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polynomial.py

import Numeric
from Numeric import *
from scimath import *

from type_check import isscalar, asarray
from matrix_base import diag
from shape_base import hstack, atleast_1d
from function_base import trim_zeros, sort_complex

__all__ = ['poly','roots','polyint','polyder','polyadd','polysub','polymul',
           'polydiv','polyval','poly1d']
 
def get_eigval_func():
    try:
        import scipy.linalg
        eigvals = scipy.linalg.eigvals
    except ImportError:
        try:
            import linalg
            eigvals = linalg.eigvals
        except ImportError:
            try:
                import LinearAlgebra
                eigvals = LinearAlgebra.eigenvalues
            except:
                raise ImportError, \
                      "You must have scipy.linalg or LinearAlgebra to "\
                      "use this function."
    return eigvals

def poly(seq_of_zeros):
    """ Return a sequence representing a polynomial given a sequence of roots.

        If the input is a matrix, return the characteristic polynomial.
    
        Example:
    
         >>> b = roots([1,3,1,5,6])
         >>> poly(b)
         array([1., 3., 1., 5., 6.])
    """
    seq_of_zeros = atleast_1d(seq_of_zeros)    
    sh = shape(seq_of_zeros)
    if len(sh) == 2 and sh[0] == sh[1]:
        eig = get_eigval_func()
        seq_of_zeros=eig(seq_of_zeros)
    elif len(sh) ==1:
        pass
    else:
        raise ValueError, "input must be 1d or square 2d array."

    if len(seq_of_zeros) == 0:
        return 1.0

    a = [1]
    for k in range(len(seq_of_zeros)):
        a = convolve(a,[1, -seq_of_zeros[k]], mode=2)

        
    if a.typecode() in ['F','D']:
        # if complex roots are all complex conjugates, the roots are real.
        roots = asarray(seq_of_zeros,'D')
        pos_roots = sort_complex(compress(roots.imag > 0,roots))
        neg_roots = conjugate(sort_complex(compress(roots.imag < 0,roots)))
        if (len(pos_roots) == len(neg_roots) and
            alltrue(neg_roots == pos_roots)):
            a = a.real.copy()

    return a

def roots(p):
    """ Return the roots of the polynomial coefficients in p.

        The values in the rank-1 array p are coefficients of a polynomial.
        If the length of p is n+1 then the polynomial is
        p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n]
    """
    # If input is scalar, this makes it an array
    eig = get_eigval_func()
    p = atleast_1d(p)
    if len(p.shape) != 1:
        raise ValueError,"Input must be a rank-1 array."
        
    # find non-zero array entries
    non_zero = nonzero(ravel(p))

    # find the number of trailing zeros -- this is the number of roots at 0.
    trailing_zeros = len(p) - non_zero[-1] - 1

    # strip leading and trailing zeros
    p = p[int(non_zero[0]):int(non_zero[-1])+1]
    
    # casting: if incoming array isn't floating point, make it floating point.
    if p.typecode() not in ['f','d','F','D']:
        p = p.astype('d')

    N = len(p)
    if N > 1:
        # build companion matrix and find its eigenvalues (the roots)
        A = diag(ones((N-2,),p.typecode()),-1)
        A[0,:] = -p[1:] / p[0]
        roots = eig(A)
    else:
        return array([])

    # tack any zeros onto the back of the array    
    roots = hstack((roots,zeros(trailing_zeros,roots.typecode())))
    return roots

def polyint(p,m=1,k=None):
    """Return the mth analytical integral of the polynomial p.

    If k is None, then zero-valued constants of integration are used.
    otherwise, k should be a list of length m (or a scalar if m=1) to
    represent the constants of integration to use for each integration
    (starting with k[0])
    """
    m = int(m)
    if m < 0:
        raise ValueError, "Order of integral must be positive (see polyder)"
    if k is None:
        k = Numeric.zeros(m)
    k = atleast_1d(k)
    if len(k) == 1 and m > 1:
        k = k[0]*Numeric.ones(m)
    if len(k) < m:
        raise ValueError, \
              "k must be a scalar or a rank-1 array of length 1 or >m."
    if m == 0:
        return p
    else:
        truepoly = isinstance(p,poly1d)
        p = asarray(p)
        y = Numeric.zeros(len(p)+1,'d')
        y[:-1] = p*1.0/Numeric.arange(len(p),0,-1)
        y[-1] = k[0]        
        val = polyint(y,m-1,k=k[1:])
        if truepoly:
            val = poly1d(val)
        return val
            
def polyder(p,m=1):
    """Return the mth derivative of the polynomial p.
    """
    m = int(m)
    truepoly = isinstance(p,poly1d)
    p = asarray(p)
    n = len(p)-1
    y = p[:-1] * Numeric.arange(n,0,-1)
    if m < 0:
        raise ValueError, "Order of derivative must be positive (see polyint)"
    if m == 0:
        return p
    else:
        val = polyder(y,m-1)
        if truepoly:
            val = poly1d(val)
        return val

def polyval(p,x):
    """Evaluate the polynomial p at x.  If x is a polynomial then composition.

