path: root/numpy/lib/function_base.py

__all__ = ['logspace', 'linspace',
           'select', 'piecewise', 'trim_zeros',
           'copy', 'iterable', #'base_repr', 'binary_repr',
           'diff', 'gradient', 'angle', 'unwrap', 'sort_complex', 'disp',
           'unique', 'extract', 'insert', 'nansum', 'nanmax', 'nanargmax',
           'nanargmin', 'nanmin', 'vectorize', 'asarray_chkfinite', 'average',
           'histogram', 'bincount', 'digitize', 'cov', 'corrcoef', 'msort',
           'median', 'sinc', 'hamming', 'hanning', 'bartlett', 'blackman',
           'kaiser', 'trapz', 'i0', 'add_newdoc', 'add_docstring', 'meshgrid'
           ]

import types
import numpy.core.numeric as _nx
from numpy.core.numeric import ones, zeros, arange, concatenate, array, \
     asarray, asanyarray, empty, empty_like, asanyarray, ndarray
from numpy.core.numeric import ScalarType, dot, where, newaxis
from numpy.core.umath import pi, multiply, add, arctan2,  \
     frompyfunc, isnan, cos, less_equal, sqrt, sin, mod, exp
from numpy.core.fromnumeric import ravel, nonzero, choose, sort
from numpy.core.numerictypes import typecodes
from numpy.lib.shape_base import atleast_1d
from numpy.lib.twodim_base import diag
from _compiled_base import _insert, add_docstring
from _compiled_base import digitize, bincount

#end Fernando's utilities

def linspace(start, stop, num=50, endpoint=True, retstep=False):
    """Return evenly spaced numbers.

    Return num evenly spaced samples from start to stop.  If
    endpoint is True, the last sample is stop. If retstep is
    True then return the step value used.
    """
    num = int(num)
    if num <= 0:
        return array([], float)
    if endpoint:
        if num == 1:
            return array([float(start)])
        step = (stop-start)/float((num-1))
    else:
        step = (stop-start)/float(num)
    y = _nx.arange(0, num) * step + start
    if retstep:
        return y, step
    else:
        return y

def logspace(start,stop,num=50,endpoint=True,base=10.0):
    """Evenly spaced numbers on a logarithmic scale.

    Computes int(num) evenly spaced exponents from base**start to
    base**stop. If endpoint=True, then last number is base**stop
    """
    y = linspace(start,stop,num=num,endpoint=endpoint)
    return _nx.power(base,y)

def iterable(y):
    try: iter(y)
    except: return 0
    return 1

def histogram(a, bins=10, range=None, normed=False):
    """histogram(sample, bins = 10, range = None, normed = False) -> H, ledges

    Return the distribution of a sample.

    Parameters
    ----------
    bins: Number of bins
    range: Lower and upper bin edges (default: [sample.min(), sample.max()]).
        Does not really work, all values greater than range are stored in
        the last bin.
    normed: If False (default), return the number of samples in each bin.
        If True, return a frequency distribution.

    Output
    ------
    histogram array, left bin edges array.
    """
    a = asarray(a).ravel()
    if not iterable(bins):
        if range is None:
            range = (a.min(), a.max())
        mn, mx = [mi+0.0 for mi in range]
        if mn == mx:
            mn -= 0.5
            mx += 0.5
        bins = linspace(mn, mx, bins, endpoint=False)

    n = sort(a).searchsorted(bins)
    n = concatenate([n, [len(a)]])
    n = n[1:]-n[:-1]

    if normed:
        db = bins[1] - bins[0]
        return 1.0/(a.size*db) * n, bins
    else:
        return n, bins

def average(a, axis=0, weights=None, returned=False):
    """average(a, axis=0, weights=None, returned=False)

    Average the array over the given axis.  If the axis is None, average
    over all dimensions of the array.  Equivalent to a.mean(axis), but
    with a default axis of 0 instead of None.

