"""
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 <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__),'lena.dat')
    f = open(fname,'rb')
    lena = array(cPickle.load(f))
    f.close()
    return lena