exp(x)*sin(x)/2 + cos(x)*exp(x)) == \
exp(x)*(sin(x) + 2*cos(x))/2
def integrate(field=None):
irreducibles = set()
for poly in reducibles:
for z in poly.atoms(Symbol):
if z in V:
break
else:
continue
irreducibles |= set(root_factors(poly, z, domain=field))
log_coeffs, log_part = [], []
B = _symbols('B', len(irreducibles))
for i, poly in enumerate(irreducibles):
if poly.has(*V):
log_coeffs.append(B[i])
log_part.append(log_coeffs[-1] * log(poly))
coeffs = poly_coeffs + log_coeffs
candidate = poly_part/poly_denom + Add(*log_part)
h = together(F - derivation(candidate) / denom)
numer = h.as_numer_denom()[0].expand()
if not numer.is_Add:
numer = [numer]
equations = {}
for term in numer.args:
coeff, dependent = term.as_independent(*V)
if dependent in equations:
equations[dependent] += coeff
else:
equations[dependent] = coeff
solution = solve(equations.values(), *coeffs)
if solution is not None:
return (solution, candidate, coeffs)
else:
return None
def integrate(field=None):
irreducibles = set()
for poly in reducibles:
for z in poly.atoms(Symbol):
if z in V:
break
else:
continue
irreducibles |= set(root_factors(poly, z, domain=field))
log_coeffs, log_part = [], []
B = _symbols('B', len(irreducibles))
for i, poly in enumerate(irreducibles):
if poly.has(*V):
log_coeffs.append(B[i])
log_part.append(log_coeffs[-1] * log(poly))
coeffs = poly_coeffs + log_coeffs
candidate = poly_part / poly_denom + Add(*log_part)
h = together(F - derivation(candidate) / denom)
numer = h.as_numer_denom()[0].expand()
if not numer.is_Add:
numer = [numer]
equations = {}
for term in numer.args:
coeff, dependent = term.as_independent(*V)
if dependent in equations:
equations[dependent] += coeff
else:
equations[dependent] = coeff
solution = solve(equations.values(), *coeffs)
if solution is not None:
return (solution, candidate, coeffs)
else:
return None
def together(self, *args, **kwargs):
"""See the together function in sympy.simplify"""
from sympy.simplify import together
return together(self, *args, **kwargs)
def together(self, *args, **kwargs):
"""See the together function in sympy.simplify"""
from sympy.simplify import together
return together(self, *args, **kwargs)
def trim(f, *symbols, **flags):
"""Cancel common factors in a given formal rational expression.
Given an arbitrary expression, map all functional components
to temporary symbols, rewriting this expression to rational
function form and perform cancelation of common factors.
When given a rational function or a list of symbols discards
all functional components, then this procedure is equivalent
to cancel().
Note that this procedure can thread over composite objects
like big operators, matrices, relational operators etc. It
can be also called recursively (to change this behaviour
unset 'recursive' flag).
>>> from sympy import *
>>> x,y = symbols('xy')
>>> f = Function('f')
>>> trim((f(x)**2+f(x))/f(x))
1 + f(x)
>>> trim((x**2+x)/x)
1 + x
Recursively simplify expressions:
>>> trim(sin((f(x)**2+f(x))/f(x)))
sin(1 + f(x))
"""
f = sympify(f)
if isinstance(f, Relational):
return Relational(trim(f.lhs, *symbols, **flags),
trim(f.rhs, *symbols, **flags), f.rel_op)
#elif isinstance(f, Matrix):
# return f.applyfunc(lambda g: trim(g, *symbols, **flags))
else:
recursive = flags.get('recursive', True)
def is_functional(g):
return not (g.is_Atom or g.is_number) \
and (not symbols or g.has(*symbols))
def components(g):
result = set()
if is_functional(g):
if g.is_Add or g.is_Mul:
args = []
for h in g.args:
h, terms = components(h)
result |= terms
args.append(h)
g = g.__class__(*args)
elif g.is_Pow:
if recursive:
base = trim(g.base, *symbols, **flags)
else:
base = g.base
if g.exp.is_Rational:
if g.exp.is_Integer:
