def subsets(seq, k=None, repetition=False):
"""Generates all k-subsets (combinations) from an n-element set, seq.
A k-subset of an n-element set is any subset of length exactly k. The
number of k-subsets of an n-element set is given by binomial(n, k),
whereas there are 2**n subsets all together. If k is None then all
2**n subsets will be returned from shortest to longest.
Examples
========
>>> from sympy.utilities.iterables import subsets
subsets(seq, k) will return the n!/k!/(n - k)! k-subsets (combinations)
without repetition, i.e. once an item has been removed, it can no
longer be "taken":
>>> list(subsets([1, 2], 2))
[(1, 2)]
>>> list(subsets([1, 2]))
[(), (1,), (2,), (1, 2)]
>>> list(subsets([1, 2, 3], 2))
[(1, 2), (1, 3), (2, 3)]
subsets(seq, k, repetition=True) will return the (n - 1 + k)!/k!/(n - 1)!
combinations *with* repetition:
>>> list(subsets([1, 2], 2, repetition=True))
[(1, 1), (1, 2), (2, 2)]
If you ask for more items than are in the set you get the empty set unless
you allow repetitions:
>>> list(subsets([0, 1], 3, repetition=False))
[]
>>> list(subsets([0, 1], 3, repetition=True))
[(0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)]
"""
if k is None:
for k in range(len(seq) + 1):
for i in subsets(seq, k, repetition):
yield i
else:
if not repetition:
for i in combinations(seq, k):
yield i
else:
for i in combinations_with_replacement(seq, k):
yield i
def staircase(n):
"""
Compute all monomials with degree less than ``n`` that are
not divisible by any element of ``leading_monomials``.
"""
if n == 0:
return [1]
S = []
for mi in combinations_with_replacement(range(len(opt.gens)), n):
m = [0]*len(opt.gens)
for i in mi:
m[i] += 1
if all([monomial_div(m, lmg) is None for lmg in
leading_monomials]):
S.append(m)
return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)
def classify_pde(eq, func=None, dict=False, **kwargs):
"""
Returns a tuple of possible pdsolve() classifications for a PDE.
The tuple is ordered so that first item is the classification that
pdsolve() uses to solve the PDE by default. In general,
classifications at the near the beginning of the list will produce
better solutions faster than those near the end, thought there are
always exceptions. To make pdsolve use a different classification,
use pdsolve(PDE, func, hint=<classification>). See also the pdsolve()
docstring for different meta-hints you can use.
If ``dict`` is true, classify_pde() will return a dictionary of
hint:match expression terms. This is intended for internal use by
pdsolve(). Note that because dictionaries are ordered arbitrarily,
this will most likely not be in the same order as the tuple.
You can get help on different hints by doing help(pde.pde_hintname),
where hintname is the name of the hint without "_Integral".
See sympy.pde.allhints or the sympy.pde docstring for a list of all
supported hints that can be returned from classify_pde.
Examples
========
>>> from sympy.solvers.pde import classify_pde
>>> from sympy import Function, diff, Eq
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> u = f(x, y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)))
>>> classify_pde(eq)
('1st_linear_constant_coeff_homogeneous',)
"""
prep = kwargs.pop('prep', True)
if func and len(func.args) != 2:
raise NotImplementedError("Right now only partial "
"differential equations of two variables are supported")
if prep or func is None:
prep, func_ = _preprocess(eq, func)
if func is None:
func = func_
if isinstance(eq, Equality):
if eq.rhs != 0:
return classify_pde(eq.lhs - eq.rhs, func)
eq = eq.lhs
f = func.func
x = func.args[0]
y = func.args[1]
fx = f(x,y).diff(x)
fy = f(x,y).diff(y)
# TODO : For now pde.py uses support offered by the ode_order function
# to find the order with respect to a multi-variable function. An
# improvement could be to classify the order of the PDE on the basis of
# individual variables.
order = ode_order(eq, f(x,y))
# hint:matchdict or hint:(tuple of matchdicts)
# Also will contain "default":<default hint> and "order":order items.
matching_hints = {'order': order}
if not order:
if dict:
matching_hints["default"] = None
return matching_hints
else:
return ()
eq = expand(eq)
a = Wild('a', exclude = [f(x,y)])
b = Wild('b', exclude = [f(x,y), fx, fy, x, y])
c = Wild('c', exclude = [f(x,y), fx, fy, x, y])
d = Wild('d', exclude = [f(x,y), fx, fy, x, y])
e = Wild('e', exclude = [f(x,y), fx, fy])
n = Wild('n', exclude = [x, y])
# Try removing the smallest power of f(x,y)
# from the highest partial derivatives of f(x,y)
reduced_eq = None
if eq.is_Add:
var = set(combinations_with_replacement((x,y), order))
dummyvar = deepcopy(var)
power = None
for i in var:
coeff = eq.coeff(f(x,y).diff(*i))
if coeff != 1:
match = coeff.match(a*f(x,y)**n)
if match and match[a]:
power = match[n]
dummyvar.remove(i)
break
dummyvar.remove(i)
for i in dummyvar:
coeff = eq.coeff(f(x,y).diff(*i))
if coeff != 1:
match = coeff.match(a*f(x,y)**n)
if match and match[a] and match[n] < power:
power = match[n]
if power:
den = f(x,y)**power
reduced_eq = Add(*[arg/den for arg in eq.args])
if not reduced_eq:
reduced_eq = eq
if order == 1:
reduced_eq = collect(reduced_eq, f(x, y))
r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e)
if r:
if not r[e]:
## Linear first-order homogeneous partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d})
matching_hints["1st_linear_constant_coeff_homogeneous"] = r
else:
if r[b]**2 + r[c]**2 != 0:
## Linear first-order general partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_constant_coeff"] = r
matching_hints[
"1st_linear_constant_coeff_Integral"] = r
else:
b = Wild('b', exclude=[f(x, y), fx, fy])
c = Wild('c', exclude=[f(x, y), fx, fy])
d = Wild('d', exclude=[f(x, y), fx, fy])
r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e)
if r:
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_variable_coeff"] = r
# Order keys based on allhints.
retlist = []
for i in allhints:
if i in matching_hints:
retlist.append(i)
if dict:
# Dictionaries are ordered arbitrarily, so make note of which
# hint would come first for pdsolve(). Use an ordered dict in Py 3.
matching_hints["default"] = None
matching_hints["ordered_hints"] = tuple(retlist)
for i in allhints:
if i in matching_hints:
matching_hints["default"] = i
break
return matching_hints
else:
return tuple(retlist)
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 autodetected (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)
[-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])
[-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 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 autodetected (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)
[-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])
[-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
