def matsimp(expr):
"""do indexing, Trace, Determinant and expand matrix equation
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> A = ImmutableMatrix(2, 2, symbols('A(:2)(:2)'))
>>> B = ImmutableMatrix(2, 2, symbols('B(:2)(:2)'))
>>> C = ImmutableMatrix(2, 2, symbols('C(:2)(:2)'))
>>> M = MatrixSymbol('M', 2, 2)
>>> M[1,1].xreplace({M: B*C})
<BLANKLINE>
[B00*C00 + B01*C10 B00*C01 + B01*C11]
[B10*C00 + B11*C10 B10*C01 + B11*C11][1, 1]
>>> matsimp(_)
B10*C01 + B11*C11
>>> Eq(Trace(B+C), 0)
0 == Trace(
[B00 + C00 B01 + C01]
[B10 + C10 B11 + C11])
>>> matsimp(_)
0 == B00 + B11 + C00 + C11
>>> Eq(A.T-A, ZeroMatrix(2,2))
<BLANKLINE>
[ 0 -A01 + A10]
[A01 - A10 0] == 0
>>> matsimp(_)
(-A01 + A10 == 0) /\ (0 == A01 - A10)
"""
from sympy.simplify.simplify import bottom_up
# do indexing: [.., aij ,..][i,j] -> aij
expr = do_indexing(expr)
# deep doit: Trace([.., aij ,..]) -> ..+ aii +..
expr = bottom_up(expr, lambda e: e.doit())
def mateq_expand(m1, m2):
if not is_Matrix(m1) and not is_Matrix(m2):
return Eq(m1, m2)
if not is_Matrix(m1) or not is_Matrix(m2):
return false
if m1.shape != m2.shape:
return false
return And(*[Eq(e1, e2) for e1, e2 in zip(m1, m2)])
def matne_expand(m1, m2):
if not is_Matrix(m1) and not is_Matrix(m2):
return Ne(m1, m2)
if not is_Matrix(m1) or not is_Matrix(m2):
return true
if m1.shape != m2.shape:
return true
return Or(*[Ne(e1, e2) for e1, e2 in zip(m1, m2)])
# expand matrix equation: [.., aij ,..] == [.., bij ,..] -> ..& aij == bij &..
# [.., aij ,..] != [.., bij ,..] -> ..| aij != bij |..
expr = expr.replace(Eq, mateq_expand)
expr = expr.replace(Ne, matne_expand)
return expr
def futrig(e, **kwargs):
"""Return simplified ``e`` using Fu-like transformations.
This is not the "Fu" algorithm. This is called by default
from ``trigsimp``. By default, hyperbolics subexpressions
will be simplified, but this can be disabled by setting
``hyper=False``.
Examples
========
>>> from sympy import trigsimp, tan, sinh, tanh
>>> from sympy.simplify.trigsimp import futrig
>>> from sympy.abc import x
>>> trigsimp(1/tan(x)**2)
tan(x)**(-2)
>>> futrig(sinh(x)/tanh(x))
cosh(x)
"""
from sympy.simplify.fu import hyper_as_trig
from sympy.simplify.simplify import bottom_up
e = sympify(e)
if not isinstance(e, Basic):
return e
if not e.args:
return e
old = e
e = bottom_up(e, lambda x: _futrig(x, **kwargs))
if kwargs.pop('hyper', True) and e.has(HyperbolicFunction):
e, f = hyper_as_trig(e)
e = f(_futrig(e))
if e != old and e.is_Mul and e.args[0].is_Rational:
# redistribute leading coeff on 2-arg Add
e = Mul(*e.as_coeff_Mul())
return e
def exptrigsimp(expr):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
def f(rv):
if not rv.is_Mul:
return rv
commutative_part, noncommutative_part = rv.args_cnc()
