857 lines
38 KiB
Python
857 lines
38 KiB
Python
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# mypy: disable-error-code="attr-defined"
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import numpy as np
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import scipy._lib._elementwise_iterative_method as eim
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from scipy._lib._util import _RichResult
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_EERRORINCREASE = -1 # used in _differentiate
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def _differentiate_iv(func, x, args, atol, rtol, maxiter, order, initial_step,
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step_factor, step_direction, preserve_shape, callback):
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# Input validation for `_differentiate`
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if not callable(func):
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raise ValueError('`func` must be callable.')
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# x has more complex IV that is taken care of during initialization
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x = np.asarray(x)
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dtype = x.dtype if np.issubdtype(x.dtype, np.inexact) else np.float64
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if not np.iterable(args):
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args = (args,)
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if atol is None:
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atol = np.finfo(dtype).tiny
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if rtol is None:
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rtol = np.sqrt(np.finfo(dtype).eps)
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message = 'Tolerances and step parameters must be non-negative scalars.'
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tols = np.asarray([atol, rtol, initial_step, step_factor])
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if (not np.issubdtype(tols.dtype, np.number)
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or np.any(tols < 0)
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or tols.shape != (4,)):
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raise ValueError(message)
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initial_step, step_factor = tols[2:].astype(dtype)
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maxiter_int = int(maxiter)
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if maxiter != maxiter_int or maxiter <= 0:
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raise ValueError('`maxiter` must be a positive integer.')
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order_int = int(order)
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if order_int != order or order <= 0:
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raise ValueError('`order` must be a positive integer.')
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step_direction = np.sign(step_direction).astype(dtype)
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x, step_direction = np.broadcast_arrays(x, step_direction)
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x, step_direction = x[()], step_direction[()]
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message = '`preserve_shape` must be True or False.'
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if preserve_shape not in {True, False}:
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raise ValueError(message)
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if callback is not None and not callable(callback):
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raise ValueError('`callback` must be callable.')
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return (func, x, args, atol, rtol, maxiter_int, order_int, initial_step,
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step_factor, step_direction, preserve_shape, callback)
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def _differentiate(func, x, *, args=(), atol=None, rtol=None, maxiter=10,
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order=8, initial_step=0.5, step_factor=2.0,
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step_direction=0, preserve_shape=False, callback=None):
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"""Evaluate the derivative of an elementwise scalar function numerically.
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Parameters
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----------
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func : callable
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The function whose derivative is desired. The signature must be::
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func(x: ndarray, *fargs) -> ndarray
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where each element of ``x`` is a finite real number and ``fargs`` is a tuple,
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which may contain an arbitrary number of arrays that are broadcastable
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with `x`. ``func`` must be an elementwise function: each element
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``func(x)[i]`` must equal ``func(x[i])`` for all indices ``i``.
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x : array_like
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Abscissae at which to evaluate the derivative.
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args : tuple, optional
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Additional positional arguments to be passed to `func`. Must be arrays
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broadcastable with `x`. If the callable to be differentiated requires
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arguments that are not broadcastable with `x`, wrap that callable with
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`func`. See Examples.
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atol, rtol : float, optional
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Absolute and relative tolerances for the stopping condition: iteration
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will stop when ``res.error < atol + rtol * abs(res.df)``. The default
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`atol` is the smallest normal number of the appropriate dtype, and
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the default `rtol` is the square root of the precision of the
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appropriate dtype.
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order : int, default: 8
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The (positive integer) order of the finite difference formula to be
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used. Odd integers will be rounded up to the next even integer.
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initial_step : float, default: 0.5
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The (absolute) initial step size for the finite difference derivative
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approximation.
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step_factor : float, default: 2.0
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The factor by which the step size is *reduced* in each iteration; i.e.
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the step size in iteration 1 is ``initial_step/step_factor``. If
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``step_factor < 1``, subsequent steps will be greater than the initial
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step; this may be useful if steps smaller than some threshold are
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undesirable (e.g. due to subtractive cancellation error).
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maxiter : int, default: 10
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The maximum number of iterations of the algorithm to perform. See
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notes.
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step_direction : array_like
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An array representing the direction of the finite difference steps (for
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use when `x` lies near to the boundary of the domain of the function.)
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Must be broadcastable with `x` and all `args`.
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Where 0 (default), central differences are used; where negative (e.g.
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-1), steps are non-positive; and where positive (e.g. 1), all steps are
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non-negative.
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preserve_shape : bool, default: False
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In the following, "arguments of `func`" refers to the array ``x`` and
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any arrays within ``fargs``. Let ``shape`` be the broadcasted shape
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of `x` and all elements of `args` (which is conceptually
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distinct from ``fargs`` passed into `f`).
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- When ``preserve_shape=False`` (default), `f` must accept arguments
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of *any* broadcastable shapes.
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- When ``preserve_shape=True``, `f` must accept arguments of shape
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``shape`` *or* ``shape + (n,)``, where ``(n,)`` is the number of
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abscissae at which the function is being evaluated.
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In either case, for each scalar element ``xi`` within `x`, the array
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returned by `f` must include the scalar ``f(xi)`` at the same index.
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Consequently, the shape of the output is always the shape of the input
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``x``.
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See Examples.
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callback : callable, optional
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An optional user-supplied function to be called before the first
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iteration and after each iteration.
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Called as ``callback(res)``, where ``res`` is a ``_RichResult``
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similar to that returned by `_differentiate` (but containing the
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current iterate's values of all variables). If `callback` raises a
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``StopIteration``, the algorithm will terminate immediately and
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`_differentiate` will return a result.
