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differentiation.py
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differentiation.py
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"""Numerical differentiation."""
import numpy as np
def backward_difference(x, y):
"""Calculate the first derivative.
All values in 'x' must be equally spaced.
Args:
x (numpy.ndarray): x values.
y (numpy.ndarray): y values.
Returns:
dy (numpy.ndarray): the first derivative values.
"""
if x.size < 2 or y.size < 2:
raise ValueError("'x' and 'y' arrays must have 2 values or more.")
if x.size != y.size:
raise ValueError("'x' and 'y' must have same size.")
def dy_difference(h, y0, y1):
return (y1 - y0) / h
n = x.size
dy = np.zeros(n)
for i in range(0, n):
if i == n - 1:
hx = x[i] - x[i - 1]
dy[i] = dy_difference(-hx, y[i], y[i - 1])
else:
hx = x[i + 1] - x[i]
dy[i] = dy_difference(hx, y[i], y[i + 1])
return dy
def three_point(x, y):
"""Calculate the first derivative.
All values in 'x' must be equally spaced.
Args:
x (numpy.ndarray): x values.
y (numpy.ndarray): y values.
Returns:
dy (numpy.ndarray): the first derivative values.
"""
if x.size < 3 or y.size < 3:
raise ValueError("'x' and 'y' arrays must have 3 values or more.")
if x.size != y.size:
raise ValueError("'x' and 'y' must have same size.")
def dy_mid(h, y0, y2):
return (1 / (2 * h)) * (y2 - y0)
def dy_end(h, y0, y1, y2):
return (1 / (2 * h)) * (-3 * y0 + 4 * y1 - y2)
hx = x[1] - x[0]
n = x.size
dy = np.zeros(n)
for i in range(0, n):
if i == 0:
dy[i] = dy_end(hx, y[i], y[i + 1], y[i + 2])
elif i == n - 1:
dy[i] = dy_end(-hx, y[i], y[i - 1], y[i - 2])
else:
dy[i] = dy_mid(hx, y[i - 1], y[i + 1])
return dy
def five_point(x, y):
"""Calculate the first derivative.
All values in 'x' must be equally spaced.
Args:
x (numpy.ndarray): x values.
y (numpy.ndarray): y values.
Returns:
dy (numpy.ndarray): the first derivative values.
"""
if x.size < 6 or y.size < 6:
raise ValueError("'x' and 'y' arrays must have 6 values or more.")
if x.size != y.size:
raise ValueError("'x' and 'y' must have same size.")
def dy_mid(h, y0, y1, y3, y4):
return (1 / (12 * h)) * (y0 - 8 * y1 + 8 * y3 - y4)
def dy_end(h, y0, y1, y2, y3, y4):
return (1 / (12 * h)) * \
(-25 * y0 + 48 * y1 - 36 * y2 + 16 * y3 - 3 * y4)
hx = x[1] - x[0]
n = x.size
dy = np.zeros(n)
for i in range(0, n):
if i in (0, 1):
dy[i] = dy_end(hx, y[i], y[i + 1], y[i + 2], y[i + 3], y[i + 4])
elif i in (n - 1, n - 2):
dy[i] = dy_end(-hx, y[i], y[i - 1], y[i - 2], y[i - 3], y[i - 4])
else:
dy[i] = dy_mid(hx, y[i - 2], y[i - 1], y[i + 1], y[i + 2])
return dy