    Description:

      If p is of length N, this function returns the value:
      p[0]*(x**N-1) + p[1]*(x**N-2) + ... + p[N-2]*x + p[N-1]

      x can be a sequence and p(x) will be returned for all elements of x.
      or x can be another polynomial and the composite polynomial p(x) will be
      returned.
    """
    p = asarray(p)
    if isinstance(x,poly1d):
        y = 0
    else:
        x = asarray(x)
        y = Numeric.zeros(x.shape,x.typecode())
    for i in range(len(p)):
        y = x * y + p[i]
    return y

def polyadd(a1,a2):
    """Adds two polynomials represented as lists
    """
    truepoly = (isinstance(a1,poly1d) or isinstance(a2,poly1d))
    a1,a2 = map(atleast_1d,(a1,a2))
    diff = len(a2) - len(a1)
    if diff == 0:
        return a1 + a2
    elif diff > 0:
        zr = Numeric.zeros(diff)
        val = Numeric.concatenate((zr,a1)) + a2
    else:
        zr = Numeric.zeros(abs(diff))
        val = a1 + Numeric.concatenate((zr,a2))
    if truepoly:
        val = poly1d(val)
    return val

def polysub(a1,a2):
    """Subtracts two polynomials represented as lists
    """
    truepoly = (isinstance(a1,poly1d) or isinstance(a2,poly1d))
    a1,a2 = map(atleast_1d,(a1,a2))
    diff = len(a2) - len(a1)
    if diff == 0:
        return a1 - a2
    elif diff > 0:
        zr = Numeric.zeros(diff)
        val = Numeric.concatenate((zr,a1)) - a2
    else:
        zr = Numeric.zeros(abs(diff))
        val = a1 - Numeric.concatenate((zr,a2))
    if truepoly:
        val = poly1d(val)
    return val


def polymul(a1,a2):
    """Multiplies two polynomials represented as lists.
    """
    truepoly = (isinstance(a1,poly1d) or isinstance(a2,poly1d))
    val = Numeric.convolve(a1,a2)
    if truepoly:
        val = poly1d(val)
    return val

def polydiv(a1,a2):
    """Computes q and r polynomials so that a1(s) = q(s)*a2(s) + r(s)
    """
    truepoly = (isinstance(a1,poly1d) or isinstance(a2,poly1d))
    q, r = deconvolve(a1,a2)
    while Numeric.allclose(r[0], 0, rtol=1e-14) and (r.shape[-1] > 1):
        r = r[1:]
    if truepoly:
        q, r = map(poly1d,(q,r))
    return q, r

def deconvolve(signal, divisor):
    """Deconvolves divisor out of signal.
    """
    try:
        import scipy.signal
    except:
        print "You need scipy.signal to use this function."
    num = atleast_1d(signal)
    den = atleast_1d(divisor)
    N = len(num)
    D = len(den)
    if D > N:
        quot = [];
        rem = num;
    else:
        input = Numeric.ones(N-D+1,Numeric.Float)
        input[1:] = 0
        quot = scipy.signal.lfilter(num, den, input)
        rem = num - Numeric.convolve(den,quot,mode=2)
    return quot, rem

import re
_poly_mat = re.compile(r"[*][*]([0-9]*)")
def _raise_power(astr, wrap=70):
    n = 0
    line1 = ''
    line2 = ''
    output = ' '
    while 1:
        mat = _poly_mat.search(astr,n)
        if mat is None:
            break
        span = mat.span()
        power = mat.groups()[0]
        partstr = astr[n:span[0]]
        n = span[1]
        toadd2 = partstr + ' '*(len(power)-1)
        toadd1 = ' '*(len(partstr)-1) + power
        if ((len(line2)+len(toadd2) > wrap) or \
            (len(line1)+len(toadd1) > wrap)):
            output += line1 + "\n" + line2 + "\n "
            line1 = toadd1
            line2 = toadd2
        else:                
            line2 += partstr + ' '*(len(power)-1)
            line1 += ' '*(len(partstr)-1) + power
    output += line1 + "\n" + line2
    return output + astr[n:]
    
                       
00289 class poly1d:
    """A one-dimensional polynomial class.

    p = poly1d([1,2,3]) constructs the polynomial x**2 + 2 x + 3

    p(0.5) evaluates the polynomial at the location
    p.r  is a list of roots
    p.c  is the coefficient array [1,2,3]
    p.order is the polynomial order (after leading zeros in p.c are removed)
    p[k] is the coefficient on the kth power of x (backwards from
         sequencing the coefficient array.

