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_numdiff.py
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"""Routines for numerical differentiation.""" import numpy as np from numpy.linalg import norm from scipy.sparse.linalg import LinearOperator from ..sparse import issparse, csc_matrix, csr_matrix, coo_matrix, find from ._group_columns import group_dense, group_sparse EPS = np.finfo(np.float64).eps def _adjust_scheme_to_bounds(x0, h, num_steps, scheme, lb, ub): """Adjust final difference scheme to the presence of bounds. Parameters ---------- x0 : ndarray, shape (n,) Point at which we wish to estimate derivative. h : ndarray, shape (n,) Desired absolute finite difference steps. num_steps : int Number of `h` steps in one direction required to implement finite difference scheme. For example, 2 means that we need to evaluate f(x0 + 2 * h) or f(x0 - 2 * h) scheme : {'1-sided', '2-sided'} Whether steps in one or both directions are required. In other words '1-sided' applies to forward and backward schemes, '2-sided' applies to center schemes. lb : ndarray, shape (n,) Lower bounds on independent variables. ub : ndarray, shape (n,) Upper bounds on independent variables. Returns ------- h_adjusted : ndarray, shape (n,) Adjusted absolute step sizes. Step size decreases only if a sign flip or switching to one-sided scheme doesn't allow to take a full step. use_one_sided : ndarray of bool, shape (n,) Whether to switch to one-sided scheme. Informative only for ``scheme='2-sided'``. """ if scheme == '1-sided': use_one_sided = np.ones_like(h, dtype=bool) elif scheme == '2-sided': h = np.abs(h) use_one_sided = np.zeros_like(h, dtype=bool) else: raise ValueError("`scheme` must be '1-sided' or '2-sided'.") if np.all((lb == -np.inf) & (ub == np.inf)): return h, use_one_sided h_total = h * num_steps h_adjusted = h.copy() lower_dist = x0 - lb upper_dist = ub - x0 if scheme == '1-sided': x = x0 + h_total violated = (x < lb) | (x > ub) fitting = np.abs(h_total) <= np.maximum(lower_dist, upper_dist) h_adjusted[violated & fitting] *= -1 forward = (upper_dist >= lower_dist) & ~fitting h_adjusted[forward] = upper_dist[forward] / num_steps backward = (upper_dist < lower_dist) & ~fitting h_adjusted[backward] = -lower_dist[backward] / num_steps elif scheme == '2-sided': central = (lower_dist >= h_total) & (upper_dist >= h_total) forward = (upper_dist >= lower_dist) & ~central h_adjusted[forward] = np.minimum( h[forward], 0.5 * upper_dist[forward] / num_steps) use_one_sided[forward] = True backward = (upper_dist < lower_dist) & ~central h_adjusted[backward] = -np.minimum( h[backward], 0.5 * lower_dist[backward] / num_steps) use_one_sided[backward] = True min_dist = np.minimum(upper_dist, lower_dist) / num_steps adjusted_central = (~central & (np.abs(h_adjusted) <= min_dist)) h_adjusted[adjusted_central] = min_dist[adjusted_central] use_one_sided[adjusted_central] = False return h_adjusted, use_one_sided relative_step = {"2-point": EPS**0.5, "3-point": EPS**(1/3), "cs": EPS**0.5} def _compute_absolute_step(rel_step, x0, method): """ Computes an absolute step from a relative step for finite difference calculation. Parameters ---------- rel_step: None or array-like Relative step for the finite difference calculation x0 : np.ndarray Parameter vector method : {'2-point', '3-point', 'cs'} """ if rel_step is None: rel_step = relative_step[method] sign_x0 = (x0 >= 0).astype(float) * 2 - 1 return rel_step * sign_x0 * np.maximum(1.0, np.abs(x0)) def _prepare_bounds(bounds, x0): """ Prepares new-style bounds from a two-tuple specifying the lower and upper limits for values in x0. If a value is not bound then the lower/upper bound will be expected to be -np.inf/np.inf. Examples -------- >>> _prepare_bounds([(0, 1, 2), (1, 2, np.inf)], [0.5, 1.5, 2.5]) (array([0., 1., 2.]), array([ 1., 2., inf])) """ lb, ub = [np.asarray(b, dtype=float) for b in bounds] if lb.ndim == 0: lb = np.resize(lb, x0.shape) if ub.ndim == 0: ub = np.resize(ub, x0.shape) return lb, ub def group_columns(A, order=0): """Group columns of a 2-D matrix for sparse finite differencing [1]_. Two columns