    If an integer axis is given, this equals:
        a.sum(axis) * 1.0 / len(a)

    If axis is None, this equals:
        a.sum(axis) * 1.0 / product(a.shape)

    If weights are given, result is:
        sum(a * weights) / sum(weights),
    where the weights must have a's shape or be 1D with length the
    size of a in the given axis. Integer weights are converted to
    Float.  Not specifying weights is equivalent to specifying
    weights that are all 1.

    If 'returned' is True, return a tuple: the result and the sum of
    the weights or count of values. The shape of these two results
    will be the same.

    Raises ZeroDivisionError if appropriate.  (The version in MA does
    not -- it returns masked values).
    """
    if axis is None:
        a = array(a).ravel()
        if weights is None:
            n = add.reduce(a)
            d = len(a) * 1.0
        else:
            w = array(weights).ravel() * 1.0
            n = add.reduce(multiply(a, w))
            d = add.reduce(w)
    else:
        a = array(a)
        ash = a.shape
        if ash == ():
            a.shape = (1,)
        if weights is None:
            n = add.reduce(a, axis)
            d = ash[axis] * 1.0
            if returned:
                d = ones(n.shape) * d
        else:
            w = array(weights, copy=False) * 1.0
            wsh = w.shape
            if wsh == ():
                wsh = (1,)
            if wsh == ash:
                n = add.reduce(a*w, axis)
                d = add.reduce(w, axis)
            elif wsh == (ash[axis],):
                ni = ash[axis]
                r = [newaxis]*ni
                r[axis] = slice(None, None, 1)
                w1 = eval("w["+repr(tuple(r))+"]*ones(ash, Float)")
                n = add.reduce(a*w1, axis)
                d = add.reduce(w1, axis)
            else:
                raise ValueError, 'averaging weights have wrong shape'

    if not isinstance(d, ndarray):
        if d == 0.0:
            raise ZeroDivisionError, 'zero denominator in average()'
    if returned:
        return n/d, d
    else:
        return n/d

def asarray_chkfinite(a):
    """Like asarray, but check that no NaNs or Infs are present.
    """
    a = asarray(a)
    if (a.dtype.char in typecodes['AllFloat']) \
           and (_nx.isnan(a).any() or _nx.isinf(a).any()):
        raise ValueError, "array must not contain infs or NaNs"
    return a

def piecewise(x, condlist, funclist, *args, **kw):
    """Return a piecewise-defined function.

    x is the domain

    condlist is a list of boolean arrays or a single boolean array
      The length of the condition list must be n2 or n2-1 where n2
      is the length of the function list.  If len(condlist)==n2-1, then
      an 'otherwise' condition is formed by |'ing all the conditions
      and inverting.

    funclist is a list of functions to call of length (n2).
      Each function should return an array output for an array input
      Each function can take (the same set) of extra arguments and
      keyword arguments which are passed in after the function list.
      A constant may be used in funclist for a function that returns a
      constant (e.g. val  and lambda x: val are equivalent in a funclist).

    The output is the same shape and type as x and is found by
      calling the functions on the appropriate portions of x.

    Note: This is similar to choose or select, except
          the the functions are only evaluated on elements of x
          that satisfy the corresponding condition.

    The result is
           |--
           |  f1(x)  for condition1
     y = --|  f2(x)  for condition2
           |   ...
           |  fn(x)  for conditionn
           |--

    """
    x = asanyarray(x)
    n2 = len(funclist)
    if not isinstance(condlist, type([])):
        condlist = [condlist]
    n = len(condlist)
    if n == n2-1:  # compute the "otherwise" condition.
        totlist = condlist[0]
        for k in range(1, n):
            totlist |= condlist[k]
        condlist.append(~totlist)
        n += 1
    if (n != n2):
        raise ValueError, "function list and condition list must be the same"
    y = empty(x.shape, x.dtype)
    for k in range(n):
        item = funclist[k]
        if not callable(item):
            y[condlist[k]] = item
        else:
            y[condlist[k]] = item(x[condlist[k]], *args, **kw)
    return y

def select(condlist, choicelist, default=0):
    """ Return an array composed of different elements of choicelist
        depending on the list of conditions.