    polynomials can be added, substracted, multplied and divided (returns
         quotient and remainder).
    asarray(p) will also give the coefficient array, so polynomials can
         be used in all functions that accept arrays.
    """
    def __init__(self, c_or_r, r=0):
        if isinstance(c_or_r,poly1d):
            for key in c_or_r.__dict__.keys():
                self.__dict__[key] = c_or_r.__dict__[key]
            return
        if r:
            c_or_r = poly(c_or_r)
        c_or_r = atleast_1d(c_or_r)
        if len(c_or_r.shape) > 1:
            raise ValueError, "Polynomial must be 1d only."
        c_or_r = trim_zeros(c_or_r, trim='f')
        if len(c_or_r) == 0:
            c_or_r = Numeric.array([0])
        self.__dict__['coeffs'] = c_or_r
        self.__dict__['order'] = len(c_or_r) - 1

    def __array__(self,t=None):
        if t:
            return asarray(self.coeffs,t)
        else:
            return asarray(self.coeffs)

    def __coerce__(self,other):
        return None
    
    def __repr__(self):
        vals = repr(self.coeffs)
        vals = vals[6:-1]
        return "poly1d(%s)" % vals

    def __len__(self):
        return self.order

    def __str__(self):
        N = self.order
        thestr = "0"
        for k in range(len(self.coeffs)):
            coefstr ='%.4g' % abs(self.coeffs[k])
            if coefstr[-4:] == '0000':
                coefstr = coefstr[:-5]
            power = (N-k)
            if power == 0:
                if coefstr != '0':
                    newstr = '%s' % (coefstr,)
                else:
                    if k == 0:
                        newstr = '0'
                    else:
                        newstr = ''
            elif power == 1:
                if coefstr == '0':
                    newstr = ''
                elif coefstr == '1':
                    newstr = 'x'
                else:                    
                    newstr = '%s x' % (coefstr,)
            else:
                if coefstr == '0':
                    newstr = ''
                elif coefstr == '1':
                    newstr = 'x**%d' % (power,)
                else:                    
                    newstr = '%s x**%d' % (coefstr, power)

            if k > 0:
                if newstr != '':
                    if self.coeffs[k] < 0:
                        thestr = "%s - %s" % (thestr, newstr)
                    else:
                        thestr = "%s + %s" % (thestr, newstr)
            elif (k == 0) and (newstr != '') and (self.coeffs[k] < 0):
                thestr = "-%s" % (newstr,)
            else:
                thestr = newstr
        return _raise_power(thestr)
        

    def __call__(self, val):
        return polyval(self.coeffs, val)

    def __mul__(self, other):
        if isscalar(other):
            return poly1d(self.coeffs * other)
        else:
            other = poly1d(other)
            return poly1d(polymul(self.coeffs, other.coeffs))

    def __rmul__(self, other):
        if isscalar(other):
            return poly1d(other * self.coeffs)
        else:
            other = poly1d(other)
            return poly1d(polymul(self.coeffs, other.coeffs))        
    
    def __add__(self, other):
        other = poly1d(other)
        return poly1d(polyadd(self.coeffs, other.coeffs))        
        
    def __radd__(self, other):
        other = poly1d(other)
        return poly1d(polyadd(self.coeffs, other.coeffs))

    def __pow__(self, val):
        if not isscalar(val) or int(val) != val or val < 0:
            raise ValueError, "Power to non-negative integers only."
        res = [1]
        for k in range(val):
            res = polymul(self.coeffs, res)
        return poly1d(res)

    def __sub__(self, other):
        other = poly1d(other)
        return poly1d(polysub(self.coeffs, other.coeffs))

    def __rsub__(self, other):
        other = poly1d(other)
        return poly1d(polysub(other.coeffs, self.coeffs))

    def __div__(self, other):
        if isscalar(other):
            return poly1d(self.coeffs/other)
        else:
            other = poly1d(other)
            return map(poly1d,polydiv(self.coeffs, other.coeffs))

    def __rdiv__(self, other):
        if isscalar(other):
            return poly1d(other/self.coeffs)
        else:
            other = poly1d(other)
            return map(poly1d,polydiv(other.coeffs, self.coeffs))

    def __setattr__(self, key, val):
        raise ValueError, "Attributes cannot be changed this way."

    def __getattr__(self, key):
        if key in ['r','roots']:
            return roots(self.coeffs)
        elif key in ['c','coef','coefficients']:
            return self.coeffs
        elif key in ['o']:
            return self.order
        else:
            return self.__dict__[key]
        
    def __getitem__(self, val):
        ind = self.order - val
        if val > self.order:
            return 0
        if val < 0:
            return 0
        return self.coeffs[ind]

    def __setitem__(self, key, val):
        ind = self.order - key
        if key < 0:
            raise ValueError, "Does not support negative powers."
        if key > self.order:
            zr = Numeric.zeros(key-self.order,self.coeffs.typecode())
            self.__dict__['coeffs'] = Numeric.concatenate((zr,self.coeffs))
            self.__dict__['order'] = key
            ind = 0
        self.__dict__['coeffs'][ind] = val
        return

    def integ(self, m=1, k=0):
        return poly1d(polyint(self.coeffs,m=m,k=k))

    def deriv(self, m=1):
        return poly1d(polyder(self.coeffs,m=m))

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