# mypy: disable-error-code="attr-defined" import numpy as np import scipy._lib._elementwise_iterative_method as eim from scipy._lib._util import _RichResult _EERRORINCREASE = -1 # used in _differentiate def _differentiate_iv(func, x, args, atol, rtol, maxiter, order, initial_step, step_factor, step_direction, preserve_shape, callback): # Input validation for `_differentiate` if not callable(func): raise ValueError('`func` must be callable.') # x has more complex IV that is taken care of during initialization x = np.asarray(x) dtype = x.dtype if np.issubdtype(x.dtype, np.inexact) else np.float64 if not np.iterable(args): args = (args,) if atol is None: atol = np.finfo(dtype).tiny if rtol is None: rtol = np.sqrt(np.finfo(dtype).eps) message = 'Tolerances and step parameters must be non-negative scalars.' tols = np.asarray([atol, rtol, initial_step, step_factor]) if (not np.issubdtype(tols.dtype, np.number) or np.any(tols < 0) or tols.shape != (4,)): raise ValueError(message) initial_step, step_factor = tols[2:].astype(dtype) maxiter_int = int(maxiter) if maxiter != maxiter_int or maxiter <= 0: raise ValueError('`maxiter` must be a positive integer.') order_int = int(order) if order_int != order or order <= 0: raise ValueError('`order` must be a positive integer.') step_direction = np.sign(step_direction).astype(dtype) x, step_direction = np.broadcast_arrays(x, step_direction) x, step_direction = x[()], step_direction[()] message = '`preserve_shape` must be True or False.' if preserve_shape not in {True, False}: raise ValueError(message) if callback is not None and not callable(callback): raise ValueError('`callback` must be callable.') return (func, x, args, atol, rtol, maxiter_int, order_int, initial_step, step_factor, step_direction, preserve_shape, callback) def _differentiate(func, x, *, args=(), atol=None, rtol=None, maxiter=10, order=8, initial_step=0.5, step_factor=2.0, step_direction=0, preserve_shape=False, callback=None): """Evaluate the derivative of an elementwise scalar function numerically. Parameters ---------- func : callable The function whose derivative is desired. The signature must be:: func(x: ndarray, *fargs) -> ndarray where each element of ``x`` is a finite real number and ``fargs`` is a tuple, which may contain an arbitrary number of arrays that are broadcastable with `x`. ``func`` must be an elementwise function: each element ``func(x)[i]`` must equal ``func(x[i])`` for all indices ``i``. x : array_like Abscissae at which to evaluate the derivative. args : tuple, optional Additional positional arguments to be passed to `func`. Must be arrays broadcastable with `x`. If the callable to be differentiated requires arguments that are not broadcastable with `x`, wrap that callable with `func`. See Examples. atol, rtol : float, optional Absolute and relative tolerances for the stopping condition: iteration will stop 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. See notes. step_direction : array_like An array representing the direction of the finite difference steps (for use when `x` lies near to the boundary of the domain of the function.) Must be broadcastable with `x` and all `args`. Where 0 (default), central differences are used; where negative (e.g. -1), steps are non-positive; and where positive (e.g. 1), all steps are non-negative. preserve_shape : bool, default: False In the following, "arguments of `func`" refers to the array ``x`` and any arrays within ``fargs``. Let ``shape`` be the broadcasted shape of `x` and all elements of `args` (which is conceptually distinct from ``fargs`` passed into `f`). - When ``preserve_shape=False`` (default), `f` must accept arguments of *any* broadcastable shapes. - When ``preserve_shape=True``, `f` must accept arguments of shape ``shape`` *or* ``shape + (n,)``, where ``(n,)`` is the number of abscissae at which the function is being evaluated. In either case, for each scalar element ``xi`` within `x`, the array returned by `f` must include the scalar ``f(xi)`` at the same index. Consequently, the shape of the output is always the shape of the input ``x``. See Examples. callback : callable, optional An optional user-supplied function to be called before the first iteration and after each iteration. Called as ``callback(res)``, where ``res`` is a ``_RichResult`` similar to that returned by `_differentiate` (but containing the current iterate's values of all variables). If `callback` raises a ``StopIteration``, the algorithm will terminate immediately and `_differentiate` will return a result. Returns ------- res : _RichResult An instance of `scipy._lib._util._RichResult` with the following attributes. (The descriptions are written as though the values will be scalars; however, if `func` returns an array, the outputs will be arrays of the same shape.) success : bool ``True`` when the algorithm terminated successfully (status ``0``). status : int 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 The derivative of `func` at `x`, if the algorithm terminated successfully. error : float 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 The number of iterations performed. nfev : int The number of points at which `func` was evaluated. x : float The value at which the derivative of `func` was evaluated (after broadcasting with `args` and `step_direction`). Notes ----- The implementation was inspired by jacobi [1]_, numdifftools [2]_, and DERIVEST [3]_, but the implementation follows the theory of Taylor series more straightforwardly (and arguably naively so). In the first iteration, the derivative is estimated using a finite difference formula of order `order` with maximum step size `initial_step`. Each subsequent iteration, the maximum step size is reduced by `step_factor`, and the derivative is estimated again until a termination condition is reached. The error estimate is the magnitude of the difference between the current derivative approximation and that of the previous iteration. The stencils of