        condlist is a list of condition arrays containing ones or zeros

        choicelist is a list of choice arrays (of the "same" size as the
        arrays in condlist).  The result array has the "same" size as the
        arrays in choicelist.  If condlist is [c0, ..., cN-1] then choicelist
        must be of length N.  The elements of the choicelist can then be
        represented as [v0, ..., vN-1]. The default choice if none of the
        conditions are met is given as the default argument.

        The conditions are tested in order and the first one statisfied is
        used to select the choice. In other words, the elements of the
        output array are found from the following tree (notice the order of
        the conditions matters):

        if c0: v0
        elif c1: v1
        elif c2: v2
        ...
        elif cN-1: vN-1
        else: default

        Note that one of the condition arrays must be large enough to handle
        the largest array in the choice list.
    """
    n = len(condlist)
    n2 = len(choicelist)
    if n2 != n:
        raise ValueError, "list of cases must be same length as list of conditions"
    choicelist.insert(0, default)
    S = 0
    pfac = 1
    for k in range(1, n+1):
        S += k * pfac * asarray(condlist[k-1])
        if k < n:
            pfac *= (1-asarray(condlist[k-1]))
    # handle special case of a 1-element condition but
    #  a multi-element choice
    if type(S) in ScalarType or max(asarray(S).shape)==1:
        pfac = asarray(1)
        for k in range(n2+1):
            pfac = pfac + asarray(choicelist[k])
        S = S*ones(asarray(pfac).shape)
    return choose(S, tuple(choicelist))

def _asarray1d(arr, copy=False):
    """Ensure 1D array for one array.
    """
    if copy:
        return asarray(arr).flatten()
    else:
        return asarray(arr).ravel()

def copy(a):
    """Return an array copy of the given object.
    """
    return array(a, copy=True)

# Basic operations

def gradient(f, *varargs):
    """Calculate the gradient of an N-dimensional scalar function.

    Uses central differences on the interior and first differences on boundaries
    to give the same shape.

    Inputs:

      f -- An N-dimensional array giving samples of a scalar function

      varargs -- 0, 1, or N scalars giving the sample distances in each direction

    Outputs:

      N arrays of the same shape as f giving the derivative of f with respect
       to each dimension.
    """
    N = len(f.shape)  # number of dimensions
    n = len(varargs)
    if n == 0:
        dx = [1.0]*N
    elif n == 1:
        dx = [varargs[0]]*N
    elif n == N:
        dx = list(varargs)
    else:
        raise SyntaxError, "invalid number of arguments"

    # use central differences on interior and first differences on endpoints

    outvals = []

    # create slice objects --- initially all are [:, :, ..., :]
    slice1 = [slice(None)]*N
    slice2 = [slice(None)]*N
    slice3 = [slice(None)]*N

    otype = f.dtype.char
    if otype not in ['f', 'd', 'F', 'D']:
        otype = 'd'

    for axis in range(N):
        # select out appropriate parts for this dimension
        out = zeros(f.shape, f.dtype.char)
        slice1[axis] = slice(1, -1)
        slice2[axis] = slice(2, None)
        slice3[axis] = slice(None, -2)
        # 1D equivalent -- out[1:-1] = (f[2:] - f[:-2])/2.0
        out[slice1] = (f[slice2] - f[slice3])/2.0
        slice1[axis] = 0
        slice2[axis] = 1
        slice3[axis] = 0
        # 1D equivalent -- out[0] = (f[1] - f[0])
        out[slice1] = (f[slice2] - f[slice3])
        slice1[axis] = -1
        slice2[axis] = -1
        slice3[axis] = -2
        # 1D equivalent -- out[-1] = (f[-1] - f[-2])
        out[slice1] = (f[slice2] - f[slice3])

