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calculus.py
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calculus.py
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# Copyright (c) 2017, Michael Boyle
# See LICENSE file for details: <https://github.com/moble/quaternion/blob/master/LICENSE>
from __future__ import division, print_function, absolute_import
import numpy as np
from quaternion.numba_wrapper import njit, jit, xrange
def derivative(f, t):
"""Fourth-order finite-differencing with non-uniform time steps
The formula for this finite difference comes from Eq. (A 5b) of "Derivative formulas and errors for non-uniformly
spaced points" by M. K. Bowen and Ronald Smith. As explained in their Eqs. (B 9b) and (B 10b), this is a
fourth-order formula -- though that's a squishy concept with non-uniform time steps.
TODO: If there are fewer than five points, the function should revert to simpler (lower-order) formulas.
"""
dfdt = np.empty_like(f)
if (f.ndim == 1):
_derivative(f, t, dfdt)
elif (f.ndim == 2):
_derivative_2d(f, t, dfdt)
elif (f.ndim == 3):
_derivative_3d(f, t, dfdt)
else:
raise NotImplementedError("Taking derivatives of {0}-dimensional arrays is not yet implemented".format(f.ndim))
return dfdt
@njit
def _derivative(f, t, dfdt):
for i in xrange(2):
t_i = t[i]
t1 = t[0]
t2 = t[1]
t3 = t[2]
t4 = t[3]
t5 = t[4]
h1 = t1 - t_i
h2 = t2 - t_i
h3 = t3 - t_i
h4 = t4 - t_i
h5 = t5 - t_i
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
dfdt[i] = (-((h2 * h3 * h4 + h2 * h3 * h5 + h2 * h4 * h5 + h3 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[0]
+ ((h1 * h3 * h4 + h1 * h3 * h5 + h1 * h4 * h5 + h3 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[1]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[2]
+ ((h1 * h2 * h3 + h1 * h2 * h5 + h1 * h3 * h5 + h2 * h3 * h5) / (h14 * h24 * h34 * h45)) * f[3]
- ((h1 * h2 * h3 + h1 * h2 * h4 + h1 * h3 * h4 + h2 * h3 * h4) / (h15 * h25 * h35 * h45)) * f[4])
for i in xrange(2, len(t) - 2):
t1 = t[i - 2]
t2 = t[i - 1]
t3 = t[i]
t4 = t[i + 1]
t5 = t[i + 2]
h1 = t1 - t3
h2 = t2 - t3
h4 = t4 - t3
h5 = t5 - t3
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
dfdt[i] = (-((h2 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[i - 2]
+ ((h1 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[i - 1]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[i]
+ ((h1 * h2 * h5) / (h14 * h24 * h34 * h45)) * f[i + 1]
- ((h1 * h2 * h4) / (h15 * h25 * h35 * h45)) * f[i + 2])
for i in xrange(len(t) - 2, len(t)):
t_i = t[i]
t1 = t[-5]
t2 = t[-4]
t3 = t[-3]
t4 = t[-2]
t5 = t[-1]
h1 = t1 - t_i
h2 = t2 - t_i
h3 = t3 - t_i
h4 = t4 - t_i
h5 = t5 - t_i
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
dfdt[i] = (-((h2 * h3 * h4 + h2 * h3 * h5 + h2 * h4 * h5 + h3 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[-5]
+ ((h1 * h3 * h4 + h1 * h3 * h5 + h1 * h4 * h5 + h3 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[-4]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[-3]
+ ((h1 * h2 * h3 + h1 * h2 * h5 + h1 * h3 * h5 + h2 * h3 * h5) / (h14 * h24 * h34 * h45)) * f[-2]
- ((h1 * h2 * h3 + h1 * h2 * h4 + h1 * h3 * h4 + h2 * h3 * h4) / (h15 * h25 * h35 * h45)) * f[-1])
return
@njit
def _derivative_2d(f, t, dfdt):
for i in xrange(2):
t_i = t[i]
t1 = t[0]
t2 = t[1]
t3 = t[2]
t4 = t[3]
t5 = t[4]
h1 = t1 - t_i
h2 = t2 - t_i
h3 = t3 - t_i
h4 = t4 - t_i
h5 = t5 - t_i
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
for k in xrange(f.shape[1]):
dfdt[i, k] = (
-((h2 * h3 * h4 + h2 * h3 * h5 + h2 * h4 * h5 + h3 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[0, k]
+ ((h1 * h3 * h4 + h1 * h3 * h5 + h1 * h4 * h5 + h3 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[1, k]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[2, k]
+ ((h1 * h2 * h3 + h1 * h2 * h5 + h1 * h3 * h5 + h2 * h3 * h5) / (h14 * h24 * h34 * h45)) * f[3, k]
- ((h1 * h2 * h3 + h1 * h2 * h4 + h1 * h3 * h4 + h2 * h3 * h4) / (h15 * h25 * h35 * h45)) * f[4, k])
for i in xrange(2, len(t) - 2):
t1 = t[i - 2]
t2 = t[i - 1]
t3 = t[i]
t4 = t[i + 1]
t5 = t[i + 2]
