awips2/pythonPackages/scientific/Scientific/Functions/Polynomial.py

# This module defines a multivariate polynomial class
#
# Written by Konrad Hinsen <hinsen@cnrs-orleans.fr>
# last revision: 2006-11-24
#

"""
Polynomials in any number of variables
"""

from Scientific import N, LA; Numeric = N; LinearAlgebra = LA
from Scientific.indexing import index_expression

# Class definition

class Polynomial:

    """X{Multivariate} X{polynomial}

    Instances of this class represent polynomials of any order and
    in any number of variables. The coefficients and thus the values
    can be real or complex. Polynomials can be evaluated like functions.
    """

    def __init__(self, coefficients):
        """
        @param coefficients: an M{N}-dimnesional array for a polynomial
            in M{N} variables. C{coeffcients[i, j, ...]} is the coefficient
            of M{x_1^i x_2^j ...}
        @type coefficients: C{Numeric.array} or nested list of numbers
        """
        self.coeff = Numeric.array(coefficients)
        self.dim = len(self.coeff.shape)

    is_polynomial = 1

    def __call__(self, *args):
        """
        @param args: tuple of values, one for each variable of the
            polynomial
        @type args: C{tuple} of numbers
        @returns: the value of the polynomial at the given point
        @rtype: number
        @raise TypeError: if the number of arguments is not equal
            to the number of variable of the polynomial
        """
        if len(args) != self.dim:
            raise TypeError('Wrong number of arguments')
        p = _powers(args, self.coeff.shape)
        return Numeric.add.reduce(Numeric.ravel(p*self.coeff))

    def __repr__(self):
        if self.dim == 1:
            return "Polynomial(%s)" % repr(list(self.coeff))
        else:
            return "Polynomial(%s)" % repr(self.coeff)

    def __coerce__(self, other):
        if hasattr(other, 'is_polynomial'):
            return (self, other)
        elif hasattr(other, 'is_rational_function'):
            return None
        else:
            return (self, Polynomial([other]))

    def __add__(self, other):
        dim = max(self.dim, other.dim)
        shape = Numeric.zeros((dim,), Numeric.Int)
        shape[:self.dim] = self.coeff.shape
        shape[:other.dim] = Numeric.maximum(shape[:other.dim],
                                            other.coeff.shape)
        coeff1 = Numeric.zeros_st(shape, self.coeff)
        index = tuple(map(lambda d: slice(0, d), self.coeff.shape) + \
                      (dim-self.dim)*[0])
        coeff1[index] = self.coeff
        coeff2 = Numeric.zeros_st(shape, other.coeff)
        index = tuple(map(lambda d: slice(0, d), other.coeff.shape) + \
                      (dim-other.dim)*[0])
        coeff2[index] = other.coeff
        return Polynomial(coeff1+coeff2)

    def __mul__(self, other):
        if self.dim != 1 or other.dim != 1:
            raise ValueError("not implemented")
        c = Numeric.multiply.outer(self.coeff, other.coeff)
        temp = Numeric.concatenate((c, Numeric.zeros(2*(c.shape[0],))), 1)
        temp = Numeric.ravel(temp)[:-c.shape[0]]
        temp.shape = (c.shape[0], c.shape[0]+c.shape[1]-1)
        return Polynomial(Numeric.sum(temp))

    def __div__(self, other):
        if self.dim != 1 or other.dim != 1:
            raise ValueError("not implemented")
        if len(other.coeff) == 1:
            return Polynomial(self.coeff/other.coeff[0])
        from Rational import RationalFunction
        return RationalFunction(self, other)

    def __rdiv__(self, other):
        from Rational import RationalFunction
        return RationalFunction(other, self)