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Returns
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-------
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res : _RichResult
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An instance of `scipy._lib._util._RichResult` with the following
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attributes. (The descriptions are written as though the values will be
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scalars; however, if `func` returns an array, the outputs will be
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arrays of the same shape.)
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success : bool
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``True`` when the algorithm terminated successfully (status ``0``).
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status : int
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An integer representing the exit status of the algorithm.
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``0`` : The algorithm converged to the specified tolerances.
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``-1`` : The error estimate increased, so iteration was terminated.
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``-2`` : The maximum number of iterations was reached.
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``-3`` : A non-finite value was encountered.
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``-4`` : Iteration was terminated by `callback`.
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``1`` : The algorithm is proceeding normally (in `callback` only).
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df : float
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The derivative of `func` at `x`, if the algorithm terminated
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successfully.
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error : float
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An estimate of the error: the magnitude of the difference between
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the current estimate of the derivative and the estimate in the
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previous iteration.
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nit : int
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The number of iterations performed.
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nfev : int
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The number of points at which `func` was evaluated.
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x : float
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The value at which the derivative of `func` was evaluated
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(after broadcasting with `args` and `step_direction`).
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Notes
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-----
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The implementation was inspired by jacobi [1]_, numdifftools [2]_, and
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DERIVEST [3]_, but the implementation follows the theory of Taylor series
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more straightforwardly (and arguably naively so).
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In the first iteration, the derivative is estimated using a finite
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difference formula of order `order` with maximum step size `initial_step`.
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Each subsequent iteration, the maximum step size is reduced by
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`step_factor`, and the derivative is estimated again until a termination
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condition is reached. The error estimate is the magnitude of the difference
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between the current derivative approximation and that of the previous
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iteration.
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The stencils of the finite difference formulae are designed such that
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abscissae are "nested": after `func` is evaluated at ``order + 1``
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points in the first iteration, `func` is evaluated at only two new points
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in each subsequent iteration; ``order - 1`` previously evaluated function
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values required by the finite difference formula are reused, and two
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function values (evaluations at the points furthest from `x`) are unused.
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Step sizes are absolute. When the step size is small relative to the
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magnitude of `x`, precision is lost; for example, if `x` is ``1e20``, the
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default initial step size of ``0.5`` cannot be resolved. Accordingly,
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consider using larger initial step sizes for large magnitudes of `x`.
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The default tolerances are challenging to satisfy at points where the
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true derivative is exactly zero. If the derivative may be exactly zero,
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consider specifying an absolute tolerance (e.g. ``atol=1e-16``) to
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improve convergence.
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References
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----------
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[1]_ Hans Dembinski (@HDembinski). jacobi.
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https://github.com/HDembinski/jacobi
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[2]_ Per A. Brodtkorb and John D'Errico. numdifftools.
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https://numdifftools.readthedocs.io/en/latest/
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[3]_ John D'Errico. DERIVEST: Adaptive Robust Numerical Differentiation.
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https://www.mathworks.com/matlabcentral/fileexchange/13490-adaptive-robust-numerical-differentiation
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[4]_ Numerical Differentition. Wikipedia.
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https://en.wikipedia.org/wiki/Numerical_differentiation
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Examples
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--------
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Evaluate the derivative of ``np.exp`` at several points ``x``.
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>>> import numpy as np
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>>> from scipy.optimize._differentiate import _differentiate
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>>> f = np.exp
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>>> df = np.exp # true derivative
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>>> x = np.linspace(1, 2, 5)
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>>> res = _differentiate(f, x)
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>>> res.df # approximation of the derivative
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array([2.71828183, 3.49034296, 4.48168907, 5.75460268, 7.3890561 ])
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>>> res.error # estimate of the error
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array(
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[7.12940817e-12, 9.16688947e-12, 1.17594823e-11, 1.50972568e-11, 1.93942640e-11]
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)
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>>> abs(res.df - df(x)) # true error
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array(
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[3.06421555e-14, 3.01980663e-14, 5.06261699e-14, 6.30606678e-14, 8.34887715e-14]
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)
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Show the convergence of the approximation as the step size is reduced.
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Each iteration, the step size is reduced by `step_factor`, so for
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sufficiently small initial step, each iteration reduces the error by a
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factor of ``1/step_factor**order`` until finite precision arithmetic
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inhibits further improvement.
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>>> iter = list(range(1, 12)) # maximum iterations
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>>> hfac = 2 # step size reduction per iteration
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>>> hdir = [-1, 0, 1] # compare left-, central-, and right- steps
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>>> order = 4 # order of differentiation formula
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>>> x = 1
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>>> ref = df(x)
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>>> errors = [] # true error
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>>> for i in iter:
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... res = _differentiate(f, x, maxiter=i, step_factor=hfac,
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... step_direction=hdir, order=order,
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... atol=0, rtol=0) # prevent early termination
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... errors.append(abs(res.df - ref))
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>>> errors = np.array(errors)
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>>> plt.semilogy(iter, errors[:, 0], label='left differences')
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>>> plt.semilogy(iter, errors[:, 1], label='central differences')
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>>> plt.semilogy(iter, errors[:, 2], label='right differences')
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>>> plt.xlabel('iteration')
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>>> plt.ylabel('error')
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>>> plt.legend()
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>>> plt.show()
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>>> (errors[1, 1] / errors[0, 1], 1 / hfac**order)
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(0.06215223140159822, 0.0625)
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The implementation is vectorized over `x`, `step_direction`, and `args`.
