Compute numerical derivatives.
Project description
Fast numerical derivatives for real analytic functions with arbitrary round-off error.
Features
Robustly compute the generalised Jacobi matrix for an arbitrary real analytic mapping ℝⁿ → ℝⁱ¹ × … × ℝⁱⁿ
Derivative is either computed to specified accuracy (to save computing time) or until maximum precision of function is reached
Algorithm based on John D’Errico’s DERIVEST: works even with functions that have large round-off error
Up to 1200x faster than numdifftools at equivalent precision
Returns error estimates for derivatives
Supports arbitrary auxiliary function arguments
Lightweight package, only depends on numpy
Planned features
Compute the Hessian matrix numerically with the same algorithm
Further generalize the calculation to support function arguments with shape (N, K), in that case compute the Jacobi matrix for each of the K vectors of length N
Example
from matplotlib import pyplot as plt
import numpy as np
from jacobi import jacobi
# function of one variable with auxiliary argument; returns a vector
def f(dx, x):
y = x + dx
return np.sin(y) / y
x = np.linspace(-10, 10, 1000)
fx = f(0, x)
fdx, fdex = jacobi(f, 0, x)
plt.plot(x, fx, label="f(x) = sin(x) / x")
plt.plot(x, fdx, ls="--", label="f'(x)")
plt.legend()
Comparison to numdifftools
Speed
Jacobi makes better use of vectorised computation than numdifftools. Smaller run-time is better (and ratio > 1).
Precision
The machine precision is indicated by the dashed line.
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