# scipy/scipy

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 """ Functions which are common and require SciPy Base and Level 1 SciPy (special, linalg) """ from numpy import exp, asarray, arange, newaxis, hstack, product, array, \ where, zeros, extract, place, pi, sqrt, eye, poly1d, dot, r_ __all__ = ['factorial','factorial2','factorialk','comb', 'central_diff_weights', 'derivative', 'pade', 'lena'] # XXX: the factorial functions could move to scipy.special, and the others # to numpy perhaps? def factorial(n,exact=0): """n! = special.gamma(n+1) If exact==0, then floating point precision is used, otherwise exact long integer is computed. Notes: - Array argument accepted only for exact=0 case. - If n<0, the return value is 0. """ if exact: if n < 0: return 0L val = 1L for k in xrange(1,n+1): val *= k return val else: from scipy import special n = asarray(n) sv = special.errprint(0) vals = special.gamma(n+1) sv = special.errprint(sv) return where(n>=0,vals,0) def factorial2(n,exact=0): """n!! = special.gamma(n/2+1)*2**((m+1)/2)/sqrt(pi) n odd = 2**(n) * n! n even If exact==0, then floating point precision is used, otherwise exact long integer is computed. Notes: - Array argument accepted only for exact=0 case. - If n<0, the return value is 0. """ if exact: if n < -1: return 0L if n <= 0: return 1L val = 1L for k in xrange(n,0,-2): val *= k return val else: from scipy import special n = asarray(n) vals = zeros(n.shape,'d') cond1 = (n % 2) & (n >= -1) cond2 = (1-(n % 2)) & (n >= -1) oddn = extract(cond1,n) evenn = extract(cond2,n) nd2o = oddn / 2.0 nd2e = evenn / 2.0 place(vals,cond1,special.gamma(nd2o+1)/sqrt(pi)*pow(2.0,nd2o+0.5)) place(vals,cond2,special.gamma(nd2e+1) * pow(2.0,nd2e)) return vals def factorialk(n,k,exact=1): """n(!!...!) = multifactorial of order k k times """ if exact: if n < 1-k: return 0L if n<=0: return 1L val = 1L for j in xrange(n,0,-k): val = val*j return val else: raise NotImplementedError def comb(N,k,exact=0): """Combinations of N things taken k at a time. If exact==0, then floating point precision is used, otherwise exact long integer is computed. Notes: - Array arguments accepted only for exact=0 case. - If k > N, N < 0, or k < 0, then a 0 is returned. """ if exact: if (k > N) or (N < 0) or (k < 0): return 0L val = 1L for j in xrange(min(k, N-k)): val = (val*(N-j))//(j+1) return val else: from scipy import special k,N = asarray(k), asarray(N) lgam = special.gammaln cond = (k <= N) & (N >= 0) & (k >= 0) sv = special.errprint(0) vals = exp(lgam(N+1) - lgam(N-k+1) - lgam(k+1)) sv = special.errprint(sv) return where(cond, vals, 0.0) def central_diff_weights(Np,ndiv=1): """Return weights for an Np-point central derivative of order ndiv assuming 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) Can be inaccurate for large number of points. """ assert (Np >= ndiv+1), "Number of points must be at least the derivative order + 1." assert (Np % 2 == 1), "Odd-number of points only." 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): """Given a function, use a central difference formula with spacing dx to compute the nth derivative at x0. order is the number of points to use and must be odd. Warning: Decreasing the step size too small can result in round-off error. """ assert (order >= n+1), "Number of points must be at least the derivative order + 1." assert (order % 2 == 1), "Odd number of points only." # 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 pade(an, m): """Given Taylor series coefficients in an, return a Pade approximation to the function as the ratio of two polynomials p / q where the order of q is m. """ from scipy import linalg an = asarray(an) N = len(an) - 1 n = N-m if (n < 0): raise ValueError, \ "Order of q must be smaller than len(an)-1." Akj = eye(N+1,n+1) Bkj = zeros((N+1,m),'d') for row in range(1,m+1): Bkj[row,:row] = -(an[:row])[::-1] for row in range(m+1,N+1): Bkj[row,:] = -(an[row-m:row])[::-1] C = hstack((Akj,Bkj)) pq = dot(linalg.inv(C),an) p = pq[:n+1] q = r_[1.0,pq[n+1:]] return poly1d(p[::-1]), poly1d(q[::-1]) def lena(): import cPickle, os fname = os.path.join(os.path.dirname(__file__),'lena.dat') f = open(fname,'rb') lena = array(cPickle.load(f)) f.close() return lena
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