    def derivative(self, variable=0):
        """
        @param variable: the index of the variable with respect to which
            the X{derivative} is taken
        @type variable: C{int}
        @returns: a polynomial of reduced order in one variable
        @rtype: L{Polynomial}
        """
        n = self.coeff.shape[variable]
        if n == 1:
            return Polynomial(apply(Numeric.zeros, self.dim*(1,)))
        index = variable*index_expression[::] + \
                index_expression[1::] + index_expression[...]
        factor = Numeric.arange(1.,n)
        factor = factor[index_expression[::] +
                        (self.dim-variable-1) * \
                        index_expression[Numeric.NewAxis]]
        return Polynomial(factor*self.coeff[index])

    def integral(self, variable=0):
        """
        @param variable: the index of the variable with respect to which
            the X{integral} is computed
        @type variable: C{int}
        @returns: a polynomial of higher order in one variable
        @rtype: L{Polynomial}
        """
        n = self.coeff.shape[variable]
        factor = 1./Numeric.arange(1.,n+1)
        factor = factor[index_expression[::] +
                        (self.dim-variable-1) * \
                        index_expression[Numeric.NewAxis]]
        s = map(None, self.coeff.shape)
        s[variable] = 1
        z = apply(Numeric.zeros, tuple(s))
        intcoeff = Numeric.concatenate((z, factor*self.coeff), variable)
        return Polynomial(intcoeff)

    def zeros(self):
        """
        Find the X{zeros} (X{roots}) of the polynomial by diagonalization
        of the associated Frobenius matrix.

        @returns: an array containing the zeros
        @rtype: C{Numeric.array}
        @note: this is defined only for polynomials in one variable
        @raise ValueError: is the polynomial has more than one variable
        """
        if self.dim != 1:
            raise ValueError("not implemented")
        n = len(self.coeff)-1
        if n == 0:
            return Numeric.array([])
        a = Numeric.zeros_st((n, n), self.coeff)
        if n > 1:
            a[1:, :-1] = Numeric.identity(n-1)
        a[:, -1] = -self.coeff[:-1]/self.coeff[-1]
        from Scientific.LA import eigenvalues
        return eigenvalues(a)

# Polynomial fit constructor for use in module Interpolation

def _fitPolynomial(order, points, values):
    if len(points) != len(values):
        raise ValueError('Inconsistent arguments')
    if type(order) != type(()):
        order = (order,)
    order = tuple(map(lambda n: n+1, order))
    if not _isSequence(points[0]):
        points = map(lambda p: (p,), points)
    if len(order) != len(points[0]):
        raise ValueError('Inconsistent arguments')
    if Numeric.multiply.reduce(order) > len(points):
        raise ValueError('Not enough points')
    matrix = []
    for p in points:
        matrix.append(Numeric.ravel(_powers(p, order)))
    matrix = Numeric.array(matrix)
    values = Numeric.array(values)
    inv = LinearAlgebra.generalized_inverse(matrix)
    coeff = Numeric.dot(inv, values)
    #coeff = LinearAlgebra.linear_least_squares(matrix, values)[0]
    coeff = Numeric.reshape(coeff, order)
    return Polynomial(coeff)

# Helper functions

def _powers(x, n):
    p = 1.
    index = index_expression[::] + \
            (len(x)-1)*index_expression[Numeric.NewAxis]
    for i in range(len(x)):
        pi = Numeric.multiply.accumulate(Numeric.array([1.]+(n[i]-1)*[x[i]]))
        p = p*pi[index]
        index = index[-1:] + index[:-1]
    return p

def _isSequence(object):
    n = -1
    try: n = len(object)
    except: pass
    return n >= 0


# Clean up module (mostly for epydoc)

del index_expression

# Test code

if __name__ == '__main__':

    p1 = Polynomial([1.,0.3,1.,-0.4])
    x = -1.9
    print p1(x), ((-0.4*x+1.)*x+0.3)*x+1.
    zeros = p1.zeros()
    for z in zeros:
        print z, p1(z)
    p2 = Polynomial([[1.,0.3],[-0.2,0.5]])
    y = 0.3
    print p2(x,y), 1. + 0.3*y - 0.2*x + 0.5*x*y
    fit = fitPolynomial(2, [1.,2.,3.,4.], [2.,4.,8.,14.])
    print fit.coeff