def _create_Tderiv(self, k, n):
r"""Create the n'th derivative of the k'th Chebyshev polynomial."""
if not self._use_mp:
return ChebyshevT.basis(k).deriv(n)
else:
if n == 0:
return lambda x: mp.chebyt(k, x)
# Since the Chebyshev derivatives attain high values near the
# borders +-1 and usually are subtracted to obtain small values,
# minimizing their relative error (i.e. fixing the number of
# correct decimal places (dps)) is not enough to get precise
# results. Hence, the below `moredps` variable is set to higher
# values close to the border.
extradps = 5
pos = mp.fprod(
[mp.mpf(k**2 - p**2) / (2 * p + 1) for p in range(n)])
neg = (-1)**(k + n) * pos
if n == 1:
def dTk(x):
with mp.extradps(extradps):
x = clip(x, -1.0, 1.0)
if mp.almosteq(x, mp.one): return pos
if mp.almosteq(x, -mp.one): return neg
moredps = max(
0,
int(-math.log10(
min(abs(x - mp.one), abs(x + mp.one))) / 2))
moredps = min(moredps, 100)
with mp.extradps(moredps):
t = mp.acos(x)
return k * mp.sin(k * t) / mp.sin(t)
return dTk
if n == 2:
def ddTk(x):
with mp.extradps(extradps):
x = clip(x, -1.0, 1.0)
if mp.almosteq(x, mp.one): return pos
if mp.almosteq(x, -mp.one): return neg
moredps = max(
0,
int(-math.log10(
min(abs(x - mp.one), abs(x + mp.one))) * 1.5) +
2)
moredps = min(moredps, 100)
with mp.extradps(moredps):
t = mp.acos(x)
s = mp.sin(t)
return -k**2 * mp.cos(k * t) / s**2 + k * mp.cos(
t) * mp.sin(k * t) / s**3
return ddTk
raise NotImplementedError(
'Derivatives of order > 2 not implemented.')
def Prod(A):
assert (len(A) > 0)
return mp.fprod(A)
def Prod(A):
assert(len(A) > 0)
return mp.fprod(A)