if g.exp is S.NegativeOne:
h, terms = components(base)
return h**S.NegativeOne, terms
else:
h = base
else:
h = base**Rational(1, g.exp.q)
g = base**g.exp
else:
if recursive:
h = g = base**trim(g.exp, *symbols, **flags)
else:
h = g = base**g.exp
if is_functional(h):
result.add(h)
else:
if not recursive:
result.add(g)
else:
g = g.__class__(*[trim(h, *symbols,
**flags) for h in g.args])
if is_functional(g):
result.add(g)
return g, result
if f.is_number or not f.has_any_symbols(*symbols):
return f
f = together(f.expand())
f, terms = components(f)
if not terms:
return Poly.cancel(f, *symbols)
else:
mapping, reverse = {}, {}
for g in terms:
mapping[g] = Temporary()
reverse[mapping[g]] = g
p, q = f.as_numer_denom()
f = p.expand()/q.expand()
if not symbols:
symbols = tuple(f.atoms(Symbol))
symbols = tuple(mapping.values()) + symbols
H = Poly.cancel(f.subs(mapping), *symbols)
if not flags.get('extract', True):
return H.subs(reverse)
else:
def extract(f):
p = f.args[0]
for q in f.args[1:]:
p = gcd(p, q, *symbols)
if p.is_number:
return S.One, f
return p, Add(*[quo(g, p, *symbols) for g in f.args])
P, Q = H.as_numer_denom()
if P.is_Add:
GP, P = extract(P)
else:
GP = S.One
if Q.is_Add:
GQ, Q = extract(Q)
else:
GQ = S.One
return ((GP*P)/(GQ*Q)).subs(reverse)
def apart(f, z, **flags):
"""Compute partial fraction decomposition of a rational function.
Given a rational function 'f', performing only gcd operations
over the algebraic closue of the initial field of definition,
compute full partial fraction decomposition with fractions
having linear denominators.
For all other kinds of expressions the input is returned in an
unchanged form. Note however, that 'apart' function can thread
over sums and relational operators.
Note that no factorization of the initial denominator of 'f' is
needed. The final decomposition is formed in terms of a sum of
RootSum instances. By default RootSum tries to compute all its
roots to simplify itself. This behaviour can be however avoided
by seting the keyword flag evaluate=False, which will make this
function return a formal decomposition.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> apart(y/(x+2)/(x+1), x)
y/(1 + x) - y/(2 + x)
>>> apart(1/(1+x**5), x, evaluate=False)
RootSum(Lambda(_a, -1/5/(x - _a)*_a), x**5 + 1, x)
For more information on the implemented algorithm refer to:
[1] M. Bronstein, B. Salvy, Full partial fraction decomposition
of rational functions, in: M. Bronstein, ed., Proceedings
ISSAC '93, ACM Press, Kiev, Ukraine, 1993, pp. 157-160.
"""
if not f.has(z):
return f
f = Poly.cancel(f, z)
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
u = Function('u')(z)
a = Symbol('a', dummy=True)
for k, d in enumerate(poly_sqf(q, z)):
n, b = k + 1, d.as_basic()
U += [ u.diff(z, k) ]
h = together(Poly.cancel(f*b**n, z) / u**n)
H, subs = [h], []
for j in range(1, n):
H += [ H[-1].diff(z) / j ]
for j in range(1, n+1):
subs += [ (U[j-1], b.diff(z, j) / j) ]
for j in range(0, n):
P, Q = together(H[j]).as_numer_denom()
for i in range(0, j+1):
P = P.subs(*subs[j-i])
Q = Q.subs(*subs[0])
P, Q = Poly(P, z), Poly(Q, z)
G = poly_gcd(P, d)
D = poly_quo(d, G)
B, g = poly_half_gcdex(Q, D)
b = poly_rem(P * poly_quo(B, g), D)
numer = b.as_basic()
denom = (z-a)**(n-j)
expr = numer.subs(z, a) / denom
partial += RootSum(Lambda(a, expr), D, **flags)
return partial
def risch_norman(f, x, rewrite=False):
"""Computes indefinite integral using extended Risch-Norman algorithm,
also known as parallel Risch. This is a simplified version of full
recursive Risch algorithm. It is designed for integrating various
classes of functions including transcendental elementary or special
functions like Airy, Bessel, Whittaker and Lambert.
