def power(self, o, fp, gp):
f, g = o.ufl_operands
if not is_true_ufl_scalar(f):
error("Expecting scalar expression f in f**g.")
if not is_true_ufl_scalar(g):
error("Expecting scalar expression g in f**g.")
# Derivation of the general case: o = f(x)**g(x)
# do/df = g * f**(g-1) = g / f * o
# do/dg = ln(f) * f**g = ln(f) * o
# do/df * df + do/dg * dg = o * (g / f * df + ln(f) * dg)
if isinstance(gp, Zero):
# This probably produces better results for the common
# case of f**constant
op = fp * g * f**(g - 1)
else:
# Note: This produces expressions like (1/w)*w**5 instead of w**4
# op = o * (fp * g / f + gp * ln(f)) # This reuses o
op = f**(g - 1) * (
g * fp + f * ln(f) * gp
) # This gives better accuracy in dolfin integration test
# Example: d/dx[x**(x**3)]:
# f = x
# g = x**3
# df = 1
# dg = 3*x**2
# op1 = o * (fp * g / f + gp * ln(f))
# = x**(x**3) * (x**3/x + 3*x**2*ln(x))
# op2 = f**(g-1) * (g*fp + f*ln(f)*gp)
# = x**(x**3-1) * (x**3 + x*3*x**2*ln(x))
return op
def power(self, o, fp, gp):
f, g = o.ufl_operands
if not is_true_ufl_scalar(f):
error("Expecting scalar expression f in f**g.")
if not is_true_ufl_scalar(g):
error("Expecting scalar expression g in f**g.")
# Derivation of the general case: o = f(x)**g(x)
# do/df = g * f**(g-1) = g / f * o
# do/dg = ln(f) * f**g = ln(f) * o
# do/df * df + do/dg * dg = o * (g / f * df + ln(f) * dg)
if isinstance(gp, Zero):
# This probably produces better results for the common
# case of f**constant
op = fp * g * f**(g-1)
else:
# Note: This produces expressions like (1/w)*w**5 instead of w**4
# op = o * (fp * g / f + gp * ln(f)) # This reuses o
op = f**(g-1) * (g*fp + f*ln(f)*gp) # This gives better accuracy in dolfin integration test
# Example: d/dx[x**(x**3)]:
# f = x
# g = x**3
# df = 1
# dg = 3*x**2
# op1 = o * (fp * g / f + gp * ln(f))
# = x**(x**3) * (x**3/x + 3*x**2*ln(x))
# op2 = f**(g-1) * (g*fp + f*ln(f)*gp)
# = x**(x**3-1) * (x**3 + x*3*x**2*ln(x))
return op
def power(self, f):
"""f = x**y
d/dx x**y = y*x**(y-1) = y*f/x
d/dy x**y = ln(x)*x**y = ln(x)*f"""
x, y = f.operands()
dx = y*f/x
dy = ln(x)*f
return (dx, dy)
def power(self, o, a, b):
f, fp = a
g, gp = b
# Debugging prints, should never happen:
if not is_true_ufl_scalar(f):
print ":" * 80
print "f =", str(f)
print "g =", str(g)
print ":" * 80
ufl_assert(is_true_ufl_scalar(f),
"Expecting scalar expression f in f**g.")
ufl_assert(is_true_ufl_scalar(g),
"Expecting scalar expression g in f**g.")
# Derivation of the general case: o = f(x)**g(x)
#
#do_df = g * f**(g-1)
#do_dg = ln(f) * f**g
#op = do_df*fp + do_dg*gp
#
#do_df = o * g / f # f**g * g / f
#do_dg = ln(f) * o
#op = do_df*fp + do_dg*gp
# Got two possible alternatives here:
if True: # This version looks better.
# Rewriting o as f*f**(g-1) we can do:
f_g_m1 = f**(g - 1)
op = f_g_m1 * (fp * g + f * ln(f) * gp)
# In this case we can rewrite o using new subexpression
o = f * f_g_m1
else:
# Pulling o out gives:
op = o * (fp * g / f + ln(f) * gp)
# This produces expressions like (1/w)*w**5 instead of w**4
# If we do this, we reuse o
o = self.reuse_if_possible(o, f, g)
return (o, op)
def power(self, o, a, b):
f, fp = a
g, gp = b
# Debugging prints, should never happen:
if not is_true_ufl_scalar(f):
print ":"*80
print "f =", str(f)
print "g =", str(g)
print ":"*80
ufl_assert(is_true_ufl_scalar(f), "Expecting scalar expression f in f**g.")
ufl_assert(is_true_ufl_scalar(g), "Expecting scalar expression g in f**g.")
# Derivation of the general case: o = f(x)**g(x)
#
#do_df = g * f**(g-1)
#do_dg = ln(f) * f**g
#op = do_df*fp + do_dg*gp
#
#do_df = o * g / f # f**g * g / f
#do_dg = ln(f) * o
#op = do_df*fp + do_dg*gp
# Got two possible alternatives here:
if True: # This version looks better.
# Rewriting o as f*f**(g-1) we can do:
f_g_m1 = f**(g-1)
op = f_g_m1*(fp*g + f*ln(f)*gp)
# In this case we can rewrite o using new subexpression
o = f*f_g_m1
else:
# Pulling o out gives:
op = o*(fp*g/f + ln(f)*gp)
# This produces expressions like (1/w)*w**5 instead of w**4
# If we do this, we reuse o
o = self.reuse_if_possible(o, f, g)
return (o, op)