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The function is evaluated once before the first iteration to perform input
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validation and standardization, and once per iteration thereafter.
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>>> def f(x, p):
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... print('here')
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... f.nit += 1
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... return x**p
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>>> f.nit = 0
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>>> def df(x, p):
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... return p*x**(p-1)
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>>> x = np.arange(1, 5)
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>>> p = np.arange(1, 6).reshape((-1, 1))
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>>> hdir = np.arange(-1, 2).reshape((-1, 1, 1))
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>>> res = _differentiate(f, x, args=(p,), step_direction=hdir, maxiter=1)
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>>> np.allclose(res.df, df(x, p))
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True
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>>> res.df.shape
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(3, 5, 4)
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>>> f.nit
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2
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By default, `preserve_shape` is False, and therefore the callable
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`f` may be called with arrays of any broadcastable shapes.
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For example:
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>>> shapes = []
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>>> def f(x, c):
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... shape = np.broadcast_shapes(x.shape, c.shape)
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... shapes.append(shape)
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... return np.sin(c*x)
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>>>
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>>> c = [1, 5, 10, 20]
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>>> res = _differentiate(f, 0, args=(c,))
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>>> shapes
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[(4,), (4, 8), (4, 2), (3, 2), (2, 2), (1, 2)]
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To understand where these shapes are coming from - and to better
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understand how `_differentiate` computes accurate results - note that
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higher values of ``c`` correspond with higher frequency sinusoids.
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The higher frequency sinusoids make the function's derivative change
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faster, so more function evaluations are required to achieve the target
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accuracy:
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>>> res.nfev
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array([11, 13, 15, 17])
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The initial ``shape``, ``(4,)``, corresponds with evaluating the
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function at a single abscissa and all four frequencies; this is used
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for input validation and to determine the size and dtype of the arrays
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that store results. The next shape corresponds with evaluating the
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function at an initial grid of abscissae and all four frequencies.
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Successive calls to the function evaluate the function at two more
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abscissae, increasing the effective order of the approximation by two.
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However, in later function evaluations, the function is evaluated at
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fewer frequencies because the corresponding derivative has already
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converged to the required tolerance. This saves function evaluations to
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improve performance, but it requires the function to accept arguments of
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any shape.
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"Vector-valued" functions are unlikely to satisfy this requirement.
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For example, consider
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>>> def f(x):
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... return [x, np.sin(3*x), x+np.sin(10*x), np.sin(20*x)*(x-1)**2]
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This integrand is not compatible with `_differentiate` as written; for instance,
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the shape of the output will not be the same as the shape of ``x``. Such a
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function *could* be converted to a compatible form with the introduction of
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additional parameters, but this would be inconvenient. In such cases,
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a simpler solution would be to use `preserve_shape`.
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>>> shapes = []
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>>> def f(x):
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... shapes.append(x.shape)
|
||
|
|
... x0, x1, x2, x3 = x
|
||
|
|
... return [x0, np.sin(3*x1), x2+np.sin(10*x2), np.sin(20*x3)*(x3-1)**2]
|
||
|
|
>>>
|
||
|
|
>>> x = np.zeros(4)
|
||
|
|
>>> res = _differentiate(f, x, preserve_shape=True)
|
||
|
|
>>> shapes
|
||
|
|
[(4,), (4, 8), (4, 2), (4, 2), (4, 2), (4, 2)]
|
||
|
|
|
||
|
|
Here, the shape of ``x`` is ``(4,)``. With ``preserve_shape=True``, the
|
||
|
|
function may be called with argument ``x`` of shape ``(4,)`` or ``(4, n)``,
|
||
|
|
and this is what we observe.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# TODO (followup):
|
||
|
|
# - investigate behavior at saddle points
|
||
|
|
# - array initial_step / step_factor?
|
||
|
|
# - multivariate functions?
|
||
|
|
|
||
|
|
res = _differentiate_iv(func, x, args, atol, rtol, maxiter, order, initial_step,
|
||
|
|
step_factor, step_direction, preserve_shape, callback)
|
||
|
|
(func, x, args, atol, rtol, maxiter, order,
|
||
|
|
h0, fac, hdir, preserve_shape, callback) = res
|
||
|
|
|
||
|
|
# Initialization
|
||
|
|
# Since f(x) (no step) is not needed for central differences, it may be
|
||
|
|
# possible to eliminate this function evaluation. However, it's useful for
|
||
|
|
# input validation and standardization, and everything else is designed to
|
||
|
|
# reduce function calls, so let's keep it simple.
|
||
|
|
temp = eim._initialize(func, (x,), args, preserve_shape=preserve_shape)
|
||
|
|
func, xs, fs, args, shape, dtype, xp = temp
|
||
|
|
x, f = xs[0], fs[0]
|
||
|
|
df = np.full_like(f, np.nan)
