Fast numerical derivatives for analytic functions with arbitrary round-off error and error propagation.
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- 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 1000x faster than numdifftools at equivalent precision
- Returns error estimates for derivatives
- Supports arbitrary auxiliary function arguments
- Perform statistical error propagation based on numerically computed jacobian
- Lightweight package, only depends on numpy
- 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
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(x):
return np.sin(x) / x
x = np.linspace(-10, 10, 200)
fx = f(x)
# f(x) is a simple vectorized function, jacobian is diagonal
fdx, fdxe = jacobi(f, x, diagonal=True)
# fdxe is uncertainty estimate for derivative
plt.plot(x, fx, color="k", label="$f(x) = sin(x) / x$")
plt.plot(x, fdx, label="$f'(x)$ computed with jacobi")
scale = 14
plt.fill_between(
x,
fdx - fdxe * 10**scale,
fdx + fdxe * 10**scale,
label=f"$f'(x)$ error estimate$\\times \\, 10^{{{scale}}}$",
facecolor="C0",
alpha=0.5,
)
plt.legend()
from jacobi import propagate
import numpy as np
from scipy.special import gamma
# arbitrarily complex function that calls compiled libraries, numba-jitted code, etc.
def fn(x):
r = np.empty(3)
r[0] = 1.5 * np.exp(-x[0] ** 2)
r[1] = gamma(x[1] ** 3.1)
r[2] = np.polyval([1, 2, 3], x[0])
return r # x and r have different lengths
# fn accepts a parameter vector x, which has an associated covariance matrix xcov
x = [1.0, 2.0]
xcov = [[1.1, 0.1], [0.1, 2.3]]
y, ycov = propagate(fn, x, xcov) # y=f(x) and ycov = J xcov J^TJacobi makes better use of vectorized computation than numdifftools and converges rapidly if the derivative is trivial. This leads to a dramatic speedup in some cases.
Smaller run-time is better (and ratio > 1).
The machine precision is indicated by the dashed line. Jacobi is comparable in accuracy to numdifftools. The error estimate has the right order of magnitude but slightly underestimates the true deviation. This does not matter for most applications.
