Derivatives at multiple points#

You can evaluate derivatives at multiple expansion points by passing an array to x0. DerivKit computes the derivative independently at each point and stacks the results with leading shape x0.shape.

Each expansion point is treated independently. There is no coupling or shared sampling across points, even for adaptive or polynomial-based backends. This makes multi-point evaluation a convenience feature rather than a vectorized solver.

Finite differences (Ridders)#

>>> import numpy as np; from derivkit.derivative_kit import DerivativeKit
>>> # Define the function and multiple expansion points
>>> f = np.sin
>>> x0 = np.array([0.1, 0.5, 1.0])
>>> # Initialize DerivativeKit with array-valued x0
>>> dk = DerivativeKit(function=f, x0=x0)
>>> # Derivative at x0 using finite differences with Ridders extrapolation
>>> deriv = dk.differentiate(
...     method="finite",
...     order=1,
...     stepsize=1e-2,
...     num_points=5,
...     extrapolation="ridders"
...     )
>>> # Result has the same shape as x0
>>> print(deriv.shape)
(3,)
>>> # Compare against analytic derivative cos(x)
>>> print(bool(np.allclose(deriv, np.cos(x0), atol=1e-6, rtol=0.0)))
True

Adaptive polynomial fit (Chebyshev)#

>>> import numpy as np; from derivkit.derivative_kit import DerivativeKit
>>> # Same function and expansion points as above
>>> f = np.sin; x0 = np.array([0.1, 0.5, 1.0])
>>> dk = DerivativeKit(function=f, x0=x0)
>>> # Adaptive Chebyshev-based local polynomial fitting at each point
>>> deriv = dk.differentiate(
...     method="adaptive",
...     order=1,
...     n_points=12,
...     spacing="auto"
...     )
>>> # Output is stacked in the order of x0
>>> print(deriv.shape)
(3,)
>>> print(bool(np.allclose(deriv, np.cos(x0), atol=1e-6, rtol=0.0)))
True

Vector-valued output at multiple points#

If the function returns a vector, the stacked output has shape x0.shape + output_shape.

>>> import numpy as np
>>> from derivkit.derivative_kit import DerivativeKit
>>> # Define a vector-valued function
>>> def vec_func(x): return np.array([np.sin(x), np.cos(x)])
>>> # Multiple expansion points
>>> x0 = np.array([0.2, 0.4, 0.6])
>>> dk = DerivativeKit(function=vec_func, x0=x0)
>>> # Finite-difference derivative applied component-wise
>>> deriv = dk.differentiate(
...     method="finite",
...     order=1,
...     stepsize=1e-2,
...     num_points=5,
...     extrapolation="ridders"
...     )
>>> # Shape is (n_points, output_dim)
>>> print("shape:", deriv.shape)
shape: (3, 2)
>>> # Reference: d/dx [sin(x), cos(x)] = [cos(x), -sin(x)]
>>> ref = np.stack([np.cos(x0), -np.sin(x0)], axis=-1)
>>> print(bool(np.allclose(deriv, ref, atol=1e-6, rtol=0.0)))
True

Notes#

Each entry in x0 is treated as an independent expansion point.
There is no shared sampling or coupling between points, even for adaptive or polynomial-based backends.
The stacked output always has leading shape x0.shape.
For vector- or tensor-valued functions, output dimensions appear after the x0 axis.
This feature is intended for convenience and clarity, not for exploiting correlations or shared structure between expansion points.