are in the same group if in each row at least one of them has zero. A greedy sequential algorithm is used to construct groups. Parameters ---------- A : array_like or sparse matrix, shape (m, n) Matrix of which to group columns. order : int, iterable of int with shape (n,) or None Permutation array which defines the order of columns enumeration. If int or None, a random permutation is used with `order` used as a random seed. Default is 0, that is use a random permutation but guarantee repeatability. Returns ------- groups : ndarray of int, shape (n,) Contains values from 0 to n_groups-1, where n_groups is the number of found groups. Each value ``groups[i]`` is an index of a group to which ith column assigned. The procedure was helpful only if n_groups is significantly less than n. References ---------- .. [1] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13 (1974), pp. 117-120. """ if issparse(A): A = csc_matrix(A) else: A = np.atleast_2d(A) A = (A != 0).astype(np.int32) if A.ndim != 2: raise ValueError("`A` must be 2-dimensional.") m, n = A.shape if order is None or np.isscalar(order): rng = np.random.RandomState(order) order = rng.permutation(n) else: order = np.asarray(order) if order.shape != (n,): raise ValueError("`order` has incorrect shape.") A = A[:, order] if issparse(A): groups = group_sparse(m, n, A.indices, A.indptr) else: groups = group_dense(m, n, A) groups[order] = groups.copy() return groups def approx_derivative(fun, x0, method='3-point', rel_step=None, abs_step=None, f0=None, bounds=(-np.inf, np.inf), sparsity=None, as_linear_operator=False, args=(), kwargs={}): """Compute finite difference approximation of the derivatives of a vector-valued function. If a function maps from R^n to R^m, its derivatives form m-by-n matrix called the Jacobian, where an element (i, j) is a partial derivative of f[i] with respect to x[j]. Parameters ---------- fun : callable Function of which to estimate the derivatives. The argument x passed to this function is ndarray of shape (n,) (never a scalar even if n=1). It must return 1-D array_like of shape (m,) or a scalar. x0 : array_like of shape (n,) or float Point at which to estimate the derivatives. Float will be converted to a 1-D array. method : {'3-point', '2-point', 'cs'}, optional Finite difference method to use: - '2-point' - use the first order accuracy forward or backward difference. - '3-point' - use central difference in interior points and the second order accuracy forward or backward difference near the boundary. - 'cs' - use a complex-step finite difference scheme. This assumes that the user function is real-valued and can be analytically continued to the complex plane. Otherwise, produces bogus results. rel_step : None or array_like, optional Relative step size to use. The absolute step size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `h` is ignored. If None (default) then step is selected automatically, see Notes. abs_step : array_like, optional Absolute step size to use, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `abs_step` is ignored. By default relative steps are used, only if ``abs_step is not None`` are absolute steps used. f0 : None or array_like, optional If not None it is assumed to be equal to ``fun(x0)``, in this case the ``fun(x0)`` is not called. Default is None. bounds : tuple of array_like, optional Lower and upper bounds on independent variables. Defaults to no bounds. Each bound must match the size of `x0` or be a scalar, in the latter case the bound will be the same for all variables. Use it to limit the range of function evaluation. Bounds checking is not implemented when `as_linear_operator` is True. sparsity : {None, array_like, sparse matrix, 2-tuple}, optional Defines a sparsity structure of the Jacobian matrix. If the Jacobian matrix is known to have only few non-zero elements in each row, then it's possible to estimate its several columns by a single function evaluation [3]_. To perform such economic computations two ingredients are required: * structure : array_like or sparse matrix of shape (m, n). A zero element means that