h1 = t1 - t3
h2 = t2 - t3
h4 = t4 - t3
h5 = t5 - t3
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
for k in xrange(f.shape[1]):
dfdt[i, k] = (-((h2 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[i - 2, k]
+ ((h1 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[i - 1, k]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35))
* f[i, k]
+ ((h1 * h2 * h5) / (h14 * h24 * h34 * h45)) * f[i + 1, k]
- ((h1 * h2 * h4) / (h15 * h25 * h35 * h45)) * f[i + 2, k])
for i in xrange(len(t) - 2, len(t)):
t_i = t[i]
t1 = t[-5]
t2 = t[-4]
t3 = t[-3]
t4 = t[-2]
t5 = t[-1]
h1 = t1 - t_i
h2 = t2 - t_i
h3 = t3 - t_i
h4 = t4 - t_i
h5 = t5 - t_i
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
for k in xrange(f.shape[1]):
dfdt[i, k] = (
-((h2 * h3 * h4 + h2 * h3 * h5 + h2 * h4 * h5 + h3 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[-5, k]
+ ((h1 * h3 * h4 + h1 * h3 * h5 + h1 * h4 * h5 + h3 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[-4, k]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[-3, k]
+ ((h1 * h2 * h3 + h1 * h2 * h5 + h1 * h3 * h5 + h2 * h3 * h5) / (h14 * h24 * h34 * h45)) * f[-2, k]
- ((h1 * h2 * h3 + h1 * h2 * h4 + h1 * h3 * h4 + h2 * h3 * h4) / (h15 * h25 * h35 * h45)) * f[-1, k])
return
@njit
def _derivative_3d(f, t, dfdt):
for i in xrange(2):
t_i = t[i]
t1 = t[0]
t2 = t[1]
t3 = t[2]
t4 = t[3]
t5 = t[4]
h1 = t1 - t_i
h2 = t2 - t_i
h3 = t3 - t_i
h4 = t4 - t_i
h5 = t5 - t_i
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
for k in xrange(f.shape[1]):
for m in xrange(f.shape[1]):
dfdt[i, k, m] = (
-((h2 * h3 * h4 + h2 * h3 * h5 + h2 * h4 * h5 + h3 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[0, k, m]
+ ((h1 * h3 * h4 + h1 * h3 * h5 + h1 * h4 * h5 + h3 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[1, k, m]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[2, k, m]
+ ((h1 * h2 * h3 + h1 * h2 * h5 + h1 * h3 * h5 + h2 * h3 * h5) / (h14 * h24 * h34 * h45)) * f[3, k, m]
- ((h1 * h2 * h3 + h1 * h2 * h4 + h1 * h3 * h4 + h2 * h3 * h4) / (h15 * h25 * h35 * h45)) * f[4, k, m])
for i in xrange(2, len(t) - 2):
t1 = t[i - 2]
t2 = t[i - 1]
t3 = t[i]
t4 = t[i + 1]
t5 = t[i + 2]
h1 = t1 - t3
h2 = t2 - t3
h4 = t4 - t3
h5 = t5 - t3
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
for k in xrange(f.shape[1]):
for m in xrange(f.shape[1]):
dfdt[i, k, m] = (-((h2 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[i - 2, k, m]
+ ((h1 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[i - 1, k, m]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35))
* f[i, k, m]
+ ((h1 * h2 * h5) / (h14 * h24 * h34 * h45)) * f[i + 1, k, m]
- ((h1 * h2 * h4) / (h15 * h25 * h35 * h45)) * f[i + 2, k, m])
for i in xrange(len(t) - 2, len(t)):
t_i = t[i]
t1 = t[-5]
t2 = t[-4]
t3 = t[-3]
t4 = t[-2]
t5 = t[-1]
h1 = t1 - t_i
h2 = t2 - t_i
h3 = t3 - t_i
h4 = t4 - t_i
h5 = t5 - t_i
h12 = t1 - t2
h13 = t1 - t3
h14 = t1 - t4
h15 = t1 - t5
h23 = t2 - t3
h24 = t2 - t4
h25 = t2 - t5
h34 = t3 - t4
h35 = t3 - t5
h45 = t4 - t5
for k in xrange(f.shape[1]):
for m in xrange(f.shape[1]):
dfdt[i, k, m] = (
-((h2 * h3 * h4 + h2 * h3 * h5 + h2 * h4 * h5 + h3 * h4 * h5) / (h12 * h13 * h14 * h15)) * f[-5, k, m]
+ ((h1 * h3 * h4 + h1 * h3 * h5 + h1 * h4 * h5 + h3 * h4 * h5) / (h12 * h23 * h24 * h25)) * f[-4, k, m]
- ((h1 * h2 * h4 + h1 * h2 * h5 + h1 * h4 * h5 + h2 * h4 * h5) / (h13 * h23 * h34 * h35)) * f[-3, k, m]
+ ((h1 * h2 * h3 + h1 * h2 * h5 + h1 * h3 * h5 + h2 * h3 * h5) / (h14 * h24 * h34 * h45)) * f[-2, k, m]
- ((h1 * h2 * h3 + h1 * h2 * h4 + h1 * h3 * h4 + h2 * h3 * h4) / (h15 * h25 * h35 * h45)) * f[-1, k, m])
return
# @njit('void(f8[:,:], f8[:], f8[:,:])')
@jit
def indefinite_integral(f, t):
Sfdt = np.empty_like(f)
Sfdt[0] = 0.0
for i in xrange(1, len(t)):
for j in xrange(f.shape[1]):
Sfdt[i, j] = Sfdt[i - 1, j] + (f[i, j] + f[i - 1, j]) * ((t[i] - t[i - 1]) / 2.0)
return Sfdt
#@njit('void(f8[:,:], f8[:], f8[:])')
@jit
def definite_integral(f, t):
Sfdt = np.zeros_like(f)
for i in xrange(1, f.shape[0]):
Sfdt[i, ...] += (f[i, ...] + f[i - 1, ...]) * ((t[i] - t[i - 1]) / 2.0)
return Sfdt