package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Finite-difference approximation of the gradient of a scalar function.

Parameters ---------- xk : array_like The coordinate vector at which to determine the gradient of `f`. f : callable The function of which to determine the gradient (partial derivatives). Should take `xk` as first argument, other arguments to `f` can be supplied in ``*args``. Should return a scalar, the value of the function at `xk`. epsilon : array_like Increment to `xk` to use for determining the function gradient. If a scalar, uses the same finite difference delta for all partial derivatives. If an array, should contain one value per element of `xk`. \*args : args, optional Any other arguments that are to be passed to `f`.

Returns ------- grad : ndarray The partial derivatives of `f` to `xk`.

See Also -------- check_grad : Check correctness of gradient function against approx_fprime.

Notes ----- The function gradient is determined by the forward finite difference formula::

f(xki + epsiloni) - f(xki) f'i = --------------------------------- epsiloni

The main use of `approx_fprime` is in scalar function optimizers like `fmin_bfgs`, to determine numerically the Jacobian of a function.

Examples -------- >>> from scipy import optimize >>> def func(x, c0, c1): ... 'Coordinate vector `x` should be an array of size two.' ... return c0 * x0**2 + c1*x1**2

>>> x = np.ones(2) >>> c0, c1 = (1, 200) >>> eps = np.sqrt(np.finfo(float).eps) >>> optimize.approx_fprime(x, func, eps, np.sqrt(200) * eps, c0, c1) array( 2. , 400.00004198)

array(object, dtype=None, copy=True, order='K', subok=False, ndmin=0)

Create an array.

Parameters ---------- object : array_like An array, any object exposing the array interface, an object whose __array__ method returns an array, or any (nested) sequence. dtype : data-type, optional The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence. copy : bool, optional If true (default), then the object is copied. Otherwise, a copy will only be made if __array__ returns a copy, if obj is a nested sequence, or if a copy is needed to satisfy any of the other requirements (`dtype`, `order`, etc.). order : 'K', 'A', 'C', 'F', optional Specify the memory layout of the array. If object is not an array, the newly created array will be in C order (row major) unless 'F' is specified, in which case it will be in Fortran order (column major). If object is an array the following holds.

===== ========= =================================================== order no copy copy=True ===== ========= =================================================== 'K' unchanged F & C order preserved, otherwise most similar order 'A' unchanged F order if input is F and not C, otherwise C order 'C' C order C order 'F' F order F order ===== ========= ===================================================

When ``copy=False`` and a copy is made for other reasons, the result is the same as if ``copy=True``, with some exceptions for `A`, see the Notes section. The default order is 'K'. subok : bool, optional If True, then sub-classes will be passed-through, otherwise the returned array will be forced to be a base-class array (default). ndmin : int, optional Specifies the minimum number of dimensions that the resulting array should have. Ones will be pre-pended to the shape as needed to meet this requirement.

Returns ------- out : ndarray An array object satisfying the specified requirements.

See Also -------- empty_like : Return an empty array with shape and type of input. ones_like : Return an array of ones with shape and type of input. zeros_like : Return an array of zeros with shape and type of input. full_like : Return a new array with shape of input filled with value. empty : Return a new uninitialized array. ones : Return a new array setting values to one. zeros : Return a new array setting values to zero. full : Return a new array of given shape filled with value.

Notes ----- When order is 'A' and `object` is an array in neither 'C' nor 'F' order, and a copy is forced by a change in dtype, then the order of the result is not necessarily 'C' as expected. This is likely a bug.

Examples -------- >>> np.array(1, 2, 3) array(1, 2, 3)

Upcasting:

>>> np.array(1, 2, 3.0) array( 1., 2., 3.)

More than one dimension:

>>> np.array([1, 2], [3, 4]) array([1, 2], [3, 4])

Minimum dimensions 2:

>>> np.array(1, 2, 3, ndmin=2) array([1, 2, 3])

Type provided:

>>> np.array(1, 2, 3, dtype=complex) array( 1.+0.j, 2.+0.j, 3.+0.j)

Data-type consisting of more than one element:

>>> x = np.array((1,2),(3,4),dtype=('a','<i4'),('b','<i4')) >>> x'a' array(1, 3)

Creating an array from sub-classes:

>>> np.array(np.mat('1 2; 3 4')) array([1, 2], [3, 4])

>>> np.array(np.mat('1 2; 3 4'), subok=True) matrix([1, 2], [3, 4])

Return an array converted to a float type.

Parameters ---------- a : array_like The input array. dtype : str or dtype object, optional Float type code to coerce input array `a`. If `dtype` is one of the 'int' dtypes, it is replaced with float64.

Returns ------- out : ndarray The input `a` as a float ndarray.

Examples -------- >>> np.asfarray(2, 3) array(2., 3.) >>> np.asfarray(2, 3, dtype='float') array(2., 3.) >>> np.asfarray(2, 3, dtype='int8') array(2., 3.)

Minimize a function with variables subject to bounds, using gradient information in a truncated Newton algorithm. This method wraps a C implementation of the algorithm.