The main difference between this algorithm and the recursive one
is that rather than computing a tower of differential extensions
in a recursive way, it handles all cases in one shot. That's why
it is called parallel Risch algorithm. This makes it much faster
than the original approach.
Another benefit is that it doesn't require to rewrite expressions
in terms of complex exponentials. Rather it uses tangents and so
antiderivatives are being found in a more familliar form.
Risch-Norman algorithm can also handle special functions very
easily without any additional effort. Just differentiation
method must be known for a given function.
Note that this algorithm is not a decision procedure. If it
computes an antiderivative for a given integral then it's a
proof that such function exists. However when it fails then
there still may exist an antiderivative and a fallback to
recurrsive Risch algorithm would be necessary.
The question if this algorithm can be made a full featured
decision procedure still remains open.
For more information on the implemented algorithm refer to:
[1] K. Geddes, L.Stefanus, On the Risch-Norman Integration
Method and its Implementation in Maple, Proceedings of
ISSAC'89, ACM Press, 212-217.
[2] J. H. Davenport, On the Parallel Risch Algorithm (I),
Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157.
[3] J. H. Davenport, On the Parallel Risch Algorithm (III):
Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.
[4] J. H. Davenport, B. M. Trager, On the Parallel Risch
Algorithm (II), ACM Transactions on Mathematical
Software 11 (1985), 356-362.
"""
f = Basic.sympify(f)
if not f.has(x):
return f * x
rewritables = {
(sin, cos, cot) : tan,
(sinh, cosh, coth) : tanh,
}
if rewrite:
for candidates, rule in rewritables.iteritems():
f = f.rewrite(candidates, rule)
else:
for candidates in rewritables.iterkeys():
if f.has(*candidates):
break
else:
rewrite = True
terms = components(f)
for g in set(terms):
h = g.diff(x)
if not isinstance(h, Basic.Zero):
terms |= components(h)
terms = [ g for g in terms if g.has(x) ]
V, in_terms, out_terms = [], [], {}
for i, term in enumerate(terms):
V += [ Symbol('x%s' % i) ]
N = term.count_ops(symbolic=False)
in_terms += [ (N, term, V[-1]) ]
out_terms[V[-1]] = term
in_terms.sort(lambda u, v: int(v[0] - u[0]))
def substitute(expr):
for _, g, symbol in in_terms:
expr = expr.subs(g, symbol)
return expr
diffs = [ substitute(g.diff(x)) for g in terms ]
denoms = [ g.as_numer_denom()[1] for g in diffs ]
denom = reduce(lambda p, q: lcm(p, q, V), denoms)
numers = [ normal(denom * g, *V) for g in diffs ]
def derivation(h):
return Basic.Add(*[ d * h.diff(v) for d, v in zip(numers, V) ])
def deflation(p):
for y in p.atoms(Basic.Symbol):
if not isinstance(derivation(p), Basic.Zero):
c, q = p.as_polynomial(y).as_primitive()
return deflation(c) * gcd(q, q.diff(y))
else:
return p
def splitter(p):
for y in p.atoms(Basic.Symbol):
if not isinstance(derivation(y), Basic.Zero):
c, q = p.as_polynomial(y).as_primitive()
q = q.as_basic()
h = gcd(q, derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = splitter(c)
if s.as_polynomial(y).degree() == 0:
return (c_split[0], q * c_split[1])
q_split = splitter(normal(q / s, *V))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
else:
return (S.One, p)
special = []
for term in terms:
if isinstance(term, Basic.Function):
if isinstance(term, Basic.tan):
special += [ (1 + substitute(term)**2, False) ]
elif isinstance(term.func, tanh):
special += [ (1 + substitute(term), False),
(1 - substitute(term), False) ]
#elif isinstance(term.func, Basic.LambertW):
# special += [ (substitute(term), True) ]
ff = substitute(f)
P, Q = ff.as_numer_denom()
u_split = splitter(denom)
v_split = splitter(Q)
s = u_split[0] * Basic.Mul(*[ g for g, a in special if a ])