        # divide by step size
        outvals.append(out / dx[axis])

        # reset the slice object in this dimension to ":"
        slice1[axis] = slice(None)
        slice2[axis] = slice(None)
        slice3[axis] = slice(None)

    if N == 1:
        return outvals[0]
    else:
        return outvals


def diff(a, n=1, axis=-1):
    """Calculate the nth order discrete difference along given axis.
    """
    if n == 0:
        return a
    if n < 0:
        raise ValueError, 'order must be non-negative but got ' + repr(n)
    a = asanyarray(a)
    nd = len(a.shape)
    slice1 = [slice(None)]*nd
    slice2 = [slice(None)]*nd
    slice1[axis] = slice(1, None)
    slice2[axis] = slice(None, -1)
    slice1 = tuple(slice1)
    slice2 = tuple(slice2)
    if n > 1:
        return diff(a[slice1]-a[slice2], n-1, axis=axis)
    else:
        return a[slice1]-a[slice2]

add_docstring(digitize, 
   r"""(x,bins) --> index of the bin to which each value of x belongs.
   
    Each index i returned is such that bins[i-1] <= x < bins[i] if
    bins is monotonically increasing, or bins [i-1] > x >= bins[i] if
    bins is monotonically decreasing.

    Beyond the bounds of the bins 0 or len(bins) is returned as appropriate.
    """)

add_docstring(bincount,
   r"""(x,weights=None) --> the number of occurrences of each value in x.

    x must be a list of non-negative integers.  The output, b[i],
    represents the number of times that i is found in x.  If weights
    is specified, every occurrence of i at a position p contributes
    weights[p] instead of 1.

    See also: histogram, digitize, unique.
    """)

add_docstring(add_docstring,
   r"""(obj, docstring) --> None

   Add a docstring to a built-in obj if possible.
   If the obj already has a docstring raise a RuntimeError
   If this routine does not know how to add a docstring to the object
      raise a TypeError
   """)
    
def angle(z, deg=0):
    """Return the angle of the complex argument z.
    """
    if deg:
        fact = 180/pi
    else:
        fact = 1.0
    z = asarray(z)
    if (issubclass(z.dtype.type, _nx.complexfloating)):
        zimag = z.imag
        zreal = z.real
    else:
        zimag = 0
        zreal = z
    return arctan2(zimag, zreal) * fact

def unwrap(p, discont=pi, axis=-1):
    """Unwrap radian phase p by changing absolute jumps greater than
       'discont' to their 2*pi complement along the given axis.
    """
    p = asarray(p)
    nd = len(p.shape)
    dd = diff(p, axis=axis)
    slice1 = [slice(None, None)]*nd     # full slices
    slice1[axis] = slice(1, None)
    ddmod = mod(dd+pi, 2*pi)-pi
    _nx.putmask(ddmod, (ddmod==-pi) & (dd > 0), pi)
    ph_correct = ddmod - dd;
    _nx.putmask(ph_correct, abs(dd)<discont, 0)
    up = array(p, copy=True, dtype='d')
    up[slice1] = p[slice1] + ph_correct.cumsum(axis)
    return up

def sort_complex(a):
    """ Sort 'a' as a complex array using the real part first and then
    the imaginary part if the real part is equal (the default sort order
    for complex arrays).  This function is a wrapper ensuring a complex
    return type.
    """
    b = array(a,copy=True)
    b.sort()
    if not issubclass(b.dtype.type, _nx.complexfloating):
        if b.dtype.char in 'bhBH':
            return b.astype('F')
        elif b.dtype.char == 'g':
            return b.astype('G')
        else:
            return b.astype('D')
    else:
        return b

def trim_zeros(filt, trim='fb'):
    """ Trim the leading and trailing zeros from a 1D array.