Parameters ---------- func : callable ``func(x, *args)`` Function to minimize. Must do one of:

1. Return f and g, where f is the value of the function and g its gradient (a list of floats).

2. Return the function value but supply gradient function separately as `fprime`.

3. Return the function value and set ``approx_grad=True``.

If the function returns None, the minimization is aborted. x0 : array_like Initial estimate of minimum. fprime : callable ``fprime(x, *args)``, optional Gradient of `func`. If None, then either `func` must return the function value and the gradient (``f,g = func(x, *args)``) or `approx_grad` must be True. args : tuple, optional Arguments to pass to function. approx_grad : bool, optional If true, approximate the gradient numerically. bounds : list, optional (min, max) pairs for each element in x0, defining the bounds on that parameter. Use None or +/-inf for one of min or max when there is no bound in that direction. epsilon : float, optional Used if approx_grad is True. The stepsize in a finite difference approximation for fprime. scale : array_like, optional Scaling factors to apply to each variable. If None, the factors are up-low for interval bounded variables and 1+|x| for the others. Defaults to None. offset : array_like, optional Value to subtract from each variable. If None, the offsets are (up+low)/2 for interval bounded variables and x for the others. messages : int, optional Bit mask used to select messages display during minimization values defined in the MSGS dict. Defaults to MGS_ALL. disp : int, optional Integer interface to messages. 0 = no message, 5 = all messages maxCGit : int, optional Maximum number of hessian*vector evaluations per main iteration. If maxCGit == 0, the direction chosen is -gradient if maxCGit < 0, maxCGit is set to max(1,min(50,n/2)). Defaults to -1. maxfun : int, optional Maximum number of function evaluation. if None, maxfun is set to max(100, 10*len(x0)). Defaults to None. eta : float, optional Severity of the line search. if < 0 or > 1, set to 0.25. Defaults to -1. stepmx : float, optional Maximum step for the line search. May be increased during call. If too small, it will be set to 10.0. Defaults to 0. accuracy : float, optional Relative precision for finite difference calculations. If <= machine_precision, set to sqrt(machine_precision). Defaults to 0. fmin : float, optional Minimum function value estimate. Defaults to 0. ftol : float, optional Precision goal for the value of f in the stopping criterion. If ftol < 0.0, ftol is set to 0.0 defaults to -1. xtol : float, optional Precision goal for the value of x in the stopping criterion (after applying x scaling factors). If xtol < 0.0, xtol is set to sqrt(machine_precision). Defaults to -1. pgtol : float, optional Precision goal for the value of the projected gradient in the stopping criterion (after applying x scaling factors). If pgtol < 0.0, pgtol is set to 1e-2 * sqrt(accuracy). Setting it to 0.0 is not recommended. Defaults to -1. rescale : float, optional Scaling factor (in log10) used to trigger f value rescaling. If 0, rescale at each iteration. If a large value, never rescale. If < 0, rescale is set to 1.3. callback : callable, optional Called after each iteration, as callback(xk), where xk is the current parameter vector.

Returns ------- x : ndarray The solution. nfeval : int The number of function evaluations. rc : int Return code, see below

See also -------- minimize: Interface to minimization algorithms for multivariate functions. See the 'TNC' `method` in particular.

Notes ----- The underlying algorithm is truncated Newton, also called Newton Conjugate-Gradient. This method differs from scipy.optimize.fmin_ncg in that

1. It wraps a C implementation of the algorithm 2. It allows each variable to be given an upper and lower bound.

The algorithm incorporates the bound constraints by determining the descent direction as in an unconstrained truncated Newton, but never taking a step-size large enough to leave the space of feasible x's. The algorithm keeps track of a set of currently active constraints, and ignores them when computing the minimum allowable step size. (The x's associated with the active constraint are kept fixed.) If the maximum allowable step size is zero then a new constraint is added. At the end of each iteration one of the constraints may be deemed no longer active and removed. A constraint is considered no longer active is if it is currently active but the gradient for that variable points inward from the constraint. The specific constraint removed is the one associated with the variable of largest index whose constraint is no longer active.

Return codes are defined as follows::

-1 : Infeasible (lower bound > upper bound) 0 : Local minimum reached (|pg| ~= 0) 1 : Converged (|f_n-f_(n-1)| ~= 0) 2 : Converged (|x_n-x_(n-1)| ~= 0) 3 : Max. number of function evaluations reached 4 : Linear search failed 5 : All lower bounds are equal to the upper bounds 6 : Unable to progress 7 : User requested end of minimization

References ---------- Wright S., Nocedal J. (2006), 'Numerical Optimization'

Nash S.G. (1984), 'Newton-Type Minimization Via the Lanczos Method', SIAM Journal of Numerical Analysis 21, pp. 770-778

zeros(shape, dtype=float, order='C')

Return a new array of given shape and type, filled with zeros.

Parameters ---------- shape : int or tuple of ints Shape of the new array, e.g., ``(2, 3)`` or ``2``. dtype : data-type, optional The desired data-type for the array, e.g., `numpy.int8`. Default is `numpy.float64`. order : 'C', 'F', optional, default: 'C' Whether to store multi-dimensional data in row-major (C-style) or column-major (Fortran-style) order in memory.

Returns ------- out : ndarray Array of zeros with the given shape, dtype, and order.

See Also -------- zeros_like : Return an array of zeros with shape and type of input. empty : Return a new uninitialized array. ones : Return a new array setting values to one. full : Return a new array of given shape filled with value.

Examples -------- >>> np.zeros(5) array( 0., 0., 0., 0., 0.)

>>> np.zeros((5,), dtype=int) array(0, 0, 0, 0, 0)

>>> np.zeros((2, 1)) array([ 0.], [ 0.])

>>> s = (2,2) >>> np.zeros(s) array([ 0., 0.], [ 0., 0.])

>>> np.zeros((2,), dtype=('x', 'i4'), ('y', 'i4')) # custom dtype array((0, 0), (0, 0), dtype=('x', '<i4'), ('y', '<i4'))