a, b, c = [ p.as_polynomial(*V).degree() for p in [s, P, Q] ]
candidate_denom = s * v_split[0] * deflation(v_split[1])
monoms = monomials(V, 1 + a + max(b, c))
linear = False
while True:
coeffs, candidate, factors = [], S.Zero, set()
for i, monomial in enumerate(monoms):
coeffs += [ Symbol('A%s' % i, dummy=True) ]
candidate += coeffs[-1] * monomial
candidate /= candidate_denom
polys = [ v_split[0], v_split[1], u_split[0]] + [ s[0] for s in special ]
for irreducibles in [ factorization(p, linear) for p in polys ]:
factors |= irreducibles
for i, irreducible in enumerate(factors):
if not isinstance(irreducible, Basic.Number):
coeffs += [ Symbol('B%s' % i, dummy=True) ]
candidate += coeffs[-1] * Basic.log(irreducible)
h = together(ff - derivation(candidate) / denom)
numerator = h.as_numer_denom()[0].expand()
if not isinstance(numerator, Basic.Add):
numerator = [numerator]
collected = {}
for term in numerator:
coeff, depend = term.as_independent(*V)
if depend in collected:
collected[depend] += coeff
else:
collected[depend] = coeff
solutions = solve(collected.values(), coeffs)
if solutions is None:
if linear:
break
else:
linear = True
else:
break
if solutions is not None:
antideriv = candidate.subs_dict(solutions)
for C in coeffs:
if C not in solutions:
antideriv = antideriv.subs(C, S.Zero)
antideriv = simplify(antideriv.subs_dict(out_terms)).expand()
if isinstance(antideriv, Basic.Add):
return Basic.Add(*antideriv.as_coeff_factors()[1])
else:
return antideriv
else:
if not rewrite:
return risch_norman(f, x, rewrite=True)
else:
return None
def apart(f, z, domain=None, index=None):
"""Computes full partial fraction decomposition of a univariate
rational function over the algebraic closure of its field of
definition. Although only gcd operations over the initial
field are required, the expansion is returned in a formal
form with linear denominators.
However it is possible to force expansion of the resulting
formal summations, and so factorization over a specified
domain is performed.
To specify the desired behavior of the algorithm use the
'domain' keyword. Setting it to None, which is done be
default, will result in no factorization at all.
Otherwise it can be assigned with one of Z, Q, R, C domain
specifiers and the formal partial fraction expansion will
be rewritten using all possible roots over this domain.
If the resulting expansion contains formal summations, then
for all those a single dummy index variable named 'a' will
be generated. To change this default behavior issue new
name via 'index' keyword.
For more information on the implemented algorithm refer to:
[1] M. Bronstein, B. Salvy, Full partial fraction decomposition
of rational functions, in: M. Bronstein, ed., Proceedings
ISSAC '93, ACM Press, Kiev, Ukraine, 1993, pp. 157-160.
"""
f = Basic.sympify(f)
if isinstance(f, Basic.Add):
return Add(*[ apart(g) for g in f ])
else:
if f.is_fraction(z):
f = normal(f, z)
else:
return f
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
u = Function('u')(z)
if index is None:
A = Symbol('a', dummy=True)
else:
A = Symbol(index)
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
for k, d in enumerate(sqf(q, z)):
n, d = k + 1, d.as_basic()
U += [ u.diff(z, k) ]
h = normal(f * d**n, z) / u**n
H, subs = [h], []
for j in range(1, n):
H += [ H[-1].diff(z) / j ]
for j in range(1, n+1):
subs += [ (U[j-1], d.diff(z, j) / j) ]
for j in range(0, n):
P, Q = together(H[j]).as_numer_denom()
for i in range(0, j+1):
P = P.subs(*subs[j-i])
Q = Q.subs(*subs[0])
G = gcd(P, d, z)
D = quo(d, G, z)
g, B, _ = ext_gcd(Q, D, z)
b = rem(P * B / g, D, z)
term = b.subs(z, A) / (z - A)**(n-j)
if domain is None:
a = D.diff(z)
if not a.has(z):
partial += term.subs(A, -D.subs(z, 0) / a)
else:
partial += Basic.Sum(term, (A, Basic.RootOf(D, z)))
else:
raise NotImplementedError
return partial