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common.py
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common.py
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# Functions which are common and require SciPy Base and Level 1 SciPy
# (stats, special, linalg)
# Needs eigenvalues
import sys
import types
from scipy import special, stats, linalg
import scipy.base as Numeric
from scipy.base import exp, amin, amax, ravel, asarray, cast, arange, \
ones, NewAxis, transpose, hstack, product, array, typename, where, \
zeros, extract, insert, pi, sqrt, eye, poly1d, dot, r_
__all__ = ['factorial','factorial2','factorialk','comb','who',
'lena','central_diff_weights', 'derivative', 'pade']
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
n = long(n)
val = 1L
k = 1L
while (k < n+1L):
val = val*k
k += 1
return val
else:
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
n = long(n)
val = 1L
k = n
while (k > 0):
val = val*k
k -= 2
return val
else:
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
insert(vals,cond1,special.gamma(nd2o+1)/sqrt(pi)*pow(2.0,nd2o+0.5))
insert(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
n = long(n)
val = 1L
j = n
while (j > 0):
val = val*j
j -= k
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
N,k = map(long,(N,k))
top = N
val = 1L
while (top > (N-k)):
val *= top
top -= 1
n = 1L
while (n < k+1L):
val /= n
n += 1
return val
else:
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."
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))*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)
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.
"""
an = asarray(an)
N = len(an) - 1
n = N-m
if (n < 0):
raise ValueError, \
"Order of q <m> 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__),'plt','lena.dat')
f = open(fname,'rb')
lena = array(cPickle.load(f))
f.close()
return lena
#-----------------------------------------------------------------------------
# Matlab like functions for output and information on the variables used.
#-----------------------------------------------------------------------------
def who(vardict=None):
"""Print the Numeric arrays in the given dictionary (or globals() if None).
"""
if vardict is None:
frame = sys._getframe().f_back
vardict = frame.f_globals
sta = []
cache = {}
for name in vardict.keys():
if isinstance(vardict[name],Numeric.ArrayType):
var = vardict[name]
idv = id(var)
if idv in cache.keys():
namestr = name + " (%s)" % cache[idv]
original=0
else:
cache[idv] = name
namestr = name
original=1
shapestr = " x ".join(map(str, var.shape))
bytestr = str(var.itemsize()*Numeric.product(var.shape))
sta.append([namestr, shapestr, bytestr, typename(var.typecode()), original])
maxname = 0
maxshape = 0
maxbyte = 0
totalbytes = 0
for k in range(len(sta)):
val = sta[k]
if maxname < len(val[0]):
maxname = len(val[0])
if maxshape < len(val[1]):
maxshape = len(val[1])
if maxbyte < len(val[2]):
maxbyte = len(val[2])
if val[4]:
totalbytes += int(val[2])
max = Numeric.maximum
if len(sta) > 0:
sp1 = max(10,maxname)
sp2 = max(10,maxshape)
sp3 = max(10,maxbyte)
prval = "Name %s Shape %s Bytes %s Type" % (sp1*' ', sp2*' ', sp3*' ')
print prval + "\n" + "="*(len(prval)+5) + "\n"
for k in range(len(sta)):
val = sta[k]
print "%s %s %s %s %s %s %s" % (val[0], ' '*(sp1-len(val[0])+4),
val[1], ' '*(sp2-len(val[1])+5),
val[2], ' '*(sp3-len(val[2])+5),
val[3])
print "\nUpper bound on total bytes = %d" % totalbytes
return
#-----------------------------------------------------------------------------
def test(level=10):
from scipy_test.testing import module_test
module_test(__name__,__file__,level=level)
def test_suite(level=1):
from scipy_test.testing import module_test_suite
return module_test_suite(__name__,__file__,level=level)