a corresponding element of the Jacobian identically equals to zero. * groups : array_like of shape (n,). A column grouping for a given sparsity structure, use `group_columns` to obtain it. A single array or a sparse matrix is interpreted as a sparsity structure, and groups are computed inside the function. A tuple is interpreted as (structure, groups). If None (default), a standard dense differencing will be used. Note, that sparse differencing makes sense only for large Jacobian matrices where each row contains few non-zero elements. as_linear_operator : bool, optional When True the function returns an `scipy.sparse.linalg.LinearOperator`. Otherwise it returns a dense array or a sparse matrix depending on `sparsity`. The linear operator provides an efficient way of computing ``J.dot(p)`` for any vector ``p`` of shape (n,), but does not allow direct access to individual elements of the matrix. By default `as_linear_operator` is False. args, kwargs : tuple and dict, optional Additional arguments passed to `fun`. Both empty by default. The calling signature is ``fun(x, *args, **kwargs)``. Returns ------- J : {ndarray, sparse matrix, LinearOperator} Finite difference approximation of the Jacobian matrix. If `as_linear_operator` is True returns a LinearOperator with shape (m, n). Otherwise it returns a dense array or sparse matrix depending on how `sparsity` is defined. If `sparsity` is None then a ndarray with shape (m, n) is returned. If `sparsity` is not None returns a csr_matrix with shape (m, n). For sparse matrices and linear operators it is always returned as a 2-D structure, for ndarrays, if m=1 it is returned as a 1-D gradient array with shape (n,). See Also -------- check_derivative : Check correctness of a function computing derivatives. Notes ----- If `rel_step` is not provided, it assigned to ``EPS**(1/s)``, where EPS is machine epsilon for float64 numbers, s=2 for '2-point' method and s=3 for '3-point' method. Such relative step approximately minimizes a sum of truncation and round-off errors, see [1]_. Relative steps are used by default. However, absolute steps are used when ``abs_step is not None``. If any of the absolute steps produces an indistinguishable difference from the original `x0`, ``(x0 + abs_step) - x0 == 0``, then a relative step is substituted for that particular entry. A finite difference scheme for '3-point' method is selected automatically. The well-known central difference scheme is used for points sufficiently far from the boundary, and 3-point forward or backward scheme is used for points near the boundary. Both schemes have the second-order accuracy in terms of Taylor expansion. Refer to [2]_ for the formulas of 3-point forward and backward difference schemes. For dense differencing when m=1 Jacobian is returned with a shape (n,), on the other hand when n=1 Jacobian is returned with a shape (m, 1). Our motivation is the following: a) It handles a case of gradient computation (m=1) in a conventional way. b) It clearly separates these two different cases. b) In all cases np.atleast_2d can be called to get 2-D Jacobian with correct dimensions. References ---------- .. [1] W. H. Press et. al. "Numerical Recipes. The Art of Scientific Computing. 3rd edition", sec. 5.7. .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13 (1974), pp. 117-120. .. [3] B. Fornberg, "Generation of Finite Difference Formulas on Arbitrarily Spaced Grids", Mathematics of Computation 51, 1988. Examples -------- >>> import numpy as np >>> from scipy.optimize import approx_derivative >>> >>> def f(x, c1, c2): ... return np.array([x[0] * np.sin(c1 * x[1]), ... x[0] * np.cos(c2 * x[1])]) ... >>> x0 = np.array([1.0, 0.5 * np.pi]) >>> approx_derivative(f, x0, args=(1, 2)) array([[ 1., 0.], [-1., 0.]]) Bounds can be used to limit the region of function evaluation. In the example below we compute left and right derivative at point 1.0. >>> def g(x): ... return x**2 if x >= 1 else x ... >>> x0 = 1.0 >>> approx_derivative(g, x0, bounds=(-np.inf, 1.0)) array([ 1.]) >>> approx_derivative(g, x0, bounds=(1.0, np.inf)) array([ 2.]) """ if method