    Example:
        >>> import numpy
        >>> a = array((0, 0, 0, 1, 2, 3, 2, 1, 0))
        >>> numpy.trim_zeros(a)
        array([1, 2, 3, 2, 1])
    """
    first = 0
    trim = trim.upper()
    if 'F' in trim:
        for i in filt:
            if i != 0.: break
            else: first = first + 1
    last = len(filt)
    if 'B' in trim:
        for i in filt[::-1]:
            if i != 0.: break
            else: last = last - 1
    return filt[first:last]


import sys
if sys.hexversion < 0x2040000:
   from sets import Set as set

def unique(x):
    """Return sorted unique items from an array or sequence.

    Example:
    >>> unique([5,2,4,0,4,4,2,2,1])
    array([0,1,2,4,5])
    """
    try:
        tmp = x.flatten()
        if tmp.size == 0:
            return tmp
        tmp.sort()
        idx = concatenate(([True],tmp[1:]!=tmp[:-1]))
        return tmp[idx]
    except AttributeError:
        items = list(set(x))
        items.sort()
        return asarray(items)
        
def extract(condition, arr):
    """Return the elements of ravel(arr) where ravel(condition) is True
    (in 1D).

    Equivalent to compress(ravel(condition), ravel(arr)).
    """
    return _nx.take(ravel(arr), nonzero(ravel(condition)))

def insert(arr, mask, vals):
    """Similar to putmask arr[mask] = vals but the 1D array vals has the
    same number of elements as the non-zero values of mask. Inverse of
    extract.
    """
    return _insert(arr, mask, vals)

def nansum(a, axis=-1):
    """Sum the array over the given axis, treating NaNs as 0.
    """
    y = array(a)
    if not issubclass(y.dtype.type, _nx.integer):
        y[isnan(a)] = 0
    return y.sum(axis)

def nanmin(a, axis=-1):
    """Find the minimium over the given axis, ignoring NaNs.
    """
    y = array(a)
    if not issubclass(y.dtype.type, _nx.integer):
        y[isnan(a)] = _nx.inf
    return y.min(axis)

def nanargmin(a, axis=-1):
    """Find the indices of the minimium over the given axis ignoring NaNs.
    """
    y = array(a)
    if not issubclass(y.dtype.type, _nx.integer):
        y[isnan(a)] = _nx.inf
    return y.argmin(axis)

def nanmax(a, axis=-1):
    """Find the maximum over the given axis ignoring NaNs.
    """
    y = array(a)
    if not issubclass(y.dtype.type, _nx.integer):
        y[isnan(a)] = -_nx.inf
    return y.max(axis)

def nanargmax(a, axis=-1):
    """Find the maximum over the given axis ignoring NaNs.
    """
    y = array(a)
    if not issubclass(y.dtype.type, _nx.integer):
        y[isnan(a)] = -_nx.inf
    return y.argmax(axis)

def disp(mesg, device=None, linefeed=True):
    """Display a message to the given device (default is sys.stdout)
    with or without a linefeed.
    """
    if device is None:
        import sys
        device = sys.stdout
    if linefeed:
        device.write('%s\n' % mesg)
    else:
        device.write('%s' % mesg)
    device.flush()
    return

# return number of input arguments and
#  number of default arguments
import re
def _get_nargs(obj):
    if not callable(obj):
        raise TypeError, "Object is not callable."
    if hasattr(obj,'func_code'):
        fcode = obj.func_code
        nargs = fcode.co_argcount
        if obj.func_defaults is not None:
            ndefaults = len(obj.func_defaults)
        else:
            ndefaults = 0
        if isinstance(obj, types.MethodType):
            nargs -= 1
        return nargs, ndefaults
    terr = re.compile(r'.*? takes exactly (?P<exargs>\d+) argument(s|) \((?P<gargs>\d+) given\)')
    try:
        obj()
        return 0, 0
    except TypeError, msg:
        m = terr.match(str(msg))
        if m:
            nargs = int(m.group('exargs'))
            ndefaults = int(m.group('gargs'))
            if isinstance(obj, types.MethodType):
                nargs -= 1
            return nargs, ndefaults
    raise ValueError, 'failed to determine the number of arguments for %s' % (obj)


class vectorize(object):
    """
 vectorize(somefunction, otypes=None, doc=None)
 Generalized Function class.