|
||
|
|
# Ideally we'd broadcast the shape of `hdir` in `_elementwise_algo_init`, but
|
||
|
|
# it's simpler to do it here than to generalize `_elementwise_algo_init` further.
|
||
|
|
# `hdir` and `x` are already broadcasted in `_differentiate_iv`, so we know
|
||
|
|
# that `hdir` can be broadcasted to the final shape.
|
||
|
|
hdir = np.broadcast_to(hdir, shape).flatten()
|
||
|
|
|
||
|
|
status = np.full_like(x, eim._EINPROGRESS, dtype=int) # in progress
|
||
|
|
nit, nfev = 0, 1 # one function evaluations performed above
|
||
|
|
# Boolean indices of left, central, right, and (all) one-sided steps
|
||
|
|
il = hdir < 0
|
||
|
|
ic = hdir == 0
|
||
|
|
ir = hdir > 0
|
||
|
|
io = il | ir
|
||
|
|
|
||
|
|
# Most of these attributes are reasonably obvious, but:
|
||
|
|
# - `fs` holds all the function values of all active `x`. The zeroth
|
||
|
|
# axis corresponds with active points `x`, the first axis corresponds
|
||
|
|
# with the different steps (in the order described in
|
||
|
|
# `_differentiate_weights`).
|
||
|
|
# - `terms` (which could probably use a better name) is half the `order`,
|
||
|
|
# which is always even.
|
||
|
|
work = _RichResult(x=x, df=df, fs=f[:, np.newaxis], error=np.nan, h=h0,
|
||
|
|
df_last=np.nan, error_last=np.nan, h0=h0, fac=fac,
|
||
|
|
atol=atol, rtol=rtol, nit=nit, nfev=nfev,
|
||
|
|
status=status, dtype=dtype, terms=(order+1)//2,
|
||
|
|
hdir=hdir, il=il, ic=ic, ir=ir, io=io)
|
||
|
|
# This is the correspondence between terms in the `work` object and the
|
||
|
|
# final result. In this case, the mapping is trivial. Note that `success`
|
||
|
|
# is prepended automatically.
|
||
|
|
res_work_pairs = [('status', 'status'), ('df', 'df'), ('error', 'error'),
|
||
|
|
('nit', 'nit'), ('nfev', 'nfev'), ('x', 'x')]
|
||
|
|
|
||
|
|
def pre_func_eval(work):
|
||
|
|
"""Determine the abscissae at which the function needs to be evaluated.
|
||
|
|
|
||
|
|
See `_differentiate_weights` for a description of the stencil (pattern
|
||
|
|
of the abscissae).
|
||
|
|
|
||
|
|
In the first iteration, there is only one stored function value in
|
||
|
|
`work.fs`, `f(x)`, so we need to evaluate at `order` new points. In
|
||
|
|
subsequent iterations, we evaluate at two new points. Note that
|
||
|
|
`work.x` is always flattened into a 1D array after broadcasting with
|
||
|
|
all `args`, so we add a new axis at the end and evaluate all point
|
||
|
|
in one call to the function.
|
||
|
|
|
||
|
|
For improvement:
|
||
|
|
- Consider measuring the step size actually taken, since `(x + h) - x`
|
||
|
|
is not identically equal to `h` with floating point arithmetic.
|
||
|
|
- Adjust the step size automatically if `x` is too big to resolve the
|
||
|
|
step.
|
||
|
|
- We could probably save some work if there are no central difference
|
||
|
|
steps or no one-sided steps.
|
||
|
|
"""
|
||
|
|
n = work.terms # half the order
|
||
|
|
h = work.h # step size
|
||
|
|
c = work.fac # step reduction factor
|
||
|
|
d = c**0.5 # square root of step reduction factor (one-sided stencil)
|
||
|
|
# Note - no need to be careful about dtypes until we allocate `x_eval`
|
||
|
|
|
||
|
|
if work.nit == 0:
|
||
|
|
hc = h / c**np.arange(n)
|
||
|
|
hc = np.concatenate((-hc[::-1], hc))
|
||
|
|
else:
|
||
|
|
hc = np.asarray([-h, h]) / c**(n-1)
|
||
|
|
|
||
|
|
if work.nit == 0:
|
||
|
|
hr = h / d**np.arange(2*n)
|
||
|
|
else:
|
||
|
|
hr = np.asarray([h, h/d]) / c**(n-1)
|
||
|
|
|
||
|
|
n_new = 2*n if work.nit == 0 else 2 # number of new abscissae
|
||
|
|
x_eval = np.zeros((len(work.hdir), n_new), dtype=work.dtype)
|
||
|
|
il, ic, ir = work.il, work.ic, work.ir
|
||
|
|
x_eval[ir] = work.x[ir, np.newaxis] + hr
|
||
|
|
x_eval[ic] = work.x[ic, np.newaxis] + hc
|
||
|
|
x_eval[il] = work.x[il, np.newaxis] - hr
|
||
|
|
return x_eval
|
||
|
|
|
||
|
|
def post_func_eval(x, f, work):
|
||
|
|
""" Estimate the derivative and error from the function evaluations
|
||
|
|
|
||
|
|
As in `pre_func_eval`: in the first iteration, there is only one stored
|
||
|
|
function value in `work.fs`, `f(x)`, so we need to add the `order` new
|
||
|
|
points. In subsequent iterations, we add two new points. The tricky
|
||
|
|
part is getting the order to match that of the weights, which is
|
||
|
|
described in `_differentiate_weights`.
|
||
|
|
|
||
|
|
For improvement:
|
||
|
|
- Change the order of the weights (and steps in `pre_func_eval`) to
|
||
|
|
simplify `work_fc` concatenation and eliminate `fc` concatenation.
|
||
|
|
- It would be simple to do one-step Richardson extrapolation with `df`
|
||
|
|
and `df_last` to increase the order of the estimate and/or improve
|
||
|
|
the error estimate.
|
||
|
|
- Process the function evaluations in a more numerically favorable
|
||
|
|
way. For instance, combining the pairs of central difference evals
|
||
|
|
into a second-order approximation and using Richardson extrapolation
|
||
|
|
to produce a higher order approximation seemed to retain accuracy up
|
||
|
|
to very high order.
