def test_Function():
assert mcode(f(x, y, z)) == "f[x, y, z]"
assert mcode(sin(x)**cos(x)) == "Sin[x]^Cos[x]"
assert mcode(conjugate(x)) == "Conjugate[x]"
assert mcode(Max(x, y, z) * Min(y, z)) == "Max[x, y, z]*Min[y, z]"
assert mcode(fresnelc(x)) == "FresnelC[x]"
assert mcode(fresnels(x)) == "FresnelS[x]"
assert mcode(gamma(x)) == "Gamma[x]"
assert mcode(uppergamma(x, y)) == "Gamma[x, y]"
assert mcode(polygamma(x, y)) == "PolyGamma[x, y]"
assert mcode(loggamma(x)) == "LogGamma[x]"
assert mcode(erf(x)) == "Erf[x]"
assert mcode(erfc(x)) == "Erfc[x]"
assert mcode(erfi(x)) == "Erfi[x]"
assert mcode(erf2(x, y)) == "Erf[x, y]"
assert mcode(expint(x, y)) == "ExpIntegralE[x, y]"
assert mcode(erfcinv(x)) == "InverseErfc[x]"
assert mcode(erfinv(x)) == "InverseErf[x]"
assert mcode(erf2inv(x, y)) == "InverseErf[x, y]"
assert mcode(Ei(x)) == "ExpIntegralEi[x]"
assert mcode(Ci(x)) == "CosIntegral[x]"
assert mcode(li(x)) == "LogIntegral[x]"
assert mcode(Si(x)) == "SinIntegral[x]"
assert mcode(Shi(x)) == "SinhIntegral[x]"
assert mcode(Chi(x)) == "CoshIntegral[x]"
assert mcode(beta(x, y)) == "Beta[x, y]"
assert mcode(factorial(x)) == "Factorial[x]"
assert mcode(factorial2(x)) == "Factorial2[x]"
assert mcode(subfactorial(x)) == "Subfactorial[x]"
assert mcode(FallingFactorial(x, y)) == "FactorialPower[x, y]"
assert mcode(RisingFactorial(x, y)) == "Pochhammer[x, y]"
assert mcode(catalan(x)) == "CatalanNumber[x]"
assert mcode(harmonic(x)) == "HarmonicNumber[x]"
assert mcode(harmonic(x, y)) == "HarmonicNumber[x, y]"
def rsolve_poly(coeffs, f, n, **hints):
"""Given linear recurrence operator L of order 'k' with polynomial
coefficients and inhomogeneous equation Ly = 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 Ly = 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**2 - 2*n**3 + n**4
For more information on implemented algorithms refer to:
[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 xrange(0, r+1):
for j in xrange(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 xrange(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 = Symbol('x', dummy=True)
degree_poly = S.Zero
for i in xrange(0, r+1):
if terms[i][1] - i == b:
degree_poly += terms[i][0]*FallingFactorial(x, i)
nni_roots = 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 xrange(0, N+1):
C.append(Symbol('C'+str(i)))
y += C[i] * n**i
for i in xrange(0, r+1):
E += coeffs[i].as_basic()*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 = 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 xrange(1, k+1):
B *= -Rational(k-i+1, i)
D += B * p.subs(n, a+k-i)
return D
alpha = {}
for i in xrange(-A, d+1):
I = one_vector(d+1)
for k in xrange(1, d+1):
I[k] = I[k-1] * (x+i-k+1)/k
alpha[i] = S.Zero
for j in xrange(0, A+1):
for k in xrange(0, d+1):
B = Binomial(k, i+j)
D = delta(polys[j].as_basic(), k)
alpha[i] += I[k]*B*D
V = Matrix(U, A, lambda i, j: int(i == j))
if homogeneous:
for i in xrange(A, U):
v = zero_vector(A)
for k in xrange(1, A+b+1):
if i - k < 0:
break
B = alpha[k-A].subs(x, i-k)
for j in xrange(0, A):
v[j] += B * V[i-k, j]
denom = alpha[-A].subs(x, i)
for j in xrange(0, A):
V[i, j] = -v[j] / denom
else:
G = zero_vector(U)
for i in xrange(A, U):
v = zero_vector(A)
g = S.Zero
for k in xrange(1, A+b+1):
if i - k < 0:
break
B = alpha[k-A].subs(x, i-k)
for j in xrange(0, A):
v[j] += B * V[i-k, j]
g += B * G[i-k]
denom = alpha[-A].subs(x, i)
for j in xrange(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 xrange(1, U):
P[i] = (P[i-1] * (n-a-i+1)/i).expand()
for i in xrange(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 xrange(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 xrange(N+1, U) ]
else:
E = [ g(i) + delta(h, i) for i in xrange(N+1, U) ]
if E != []:
solutions = solve(E, *C)
if solutions is None:
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 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