  Description:

    Define a vectorized function which takes nested sequence
    objects or numpy arrays as inputs and returns a
    numpy array as output, evaluating the function over successive
    tuples of the input arrays like the python map function except it uses
    the broadcasting rules of numpy.

  Input:

    somefunction -- a Python function or method

  Example:

    def myfunc(a, b):
        if a > b:
            return a-b
        else
            return a+b

    vfunc = vectorize(myfunc)

    >>> vfunc([1, 2, 3, 4], 2)
    array([3, 4, 1, 2])

    """
    def __init__(self, pyfunc, otypes='', doc=None):
        self.thefunc = pyfunc
        self.ufunc = None
        nin, ndefault = _get_nargs(pyfunc)
        if nin == 0 and ndefault == 0:
            self.nin = None
            self.nin_wo_defaults = None
        else:
            self.nin = nin
            self.nin_wo_defaults = nin - ndefault
        self.nout = None
        if doc is None:
            self.__doc__ = pyfunc.__doc__
        else:
            self.__doc__ = doc
        if isinstance(otypes, types.StringType):
            self.otypes = otypes
        else:
            raise ValueError, "output types must be a string"
        for char in self.otypes:
            if char not in typecodes['All']:
                raise ValueError, "invalid typecode specified"
        self.lastcallargs = 0

    def __call__(self, *args):
        # get number of outputs and output types by calling
        #  the function on the first entries of args
        nargs = len(args)
        if self.nin:
            if (nargs > self.nin) or (nargs < self.nin_wo_defaults):
                raise ValueError, "mismatch between python function inputs"\
                      " and received arguments"
        if self.nout is None or self.otypes == '':
            newargs = []
            for arg in args:
                newargs.append(asarray(arg).flat[0])
            theout = self.thefunc(*newargs)
            if isinstance(theout, types.TupleType):
                self.nout = len(theout)
            else:
                self.nout = 1
                theout = (theout,)
            if self.otypes == '':
                otypes = []
                for k in range(self.nout):
                    otypes.append(asarray(theout[k]).dtype.char)
                self.otypes = ''.join(otypes)

        if (self.ufunc is None) or (self.lastcallargs != nargs):
            self.ufunc = frompyfunc(self.thefunc, nargs, self.nout)
            self.lastcallargs = nargs

        if self.nout == 1:
            return array(self.ufunc(*args),copy=False).astype(self.otypes[0])
        else:
            return tuple([array(x,copy=False).astype(c) \
                          for x, c in zip(self.ufunc(*args), self.otypes)])

def cov(m,y=None, rowvar=1, bias=0):
    """Estimate the covariance matrix.

    If m is a vector, return the variance.  For matrices return the
    covariance matrix.

    If y is given it is treated as an additional (set of)
    variable(s). 

    Normalization is by (N-1) where N is the number of observations
    (unbiased estimate).  If bias is 1 then normalization is by N.