not in ['2-point', '3-point', 'cs']: raise ValueError("Unknown method '%s'. " % method) x0 = np.atleast_1d(x0) if x0.ndim > 1: raise ValueError("`x0` must have at most 1 dimension.") lb, ub = _prepare_bounds(bounds, x0) if lb.shape != x0.shape or ub.shape != x0.shape: raise ValueError("Inconsistent shapes between bounds and `x0`.") if as_linear_operator and not (np.all(np.isinf(lb)) and np.all(np.isinf(ub))): raise ValueError("Bounds not supported when " "`as_linear_operator` is True.") def fun_wrapped(x): f = np.atleast_1d(fun(x, *args, **kwargs)) if f.ndim > 1: raise RuntimeError("`fun` return value has " "more than 1 dimension.") return f if f0 is None: f0 = fun_wrapped(x0) else: f0 = np.atleast_1d(f0) if f0.ndim > 1: raise ValueError("`f0` passed has more than 1 dimension.") if np.any((x0 < lb) | (x0 > ub)): raise ValueError("`x0` violates bound constraints.") if as_linear_operator: if rel_step is None: rel_step = relative_step[method] return _linear_operator_difference(fun_wrapped, x0, f0, rel_step, method) else: # by default we use rel_step if abs_step is None: h = _compute_absolute_step(rel_step, x0, method) else: # user specifies an absolute step sign_x0 = (x0 >= 0).astype(float) * 2 - 1 h = abs_step # cannot have a zero step. This might happen if x0 is very large # or small. In which case fall back to relative step. dx = ((x0 + h) - x0) h = np.where(dx == 0, relative_step[method] * sign_x0 * np.maximum(1.0, np.abs(x0)), h) if method == '2-point': h, use_one_sided = _adjust_scheme_to_bounds( x0, h, 1, '1-sided', lb, ub) elif method == '3-point': h, use_one_sided = _adjust_scheme_to_bounds( x0, h, 1, '2-sided', lb, ub) elif method == 'cs': use_one_sided = False if sparsity is None: return _dense_difference(fun_wrapped, x0, f0, h, use_one_sided, method) else: if not issparse(sparsity) and len(sparsity) == 2: structure, groups = sparsity else: structure = sparsity groups = group_columns(sparsity) if issparse(structure): structure = csc_matrix(structure) else: structure = np.atleast_2d(structure) groups = np.atleast_1d(groups) return _sparse_difference(fun_wrapped, x0, f0, h, use_one_sided, structure, groups, method) def _linear_operator_difference(fun, x0, f0, h, method): m = f0.size n = x0.size if method == '2-point': def matvec(p): if np.array_equal(p, np.zeros_like(p)): return np.zeros(m) dx = h / norm(p) x = x0 + dx*p df = fun(x) - f0 return df / dx elif method == '3-point': def matvec(p): if np.array_equal(p, np.zeros_like(p)): return np.zeros(m) dx = 2*h / norm(p) x1 = x0 - (dx/2)*p x2 = x0 + (dx/2)*p f1 = fun(x1) f2 = fun(x2) df = f2 - f1 return df / dx elif method == 'cs': def matvec(p): if np.array_equal(p, np.zeros_like(p)): return np.zeros(m) dx = h / norm(p) x = x0 + dx*p*1.j f1 = fun(x) df = f1.imag return df / dx else: raise RuntimeError("Never be here.") return LinearOperator((m, n), matvec) def _dense_difference(fun, x0, f0, h, use_one_sided, method): m = f0.size n = x0.size J_transposed = np.empty((n, m)) h_vecs = np.diag(h) for i in range(h.size): if method == '2-point': x = x0 + h_vecs[i] dx = x[i] - x0[i] # Recompute dx as exactly representable number. df = fun(x) - f0 elif method == '3-point' and use_one_sided[i]: x1 = x0 + h_vecs[i] x2 = x0 + 2 * h_vecs[i] dx = x2[i] - x0[i] f1 = fun(x1) f2 = fun(x2) df = -3.0 * f0 + 4 * f1 - f2 elif method == '3-point' and not use_one_sided[i]: x1 = x0 - h_vecs[i] x2 = x0 + h_vecs[i] dx = x2[i] - x1[i] f1 = fun(x1) f2 = fun(x2) df = f2 - f1 elif method == 'cs': f1 = fun(x0 + h_vecs[i]*1.j) df = f1.imag dx = h_vecs[i, i] else: raise RuntimeError("Never be here.") J_transposed[i] = df / dx if m == 1: J_transposed = np.ravel(J_transposed) return J_transposed.T def _sparse_difference(fun, x0, f0, h, use_one_sided, structure, groups, method): m = f0.size n = x0.size row_indices = [] col_indices = [] fractions = [] n_groups = np.max(groups) + 1 for group in range(n_groups): # Perturb variables which are in the same group simultaneously. e = np.equal(group, groups) h_vec = h * e if method == '2-point': x = x0 + h_vec dx = x - x0 df = fun(x) - f0 # The result is written to columns which