jacobi 0.4.0

Compute numerical derivatives.

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 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

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

Examples

from matplotlib import pyplot as plt
import numpy as np
from jacobi import jacobi


def f(x):
    return np.sin(x) / x

x = np.linspace(-10, 10, 1000)

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, label="f(x) = sin(x) / x")
plt.plot(x, fdx, ls="--", label="f'(x)")
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^T

Comparison to numdifftools

Speed

Jacobi 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).

Precision

The machine precision is indicated by the dashed line.