|
||
|
|
- Alternatively, we could use `polyfit` like Jacobi. An advantage of
|
||
|
|
fitting polynomial to more points than necessary is improved noise
|
||
|
|
tolerance.
|
||
|
|
"""
|
||
|
|
n = work.terms
|
||
|
|
n_new = n if work.nit == 0 else 1
|
||
|
|
il, ic, io = work.il, work.ic, work.io
|
||
|
|
|
||
|
|
# Central difference
|
||
|
|
# `work_fc` is *all* the points at which the function has been evaluated
|
||
|
|
# `fc` is the points we're using *this iteration* to produce the estimate
|
||
|
|
work_fc = (f[ic, :n_new], work.fs[ic, :], f[ic, -n_new:])
|
||
|
|
work_fc = np.concatenate(work_fc, axis=-1)
|
||
|
|
if work.nit == 0:
|
||
|
|
fc = work_fc
|
||
|
|
else:
|
||
|
|
fc = (work_fc[:, :n], work_fc[:, n:n+1], work_fc[:, -n:])
|
||
|
|
fc = np.concatenate(fc, axis=-1)
|
||
|
|
|
||
|
|
# One-sided difference
|
||
|
|
work_fo = np.concatenate((work.fs[io, :], f[io, :]), axis=-1)
|
||
|
|
if work.nit == 0:
|
||
|
|
fo = work_fo
|
||
|
|
else:
|
||
|
|
fo = np.concatenate((work_fo[:, 0:1], work_fo[:, -2*n:]), axis=-1)
|
||
|
|
|
||
|
|
work.fs = np.zeros((len(ic), work.fs.shape[-1] + 2*n_new))
|
||
|
|
work.fs[ic] = work_fc
|
||
|
|
work.fs[io] = work_fo
|
||
|
|
|
||
|
|
wc, wo = _differentiate_weights(work, n)
|
||
|
|
work.df_last = work.df.copy()
|
||
|
|
work.df[ic] = fc @ wc / work.h
|
||
|
|
work.df[io] = fo @ wo / work.h
|
||
|
|
work.df[il] *= -1
|
||
|
|
|
||
|
|
work.h /= work.fac
|
||
|
|
work.error_last = work.error
|
||
|
|
# Simple error estimate - the difference in derivative estimates between
|
||
|
|
# this iteration and the last. This is typically conservative because if
|
||
|
|
# convergence has begin, the true error is much closer to the difference
|
||
|
|
# between the current estimate and the *next* error estimate. However,
|
||
|
|
# we could use Richarson extrapolation to produce an error estimate that
|
||
|
|
# is one order higher, and take the difference between that and
|
||
|
|
# `work.df` (which would just be constant factor that depends on `fac`.)
|
||
|
|
work.error = abs(work.df - work.df_last)
|
||
|
|
|
||
|
|
def check_termination(work):
|
||
|
|
"""Terminate due to convergence, non-finite values, or error increase"""
|
||
|
|
stop = np.zeros_like(work.df).astype(bool)
|
||
|
|
|
||
|
|
i = work.error < work.atol + work.rtol*abs(work.df)
|
||
|
|
work.status[i] = eim._ECONVERGED
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
if work.nit > 0:
|
||
|
|
i = ~((np.isfinite(work.x) & np.isfinite(work.df)) | stop)
|
||
|
|
work.df[i], work.status[i] = np.nan, eim._EVALUEERR
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
# With infinite precision, there is a step size below which
|
||
|
|
# all smaller step sizes will reduce the error. But in floating point
|
||
|
|
# arithmetic, catastrophic cancellation will begin to cause the error
|
||
|
|
# to increase again. This heuristic tries to avoid step sizes that are
|
||
|
|
# too small. There may be more theoretically sound approaches for
|
||
|
|
# detecting a step size that minimizes the total error, but this
|
||
|
|
# heuristic seems simple and effective.
|
||
|
|
i = (work.error > work.error_last*10) & ~stop
|
||
|
|
work.status[i] = _EERRORINCREASE
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
return stop
|
||
|
|
|
||
|
|
def post_termination_check(work):
|
||
|
|
return
|
||
|
|
|
||
|
|
def customize_result(res, shape):
|
||
|
|
return shape
|
||
|
|
|
||
|
|
return eim._loop(work, callback, shape, maxiter, func, args, dtype,
|
||
|
|
pre_func_eval, post_func_eval, check_termination,
|
||
|
|
post_termination_check, customize_result, res_work_pairs,
|
||
|
|
xp, preserve_shape)
|
||
|
|
|
||
|
|
|
||
|
|
def _differentiate_weights(work, n):