    If rowvar is non-zero (default), then each row is a variable with
    observations in the columns, otherwise each column
    is a variable and the observations are in the rows.
    """

    X = array(m,ndmin=2)
    if X.shape[0] == 1:
        rowvar = 1
    if rowvar:
        axis = 0
        tup = (slice(None),newaxis)
    else:
        axis = 1
        tup = (newaxis, slice(None))

        
    if y is not None:
        y = array(y,copy=False,ndmin=2)
        X = concatenate((X,y),axis)

    X -= X.mean(axis=1-axis)[tup]
    if rowvar:
        N = X.shape[1]
    else:
        N = X.shape[0]

    if bias:
        fact = N*1.0
    else:
        fact = N-1.0

    if not rowvar:
        return (dot(X.transpose(), X.conj()) / fact).squeeze()
    else:
        return (dot(X,X.transpose().conj())/fact).squeeze()
    
def corrcoef(x, y=None, rowvar=1, bias=0):
    """The correlation coefficients
    """
    c = cov(x, y, rowvar, bias)
    try:
        d = diag(c)
    except ValueError: # scalar covariance
        return 1
    return c/sqrt(multiply.outer(d,d))

def blackman(M):
    """blackman(M) returns the M-point Blackman window.
    """
    n = arange(0,M)
    return 0.42-0.5*cos(2.0*pi*n/(M-1)) + 0.08*cos(4.0*pi*n/(M-1))

def bartlett(M):
    """bartlett(M) returns the M-point Bartlett window.
    """
    n = arange(0,M)
    return where(less_equal(n,(M-1)/2.0),2.0*n/(M-1),2.0-2.0*n/(M-1))

def hanning(M):
    """hanning(M) returns the M-point Hanning window.
    """
    n = arange(0,M)
    return 0.5-0.5*cos(2.0*pi*n/(M-1))

def hamming(M):
    """hamming(M) returns the M-point Hamming window.
    """
    n = arange(0,M)
    return 0.54-0.46*cos(2.0*pi*n/(M-1))

## Code from cephes for i0

_i0A = [
-4.41534164647933937950E-18,
 3.33079451882223809783E-17,
-2.43127984654795469359E-16,
 1.71539128555513303061E-15,
-1.16853328779934516808E-14,
 7.67618549860493561688E-14,
-4.85644678311192946090E-13,
 2.95505266312963983461E-12,
-1.72682629144155570723E-11,
 9.67580903537323691224E-11,
-5.18979560163526290666E-10,
 2.65982372468238665035E-9,
-1.30002500998624804212E-8,
 6.04699502254191894932E-8,
-2.67079385394061173391E-7,
 1.11738753912010371815E-6,
-4.41673835845875056359E-6,
 1.64484480707288970893E-5,
-5.75419501008210370398E-5,
 1.88502885095841655729E-4,
-5.76375574538582365885E-4,
 1.63947561694133579842E-3,
-4.32430999505057594430E-3,
 1.05464603945949983183E-2,
-2.37374148058994688156E-2,
 4.93052842396707084878E-2,
-9.49010970480476444210E-2,
 1.71620901522208775349E-1,
-3.04682672343198398683E-1,
 6.76795274409476084995E-1]

_i0B = [
-7.23318048787475395456E-18,
-4.83050448594418207126E-18,
 4.46562142029675999901E-17,
 3.46122286769746109310E-17,
-2.82762398051658348494E-16,
-3.42548561967721913462E-16,
 1.77256013305652638360E-15,
 3.81168066935262242075E-15,
-9.55484669882830764870E-15,
-4.15056934728722208663E-14,
 1.54008621752140982691E-14,
 3.85277838274214270114E-13,
 7.18012445138366623367E-13,
-1.79417853150680611778E-12,
-1.32158118404477131188E-11,
-3.14991652796324136454E-11,
 1.18891471078464383424E-11,
 4.94060238822496958910E-10,
 3.39623202570838634515E-9,
 2.26666899049817806459E-8,
 2.04891858946906374183E-7,
 2.89137052083475648297E-6,
 6.88975834691682398426E-5,
 3.36911647825569408990E-3,
 8.04490411014108831608E-1]

def _chbevl(x, vals):
    b0 = vals[0]
    b1 = 0.0

    for i in xrange(1,len(vals)):
        b2 = b1
        b1 = b0
        b0 = x*b1 - b2 + vals[i]

    return 0.5*(b0 - b2)

def _i0_1(x):
    return exp(x) * _chbevl(x/2.0-2, _i0A)

def _i0_2(x):
    return exp(x) * _chbevl(32.0/x - 2.0, _i0B) / sqrt(x)

def i0(x):
    x = atleast_1d(x).copy()
    y = empty_like(x)
    ind = (x<0)
    x[ind] = -x[ind]
    ind = (x<=8.0)
    y[ind] = _i0_1(x[ind])
    ind2 = ~ind
    y[ind2] = _i0_2(x[ind2])
    return y.squeeze()