correspond to perturbed # variables. cols, = np.nonzero(e) # Find all non-zero elements in selected columns of Jacobian. i, j, _ = find(structure[:, cols]) # Restore column indices in the full array. j = cols[j] elif method == '3-point': # Here we do conceptually the same but separate one-sided # and two-sided schemes. x1 = x0.copy() x2 = x0.copy() mask_1 = use_one_sided & e x1[mask_1] += h_vec[mask_1] x2[mask_1] += 2 * h_vec[mask_1] mask_2 = ~use_one_sided & e x1[mask_2] -= h_vec[mask_2] x2[mask_2] += h_vec[mask_2] dx = np.zeros(n) dx[mask_1] = x2[mask_1] - x0[mask_1] dx[mask_2] = x2[mask_2] - x1[mask_2] f1 = fun(x1) f2 = fun(x2) cols, = np.nonzero(e) i, j, _ = find(structure[:, cols]) j = cols[j] mask = use_one_sided[j] df = np.empty(m) rows = i[mask] df[rows] = -3 * f0[rows] + 4 * f1[rows] - f2[rows] rows = i[~mask] df[rows] = f2[rows] - f1[rows] elif method == 'cs': f1 = fun(x0 + h_vec*1.j) df = f1.imag dx = h_vec cols, = np.nonzero(e) i, j, _ = find(structure[:, cols]) j = cols[j] else: raise ValueError("Never be here.") # All that's left is to compute the fraction. We store i, j and # fractions as separate arrays and later construct coo_matrix. row_indices.append(i) col_indices.append(j) fractions.append(df[i] / dx[j]) row_indices = np.hstack(row_indices) col_indices = np.hstack(col_indices) fractions = np.hstack(fractions) J = coo_matrix((fractions, (row_indices, col_indices)), shape=(m, n)) return csr_matrix(J) def check_derivative(fun, jac, x0, bounds=(-np.inf, np.inf), args=(), kwargs={}): """Check correctness of a function computing derivatives (Jacobian or gradient) by comparison with a finite difference approximation. Parameters ---------- fun : callable Function of which to estimate the derivatives. The argument x passed to this function is ndarray of shape (n,) (never a scalar even if n=1). It must return 1-D array_like of shape (m,) or a scalar. jac : callable Function which computes Jacobian matrix of `fun`. It must work with argument x the same way as `fun`. The return value must be array_like or sparse matrix with an appropriate shape. x0 : array_like of shape (n,) or float Point at which to estimate the derivatives. Float will be converted to 1-D array. bounds : 2-tuple of array_like, optional Lower and upper bounds on independent variables. Defaults to no bounds. Each bound must match the size of `x0` or be a scalar, in the latter case the bound will be the same for all variables. Use it to limit the range of function evaluation. args, kwargs : tuple and dict, optional Additional arguments passed to `fun` and `jac`. Both empty by default. The calling signature is ``fun(x, *args, **kwargs)`` and the same for `jac`. Returns ------- accuracy : float The maximum among all relative errors for elements with absolute values higher than 1 and absolute errors for elements with absolute values less or equal than 1. If `accuracy` is on the order of 1e-6 or lower, then it is likely that your `jac` implementation is correct. See Also -------- approx_derivative : Compute finite difference approximation of derivative. Examples -------- >>> import numpy as np >>> from scipy.optimize import check_derivative >>> >>> >>> def f(x, c1, c2): ... return np.array([x[0] * np.sin(c1 * x[1]), ... x[0] * np.cos(c2 * x[1])]) ... >>> def jac(x, c1, c2): ... return np.array([ ... [np.sin(c1 * x[1]), c1 * x[0] * np.cos(c1 * x[1])], ... [np.cos(c2 * x[1]), -c2 * x[0] * np.sin(c2 * x[1])] ... ]) ... >>> >>> x0 = np.array([1.0, 0.5 * np.pi]) >>> check_derivative(f, jac, x0, args=(1, 2)) 2.4492935982947064e-16 """ J_to_test = jac(x0, *args, **kwargs) if issparse(J_to_test): J_diff = approx_derivative(fun, x0, bounds=bounds, sparsity=J_to_test, args=args, kwargs=kwargs) J_to_test = csr_matrix(J_to_test) abs_err = J_to_test - J_diff i, j, abs_err_data = find(abs_err) J_diff_data = np.asarray(J_diff[i, j]).ravel() return np.max(np.abs(abs_err_data) / np.maximum(1, np.abs(J_diff_data))) else: J_diff = approx_derivative(fun, x0, bounds=bounds, args=args, kwargs=kwargs) abs_err = np.abs(J_to_test - J_diff) return np.max(abs_err / np.maximum(1, np.abs(J_diff)))