# Since as_powers_dict loses order information,
# if there is more than one noncommutative factor,
# it should only be used to simplify the commutative part.
if (len(noncommutative_part) > 1):
return f(Mul(*commutative_part))*Mul(*noncommutative_part)
rvd = rv.as_powers_dict()
newd = rvd.copy()
def signlog(expr, sign=1):
if expr is S.Exp1:
return sign, 1
elif isinstance(expr, exp):
return sign, expr.args[0]
elif sign == 1:
return signlog(-expr, sign=-1)
else:
return None, None
ee = rvd[S.Exp1]
for k in rvd:
if k.is_Add and len(k.args) == 2:
# k == c*(1 + sign*E**x)
c = k.args[0]
sign, x = signlog(k.args[1]/c)
if not x:
continue
m = rvd[k]
newd[k] -= m
if ee == -x*m/2:
# sinh and cosh
newd[S.Exp1] -= ee
ee = 0
if sign == 1:
newd[2*c*cosh(x/2)] += m
else:
newd[-2*c*sinh(x/2)] += m
elif newd[1 - sign*S.Exp1**x] == -m:
# tanh
del newd[1 - sign*S.Exp1**x]
if sign == 1:
newd[-c/tanh(x/2)] += m
else:
newd[-c*tanh(x/2)] += m
else:
newd[1 + sign*S.Exp1**x] += m
newd[c] += m
return Mul(*[k**newd[k] for k in newd])
newexpr = bottom_up(newexpr, f)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
def exptrigsimp(expr, simplify=True):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
When ``simplify`` is True (default) the expression obtained after the
simplification step will be then be passed through simplify to
precondition it so the final transformations will be applied.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
if simplify:
newexpr = newexpr.simplify()
# conversion from exp to hyperbolic
ex = newexpr.atoms(exp, S.Exp1)
ex = [ei for ei in ex if 1/ei not in ex]
## sinh and cosh
for ei in ex:
e2 = ei**-2
if e2 in ex:
a = e2.args[0]/2 if not e2 is S.Exp1 else S.Half
newexpr = newexpr.subs((e2 + 1)*ei, 2*cosh(a))
newexpr = newexpr.subs((e2 - 1)*ei, 2*sinh(a))
## exp ratios to tan and tanh
for ei in ex:
n, d = ei - 1, ei + 1
et = n/d
etinv = d/n # not 1/et or else recursion errors arise
a = ei.args[0] if ei.func is exp else S.One
if a.is_Mul or a is S.ImaginaryUnit:
c = a.as_coefficient(I)
if c:
t = S.ImaginaryUnit*tan(c/2)
newexpr = newexpr.subs(etinv, 1/t)
newexpr = newexpr.subs(et, t)
continue
t = tanh(a/2)
newexpr = newexpr.subs(etinv, 1/t)
newexpr = newexpr.subs(et, t)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
def exptrigsimp(expr, simplify=True):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
When ``simplify`` is True (default) the expression obtained after the
simplification step will be then be passed through simplify to
precondition it so the final transformations will be applied.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
if simplify:
newexpr = newexpr.simplify()
# conversion from exp to hyperbolic
ex = newexpr.atoms(exp, S.Exp1)
ex = [ei for ei in ex if 1 / ei not in ex]
## sinh and cosh
for ei in ex:
e2 = ei**-2
if e2 in ex:
a = e2.args[0] / 2 if not e2 is S.Exp1 else S.Half
newexpr = newexpr.subs((e2 + 1) * ei, 2 * cosh(a))
newexpr = newexpr.subs((e2 - 1) * ei, 2 * sinh(a))
## exp ratios to tan and tanh
for ei in ex:
n, d = ei - 1, ei + 1
et = n / d
etinv = d / n # not 1/et or else recursion errors arise
a = ei.args[0] if ei.func is exp else S.One
if a.is_Mul or a is S.ImaginaryUnit:
c = a.as_coefficient(I)
if c:
t = S.ImaginaryUnit * tan(c / 2)
newexpr = newexpr.subs(etinv, 1 / t)
newexpr = newexpr.subs(et, t)
continue
t = tanh(a / 2)
newexpr = newexpr.subs(etinv, 1 / t)
newexpr = newexpr.subs(et, t)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
def exptrigsimp(expr):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
def f(rv):
if not rv.is_Mul:
return rv
commutative_part, noncommutative_part = rv.args_cnc()
# Since as_powers_dict loses order information,
# if there is more than one noncommutative factor,
# it should only be used to simplify the commutative part.
if (len(noncommutative_part) > 1):
return f(Mul(*commutative_part)) * Mul(*noncommutative_part)
rvd = rv.as_powers_dict()
newd = rvd.copy()
def signlog(expr, sign=S.One):
if expr is S.Exp1:
return sign, S.One
elif isinstance(expr, exp) or (expr.is_Pow
and expr.base == S.Exp1):
return sign, expr.exp
elif sign is S.One:
return signlog(-expr, sign=-S.One)
else:
return None, None
ee = rvd[S.Exp1]
for k in rvd:
if k.is_Add and len(k.args) == 2:
# k == c*(1 + sign*E**x)
c = k.args[0]
sign, x = signlog(k.args[1] / c)
if not x:
continue
m = rvd[k]
newd[k] -= m
if ee == -x * m / 2:
# sinh and cosh
newd[S.Exp1] -= ee
ee = 0
if sign == 1:
newd[2 * c * cosh(x / 2)] += m
else:
newd[-2 * c * sinh(x / 2)] += m
elif newd[1 - sign * S.Exp1**x] == -m:
# tanh
del newd[1 - sign * S.Exp1**x]
if sign == 1:
newd[-c / tanh(x / 2)] += m
else:
newd[-c * tanh(x / 2)] += m
else:
newd[1 + sign * S.Exp1**x] += m
newd[c] += m
return Mul(*[k**newd[k] for k in newd])
newexpr = bottom_up(newexpr, f)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