the finite difference formulae are designed such that abscissae are "nested": after `func` is evaluated at ``order + 1`` points in the first iteration, `func` is evaluated at only two new points in each subsequent iteration; ``order - 1`` previously evaluated function values required by the finite difference formula are reused, and two function values (evaluations at the points furthest from `x`) are unused. Step sizes are absolute. When the step size is small relative to the magnitude of `x`, precision is lost; for example, if `x` is ``1e20``, the default initial step size of ``0.5`` cannot be resolved. Accordingly, consider using larger initial step sizes for large magnitudes of `x`. The default tolerances are challenging to satisfy at points where the true derivative is exactly zero. If the derivative may be exactly zero, consider specifying an absolute tolerance (e.g. ``atol=1e-16``) to improve convergence. References ---------- [1]_ Hans Dembinski (@HDembinski). jacobi. https://github.com/HDembinski/jacobi [2]_ Per A. Brodtkorb and John D'Errico. numdifftools. https://numdifftools.readthedocs.io/en/latest/ [3]_ John D'Errico. DERIVEST: Adaptive Robust Numerical Differentiation. https://www.mathworks.com/matlabcentral/fileexchange/13490-adaptive-robust-numerical-differentiation [4]_ Numerical Differentition. Wikipedia. https://en.wikipedia.org/wiki/Numerical_differentiation Examples -------- Evaluate the derivative of ``np.exp`` at several points ``x``. >>> import numpy as np >>> from scipy.optimize._differentiate import _differentiate >>> f = np.exp >>> df = np.exp # true derivative >>> x = np.linspace(1, 2, 5) >>> res = _differentiate(f, x) >>> res.df # approximation of the derivative array([2.71828183, 3.49034296, 4.48168907, 5.75460268, 7.3890561 ]) >>> res.error # estimate of the error array( [7.12940817e-12, 9.16688947e-12, 1.17594823e-11, 1.50972568e-11, 1.93942640e-11] ) >>> abs(res.df - df(x)) # true error array( [3.06421555e-14, 3.01980663e-14, 5.06261699e-14, 6.30606678e-14, 8.34887715e-14] ) Show the convergence of the approximation as the step size is reduced. Each iteration, the step size is reduced by `step_factor`, so for sufficiently small initial step, each iteration reduces the error by a factor of ``1/step_factor**order`` until finite precision arithmetic inhibits further improvement. >>> iter = list(range(1, 12)) # maximum iterations >>> hfac = 2 # step size reduction per iteration >>> hdir = [-1, 0, 1] # compare left-, central-, and right- steps >>> order = 4 # order of differentiation formula >>> x = 1 >>> ref = df(x) >>> errors = [] # true error >>> for i in iter: ... res = _differentiate(f, x, maxiter=i, step_factor=hfac, ... step_direction=hdir, order=order, ... atol=0, rtol=0) # prevent early termination ... errors.append(abs(res.df - ref)) >>> errors = np.array(errors) >>> plt.semilogy(iter, errors[:, 0], label='left differences') >>> plt.semilogy(iter, errors[:, 1], label='central differences') >>> plt.semilogy(iter, errors[:, 2], label='right differences') >>> plt.xlabel('iteration') >>> plt.ylabel('error') >>> plt.legend() >>> plt.show() >>> (errors[1, 1] / errors[0, 1], 1 / hfac**order) (0.06215223140159822, 0.0625) The implementation is vectorized over `x`, `step_direction`, and `args`. The function is evaluated once before the first iteration to perform input validation and standardization, and once per iteration thereafter. >>> def f(x, p): ... print('here') ... f.nit += 1 ... return x**p >>> f.nit = 0 >>> def df(x, p): ... return p*x**(p-1) >>> x = np.arange(1, 5) >>> p = np.arange(1, 6).reshape((-1, 1)) >>> hdir = np.arange(-1, 2).reshape((-1, 1, 1)) >>> res = _differentiate(f, x, args=(p,), step_direction=hdir, maxiter=1) >>> np.allclose(res.df, df(x, p)) True >>> res.df.shape (3, 5, 4) >>> f.nit 2 By default, `preserve_shape` is False, and therefore the callable `f` may be called with arrays of any broadcastable shapes. For example: >>> shapes = [] >>> def f(x, c): ... shape = np.broadcast_shapes(x.shape, c.shape) ... shapes.append(shape) ... return np.sin(c*x) >>> >>> c = [1, 5, 10, 20] >>> res = _differentiate(f, 0, args=(c,)) >>> shapes [(4,), (4, 8), (4, 2), (3, 2), (2, 2), (1, 2)] To understand where these shapes are coming from - and to better understand how `_differentiate` computes accurate results - note that higher values of ``c`` correspond with higher frequency sinusoids. The higher frequency sinusoids make the function's derivative change faster, so more function evaluations are required to achieve the target accuracy: >>> res.nfev array([11, 13, 15, 17]) The initial ``shape``, ``(4,)``, corresponds with evaluating the function at a single abscissa and all four frequencies; this is used for input validation and to determine the size and dtype of the arrays that store results. The next shape corresponds with evaluating the function at an initial grid of abscissae and all four frequencies. Successive calls to the function evaluate the function at two more abscissae, increasing the effective order of the approximation by two. However, in later function evaluations, the function is evaluated at fewer frequencies because the corresponding derivative has already converged to the required tolerance. This saves function evaluations to improve performance, but it requires the function to accept arguments of any shape. "Vector-valued" functions are unlikely to satisfy this requirement. For example, consider >>> def f(x): ... return [x, np.sin(3*x), x+np.sin(10*x), np.sin(20*x)*(x-1)**2] This integrand is not compatible with `_differentiate` as written; for instance, the shape of the output will not be the same as the shape of ``x``. Such a function *could* be converted to a compatible form with the introduction of additional parameters, but this would be inconvenient. In such cases, a simpler solution would be to use `preserve_shape`. >>> shapes = [] >>> def f(x): ... 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