## End of cephes code for i0

def kaiser(M,beta):
    """kaiser(M, beta) returns a Kaiser window of length M with shape parameter
    beta. It depends on numpy.special (in full numpy) for the modified bessel
    function i0.
    """
    from numpy.dual import i0
    n = arange(0,M)
    alpha = (M-1)/2.0
    return i0(beta * sqrt(1-((n-alpha)/alpha)**2.0))/i0(beta)

def sinc(x):
    """sinc(x) returns sin(pi*x)/(pi*x) at all points of array x.
    """
    y = pi* where(x == 0, 1.0e-20, x)
    return sin(y)/y

def msort(a):
    b = array(a,subok=True,copy=True)
    b.sort(0)
    return b

def median(m):
    """median(m) returns a median of m along the first dimension of m.
    """
    sorted = msort(m)
    if sorted.shape[0] % 2 == 1:
        return sorted[int(sorted.shape[0]/2)]
    else:
        sorted = msort(m)
        index = sorted.shape[0]/2
        return (sorted[index-1]+sorted[index])/2.0

def trapz(y, x=None, dx=1.0, axis=-1):
    """Integrate y(x) using samples along the given axis and the composite
    trapezoidal rule.  If x is None, spacing given by dx is assumed.
    """
    y = asarray(y)
    if x is None:
        d = dx
    else:
        d = diff(x,axis=axis)
    nd = len(y.shape)
    slice1 = [slice(None)]*nd
    slice2 = [slice(None)]*nd
    slice1[axis] = slice(1,None)
    slice2[axis] = slice(None,-1)
    return add.reduce(d * (y[slice1]+y[slice2])/2.0,axis)

#always succeed
def add_newdoc(place, obj, doc):
    """Adds documentation to obj which is in module place. 

    If doc is a string add it to obj as a docstring

    If doc is a tuple, then the first element is interpreted as
       an attribute of obj and the second as the docstring
          (method, docstring)
          
    If doc is a list, then each element of the list should be a
       sequence of length two --> [(method1, docstring1),
       (method2, docstring2), ...]

    This routine never raises an error. 
       """
    try:
        new = {}
        exec 'from %s import %s' % (place, obj) in new
        if isinstance(doc, str):
            add_docstring(new[obj], doc.strip())
        elif isinstance(doc, tuple):
            add_docstring(getattr(new[obj], doc[0]), doc[1].strip())
        elif isinstance(doc, list):
            for val in doc:
                add_docstring(getattr(new[obj], val[0]), val[1].strip())
    except:
        pass


# From matplotlib
def meshgrid(x,y):
    """
    For vectors x, y with lengths Nx=len(x) and Ny=len(y), return X, Y
    where X and Y are (Ny, Nx) shaped arrays with the elements of x
    and y repeated to fill the matrix

    EG,

      [X, Y] = meshgrid([1,2,3], [4,5,6,7])

       X =
         1   2   3
         1   2   3
         1   2   3
         1   2   3


       Y =
         4   4   4
         5   5   5
         6   6   6
         7   7   7
  """
    x = asarray(x)
    y = asarray(y)
    numRows, numCols = len(y), len(x)  # yes, reversed
    x = x.reshape(1,numCols)
    X = x.repeat(numRows, axis=0)
    
    y = y.reshape(numRows,1)
    Y = y.repeat(numCols, axis=1)
    return X, Y