|
||
|
|
# This produces the weights of the finite difference formula for a given
|
||
|
|
# stencil. In experiments, use of a second-order central difference formula
|
||
|
|
# with Richardson extrapolation was more accurate numerically, but it was
|
||
|
|
# more complicated, and it would have become even more complicated when
|
||
|
|
# adding support for one-sided differences. However, now that all the
|
||
|
|
# function evaluation values are stored, they can be processed in whatever
|
||
|
|
# way is desired to produce the derivative estimate. We leave alternative
|
||
|
|
# approaches to future work. To be more self-contained, here is the theory
|
||
|
|
# for deriving the weights below.
|
||
|
|
#
|
||
|
|
# Recall that the Taylor expansion of a univariate, scalar-values function
|
||
|
|
# about a point `x` may be expressed as:
|
||
|
|
# f(x + h) = f(x) + f'(x)*h + f''(x)/2!*h**2 + O(h**3)
|
||
|
|
# Suppose we evaluate f(x), f(x+h), and f(x-h). We have:
|
||
|
|
# f(x) = f(x)
|
||
|
|
# f(x + h) = f(x) + f'(x)*h + f''(x)/2!*h**2 + O(h**3)
|
||
|
|
# f(x - h) = f(x) - f'(x)*h + f''(x)/2!*h**2 + O(h**3)
|
||
|
|
# We can solve for weights `wi` such that:
|
||
|
|
# w1*f(x) = w1*(f(x))
|
||
|
|
# + w2*f(x + h) = w2*(f(x) + f'(x)*h + f''(x)/2!*h**2) + O(h**3)
|
||
|
|
# + w3*f(x - h) = w3*(f(x) - f'(x)*h + f''(x)/2!*h**2) + O(h**3)
|
||
|
|
# = 0 + f'(x)*h + 0 + O(h**3)
|
||
|
|
# Then
|
||
|
|
# f'(x) ~ (w1*f(x) + w2*f(x+h) + w3*f(x-h))/h
|
||
|
|
# is a finite difference derivative approximation with error O(h**2),
|
||
|
|
# and so it is said to be a "second-order" approximation. Under certain
|
||
|
|
# conditions (e.g. well-behaved function, `h` sufficiently small), the
|
||
|
|
# error in the approximation will decrease with h**2; that is, if `h` is
|
||
|
|
# reduced by a factor of 2, the error is reduced by a factor of 4.
|
||
|
|
#
|
||
|
|
# By default, we use eighth-order formulae. Our central-difference formula
|
||
|
|
# uses abscissae:
|
||
|
|
# x-h/c**3, x-h/c**2, x-h/c, x-h, x, x+h, x+h/c, x+h/c**2, x+h/c**3
|
||
|
|
# where `c` is the step factor. (Typically, the step factor is greater than
|
||
|
|
# one, so the outermost points - as written above - are actually closest to
|
||
|
|
# `x`.) This "stencil" is chosen so that each iteration, the step can be
|
||
|
|
# reduced by the factor `c`, and most of the function evaluations can be
|
||
|
|
# reused with the new step size. For example, in the next iteration, we
|
||
|
|
# will have:
|
||
|
|
# x-h/c**4, x-h/c**3, x-h/c**2, x-h/c, x, x+h/c, x+h/c**2, x+h/c**3, x+h/c**4
|
||
|
|
# We do not reuse `x-h` and `x+h` for the new derivative estimate.
|
||
|
|
# While this would increase the order of the formula and thus the
|
||
|
|
# theoretical convergence rate, it is also less stable numerically.
|
||
|
|
# (As noted above, there are other ways of processing the values that are
|
||
|
|
# more stable. Thus, even now we store `f(x-h)` and `f(x+h)` in `work.fs`
|
||
|
|
# to simplify future development of this sort of improvement.)
|
||
|
|
#
|
||
|
|
# The (right) one-sided formula is produced similarly using abscissae
|
||
|
|
# x, x+h, x+h/d, x+h/d**2, ..., x+h/d**6, x+h/d**7, x+h/d**7
|
||
|
|
# where `d` is the square root of `c`. (The left one-sided formula simply
|
||
|
|
# uses -h.) When the step size is reduced by factor `c = d**2`, we have
|
||
|
|
# abscissae:
|
||
|
|
# x, x+h/d**2, x+h/d**3..., x+h/d**8, x+h/d**9, x+h/d**9
|
||
|
|
# `d` is chosen as the square root of `c` so that the rate of the step-size
|
||
|
|
# reduction is the same per iteration as in the central difference case.
|
||
|
|
# Note that because the central difference formulas are inherently of even
|
||
|
|
# order, for simplicity, we use only even-order formulas for one-sided
|
||
|
|
# differences, too.
|
||
|
|
|
||
|
|
# It's possible for the user to specify `fac` in, say, double precision but
|
||
|
|
# `x` and `args` in single precision. `fac` gets converted to single
|
||
|
|
# precision, but we should always use double precision for the intermediate
|
||
|
|
# calculations here to avoid additional error in the weights.
|
||
|
|
fac = work.fac.astype(np.float64)
|
||
|
|
|
||
|
|
# Note that if the user switches back to floating point precision with
|
||
|
|
# `x` and `args`, then `fac` will not necessarily equal the (lower
|
||
|
|
# precision) cached `_differentiate_weights.fac`, and the weights will
|
||
|
|
# need to be recalculated. This could be fixed, but it's late, and of
|
||
|
|
# low consequence.
|
||
|
|
if fac != _differentiate_weights.fac:
|
||
|
|
_differentiate_weights.central = []
|
||
|
|
_differentiate_weights.right = []
|
||
|
|
_differentiate_weights.fac = fac
|
||
|
|
|
||
|
|
if len(_differentiate_weights.central) != 2*n + 1:
|
||
|
|
# Central difference weights. Consider refactoring this; it could
|
||
|
|
# probably be more compact.
|
||
|
|
i = np.arange(-n, n + 1)
|
||
|
|
p = np.abs(i) - 1. # center point has power `p` -1, but sign `s` is 0
|
||
|
|
s = np.sign(i)
|
||
|
|
|
||
|
|
h = s / fac ** p
|
||
|
|
A = np.vander(h, increasing=True).T
|
||
|
|
b = np.zeros(2*n + 1)
|
||
|
|
b[1] = 1
|
||
|
|
weights = np.linalg.solve(A, b)
|
||
|
|
|
||
|
|
# Enforce identities to improve accuracy
|
||
|
|
weights[n] = 0
|
||
|
|
for i in range(n):
|
||
|
|
weights[-i-1] = -weights[i]
|
||
|
|
|
||
|
|
# Cache the weights. We only need to calculate them once unless
|
||
|
|
# the step factor changes.
|
||
|
|
_differentiate_weights.central = weights
|
||
|
|
|
||
|
|
# One-sided difference weights. The left one-sided weights (with
|
||
|
|
# negative steps) are simply the negative of the right one-sided
|
||
|
|
# weights, so no need to compute them separately.
|
||
|
|
i = np.arange(2*n + 1)
|
||
|
|
p = i - 1.
