# scipy/scipy

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 """ Functions which are common and require SciPy Base and Level 1 SciPy (special, linalg) """ from __future__ import division, print_function, absolute_import from numpy import arange, newaxis, hstack, product, array, fromstring __all__ = ['central_diff_weights', 'derivative', 'ascent', 'face'] def central_diff_weights(Np, ndiv=1): """ Return weights for an Np-point central derivative. Assumes equally-spaced function points. If weights are in the vector w, then derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx) Parameters ---------- Np : int Number of points for the central derivative. ndiv : int, optional Number of divisions. Default is 1. Notes ----- Can be inaccurate for large number of points. """ if Np < ndiv + 1: raise ValueError("Number of points must be at least the derivative order + 1.") if Np % 2 == 0: raise ValueError("The number of points must be odd.") from scipy import linalg ho = Np >> 1 x = arange(-ho,ho+1.0) x = x[:,newaxis] X = x**0.0 for k in range(1,Np): X = hstack([X,x**k]) w = product(arange(1,ndiv+1),axis=0)*linalg.inv(X)[ndiv] return w def derivative(func, x0, dx=1.0, n=1, args=(), order=3): """ Find the n-th derivative of a function at a point. Given a function, use a central difference formula with spacing `dx` to compute the `n`-th derivative at `x0`. Parameters ---------- func : function Input function. x0 : float The point at which `n`-th derivative is found. dx : float, optional Spacing. n : int, optional Order of the derivative. Default is 1. args : tuple, optional Arguments order : int, optional Number of points to use, must be odd. Notes ----- Decreasing the step size too small can result in round-off error. Examples -------- >>> from scipy.misc import derivative >>> def f(x): ... return x**3 + x**2 >>> derivative(f, 1.0, dx=1e-6) 4.9999999999217337 """ if order < n + 1: raise ValueError("'order' (the number of points used to compute the derivative), " "must be at least the derivative order 'n' + 1.") if order % 2 == 0: raise ValueError("'order' (the number of points used to compute the derivative) " "must be odd.") # pre-computed for n=1 and 2 and low-order for speed. if n == 1: if order == 3: weights = array([-1,0,1])/2.0 elif order == 5: weights = array([1,-8,0,8,-1])/12.0 elif order == 7: weights = array([-1,9,-45,0,45,-9,1])/60.0 elif order == 9: weights = array([3,-32,168,-672,0,672,-168,32,-3])/840.0 else: weights = central_diff_weights(order,1) elif n == 2: if order == 3: weights = array([1,-2.0,1]) elif order == 5: weights = array([-1,16,-30,16,-1])/12.0 elif order == 7: weights = array([2,-27,270,-490,270,-27,2])/180.0 elif order == 9: weights = array([-9,128,-1008,8064,-14350,8064,-1008,128,-9])/5040.0 else: weights = central_diff_weights(order,2) else: weights = central_diff_weights(order, n) val = 0.0 ho = order >> 1 for k in range(order): val += weights[k]*func(x0+(k-ho)*dx,*args) return val / product((dx,)*n,axis=0) def ascent(): """ Get an 8-bit grayscale bit-depth, 512 x 512 derived image for easy use in demos The image is derived from accent-to-the-top.jpg at http://www.public-domain-image.com/people-public-domain-images-pictures/ Parameters ---------- None Returns ------- ascent : ndarray convenient image to use for testing and demonstration Examples -------- >>> import scipy.misc >>> ascent = scipy.misc.ascent() >>> ascent.shape (512, 512) >>> ascent.max() 255 >>> import matplotlib.pyplot as plt >>> plt.gray() >>> plt.imshow(ascent) >>> plt.show() """ import pickle import os fname = os.path.join(os.path.dirname(__file__),'ascent.dat') with open(fname, 'rb') as f: ascent = array(pickle.load(f)) return ascent def face(gray=False): """ Get a 1024 x 768, color image of a raccoon face. raccoon-procyon-lotor.jpg at http://www.public-domain-image.com Parameters ---------- gray : bool, optional If True return 8-bit grey-scale image, otherwise return a color image Returns ------- face : ndarray image of a racoon face Examples -------- >>> import scipy.misc >>> face = scipy.misc.face() >>> face.shape (768, 1024, 3) >>> face.max() 255 >>> face.dtype dtype('uint8') >>> import matplotlib.pyplot as plt >>> plt.gray() >>> plt.imshow(face) >>> plt.show() """ import bz2 import os with open(os.path.join(os.path.dirname(__file__), 'face.dat'), 'rb') as f: rawdata = f.read() data = bz2.decompress(rawdata) face = fromstring(data, dtype='uint8') face.shape = (768, 1024, 3) if gray is True: face = (0.21 * face[:,:,0] + 0.71 * face[:,:,1] + 0.07 * face[:,:,2]).astype('uint8') return face