|
||
|
|
s = np.sign(i)
|
||
|
|
|
||
|
|
h = s / np.sqrt(fac) ** p
|
||
|
|
A = np.vander(h, increasing=True).T
|
||
|
|
b = np.zeros(2 * n + 1)
|
||
|
|
b[1] = 1
|
||
|
|
weights = np.linalg.solve(A, b)
|
||
|
|
|
||
|
|
_differentiate_weights.right = weights
|
||
|
|
|
||
|
|
return (_differentiate_weights.central.astype(work.dtype, copy=False),
|
||
|
|
_differentiate_weights.right.astype(work.dtype, copy=False))
|
||
|
|
_differentiate_weights.central = []
|
||
|
|
_differentiate_weights.right = []
|
||
|
|
_differentiate_weights.fac = None
|
||
|
|
|
||
|
|
|
||
|
|
def _jacobian(func, x, *, atol=None, rtol=None, maxiter=10,
|
||
|
|
order=8, initial_step=0.5, step_factor=2.0):
|
||
|
|
r"""Evaluate the Jacobian of a function numerically.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable
|
||
|
|
The function whose Jacobian is desired. The signature must be::
|
||
|
|
|
||
|
|
func(x: ndarray) -> ndarray
|
||
|
|
|
||
|
|
where each element of ``x`` is a finite real. If the function to be
|
||
|
|
differentiated accepts additional, arguments wrap it (e.g. using
|
||
|
|
`functools.partial` or ``lambda``) and pass the wrapped callable
|
||
|
|
into `_jacobian`. See Notes regarding vectorization and the dimensionality
|
||
|
|
of the input and output.
|
||
|
|
x : array_like
|
||
|
|
Points at which to evaluate the Jacobian. Must have at least one dimension.
|
||
|
|
See Notes regarding the dimensionality and vectorization.
|
||
|
|
atol, rtol : float, optional
|
||
|
|
Absolute and relative tolerances for the stopping condition: iteration
|
||
|
|
will stop for each element of the Jacobian when
|
||
|
|
``res.error < atol + rtol * abs(res.df)``. The default `atol` is the
|
||
|
|
smallest normal number of the appropriate dtype, and the default `rtol`
|
||
|
|
is the square root of the precision of the appropriate dtype.
|
||
|
|
order : int, default: 8
|
||
|
|
The (positive integer) order of the finite difference formula to be
|
||
|
|
used. Odd integers will be rounded up to the next even integer.
|
||
|
|
initial_step : float, default: 0.5
|
||
|
|
The (absolute) initial step size for the finite difference derivative
|
||
|
|
approximation.
|
||
|
|
step_factor : float, default: 2.0
|
||
|
|
The factor by which the step size is *reduced* in each iteration; i.e.
|
||
|
|
the step size in iteration 1 is ``initial_step/step_factor``. If
|
||
|
|
``step_factor < 1``, subsequent steps will be greater than the initial
|
||
|
|
step; this may be useful if steps smaller than some threshold are
|
||
|
|
undesirable (e.g. due to subtractive cancellation error).
|
||
|
|
maxiter : int, default: 10
|
||
|
|
The maximum number of iterations of the algorithm to perform.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : _RichResult
|
||
|
|
An instance of `scipy._lib._util._RichResult` with the following
|
||
|
|
attributes.
|
||
|
|
|
||
|
|
success : bool array
|
||
|
|
``True`` when the algorithm terminated successfully (status ``0``).
|
||
|
|
status : int array
|
||
|
|
An integer representing the exit status of the algorithm.
|
||
|
|
``0`` : The algorithm converged to the specified tolerances.
|
||
|
|
``-1`` : The error estimate increased, so iteration was terminated.
|
||
|
|
``-2`` : The maximum number of iterations was reached.
|
||
|
|
``-3`` : A non-finite value was encountered.
|
||
|
|
``-4`` : Iteration was terminated by `callback`.
|
||
|
|
``1`` : The algorithm is proceeding normally (in `callback` only).
|
||
|
|
df : float array
|
||
|
|
The Jacobian of `func` at `x`, if the algorithm terminated
|
||
|
|
successfully.
|
||
|
|
error : float array
|
||
|
|
An estimate of the error: the magnitude of the difference between
|
||
|
|
the current estimate of the derivative and the estimate in the
|
||
|
|
previous iteration.
|
||
|
|
nit : int array
|
||
|
|
The number of iterations performed.
|
||
|
|
nfev : int array
|
||
|
|
The number of points at which `func` was evaluated.
|
||
|
|
x : float array
|
||
|
|
The value at which the derivative of `func` was evaluated.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
_differentiate
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Suppose we wish to evaluate the Jacobian of a function
|
||
|
|
:math:`f: \mathbf{R^m} \rightarrow \mathbf{R^n}`, and assign to variables
|
||
|
|
``m`` and ``n`` the positive integer values of :math:`m` and :math:`n`,
|
||
|
|
respectively. If we wish to evaluate the Jacobian at a single point,
|
||
|
|
then:
|
||
|
|
|
||
|
|
- argument `x` must be an array of shape ``(m,)``
|
||
|
|
- argument `func` must be vectorized to accept an array of shape ``(m, p)``.
|
||
|
|
The first axis represents the :math:`m` inputs of :math:`f`; the second
|
||
|
|
is for evaluating the function at multiple points in a single call.
|
||
|
|
- argument `func` must return an array of shape ``(n, p)``. The first
|
||
|
|
axis represents the :math:`n` outputs of :math:`f`; the second
|
||
|
|
is for the result of evaluating the function at multiple points.
|
||
|
|
- attribute ``df`` of the result object will be an array of shape ``(n, m)``,
|
||
|
|
the Jacobian.
|
||
|
|
|
||
|
|
This function is also vectorized in the sense that the Jacobian can be
|
||
|
|
evaluated at ``k`` points in a single call. In this case, `x` would be an
|
||
|
|
array of shape ``(m, k)``, `func` would accept an array of shape
|
||
|
|
``(m, k, p)`` and return an array of shape ``(n, k, p)``, and the ``df``
|
||
|
|
attribute of the result would have shape ``(n, m, k)``.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Jacobian matrix and determinant, *Wikipedia*,
|
||
|
|
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
The Rosenbrock function maps from :math:`\mathbf{R}^m \righarrow \mathbf{R}`;
|
||
|
|
the SciPy implementation `scipy.optimize.rosen` is vectorized to accept an
|
||
|
|
array of shape ``(m, p)`` and return an array of shape ``m``. Suppose we wish
|
||
|
|
to evaluate the Jacobian (AKA the gradient because the function returns a scalar)
|
||
|
|
at ``[0.5, 0.5, 0.5]``.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy.optimize._differentiate import _jacobian as jacobian
|
||
|
|
>>> from scipy.optimize import rosen, rosen_der
|
||
|
|
>>> m = 3
|
||
|
|
>>> x = np.full(m, 0.5)
|
||
|
|
>>> res = jacobian(rosen, x)
|
||
|
|
>>> ref = rosen_der(x) # reference value of the gradient
|
||
|
|
>>> res.df, ref
|
||
|
|
(array([-51., -1., 50.]), array([-51., -1., 50.]))
|
||
|
|
|
||
|
|
As an example of a function with multiple outputs, consider Example 4
|
||
|
|
from [1]_.
|
||
|
|
|
||
|
|
>>> def f(x):
|
||
|
|
... x1, x2, x3 = x ...
|
||
|
|
... return [x1, 5*x3, 4*x2**2 - 2*x3, x3*np.sin(x1)]
|
||
|
|
|
||
|
|
The true Jacobian is given by:
|
||
|
|
|
||
|
|
>>> def df(x):
|
||
|
|
... x1, x2, x3 = x
|
||
|
|
... one = np.ones_like(x1)
|
||
|
|
... return [[one, 0*one, 0*one],
|
||
|
|
... [0*one, 0*one, 5*one],
|
||
|
|
... [0*one, 8*x2, -2*one],
|
||
|
|
... [x3*np.cos(x1), 0*one, np.sin(x1)]]
|
||
|
|
|
||
|
|
Evaluate the Jacobian at an arbitrary point.
|
||
|
|
|
||
|
|
>>> rng = np.random.default_rng(389252938452)
|
||
|
|
>>> x = rng.random(size=3)
|
||
|
|
>>> res = jacobian(f, x)
|
||
|
|
>>> ref = df(x)
|
||
|
|
>>> res.df.shape == (4, 3)
|
||
|
|
True
|
||
|
|
>>> np.allclose(res.df, ref)
|
||
|
|
True
|
||
|
|
|
||
|
|
Evaluate the Jacobian at 10 arbitrary points in a single call.
|
||
|
|
|
||
|
|
>>> x = rng.random(size=(3, 10))
|
||
|
|
>>> res = jacobian(f, x)
|
||
|
|
>>> ref = df(x)
|
||
|
|
>>> res.df.shape == (4, 3, 10)
|
||
|
|
True
|
||
|
|
>>> np.allclose(res.df, ref)
|
||
|
|
True
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = np.asarray(x)
|
||
|
|
int_dtype = np.issubdtype(x.dtype, np.integer)
|
||
|
|
x0 = np.asarray(x, dtype=float) if int_dtype else x
|
||
|
|
|
||
|
|
if x0.ndim < 1:
|
||
|
|
message = "Argument `x` must be at least 1-D."
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
m = x0.shape[0]
|
||
|
|
i = np.arange(m)
|
||
|
|
|
||
|
|
def wrapped(x):
|
||
|
|
p = () if x.ndim == x0.ndim else (x.shape[-1],) # number of abscissae
|
||
|
|
new_dims = (1,) if x.ndim == x0.ndim else (1, -1)
|
||
|
|
new_shape = (m, m) + x0.shape[1:] + p
|
||
|
|
xph = np.expand_dims(x0, new_dims)
|
||
|
|
xph = np.broadcast_to(xph, new_shape).copy()
|
||
|
|
xph[i, i] = x
|
||
|
|
return func(xph)
|
||
|
|
|
||
|
|
res = _differentiate(wrapped, x, atol=atol, rtol=rtol,
|
||
|
|
maxiter=maxiter, order=order, initial_step=initial_step,
|
||
|
|
step_factor=step_factor, preserve_shape=True)
|
||
|
|
del res.x # the user knows `x`, and the way it gets broadcasted is meaningless here
|
||
|
|
return res
|