""" Functions to operate on polynomials. """ __all__ = ['poly', 'roots', 'polyint', 'polyder', 'polyadd', 'polysub', 'polymul', 'polydiv', 'polyval', 'poly1d', 'polyfit', 'RankWarning'] import re import warnings import numpy.core.numeric as NX from numpy.core import isscalar, abs from numpy.lib.getlimits import finfo from numpy.lib.twodim_base import diag, vander from numpy.lib.shape_base import hstack, atleast_1d from numpy.lib.function_base import trim_zeros, sort_complex eigvals = None lstsq = None _single_eps = finfo(NX.single).eps _double_eps = finfo(NX.double).eps class RankWarning(UserWarning): """Issued by polyfit when Vandermonde matrix is rank deficient. """ pass def get_linalg_funcs(): "Look for linear algebra functions in numpy" global eigvals, lstsq from numpy.dual import eigvals, lstsq return def _eigvals(arg): "Return the eigenvalues of the argument" try: return eigvals(arg) except TypeError: get_linalg_funcs() return eigvals(arg) def _lstsq(X, y, rcond): "Do least squares on the arguments" try: return lstsq(X, y, rcond) except TypeError: get_linalg_funcs() return lstsq(X, y, rcond) 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 = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1]: seq_of_zeros = _eigvals(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 = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate(sort_complex( NX.compress(roots.imag < 0,roots))) if (len(pos_roots) == len(neg_roots) and NX.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 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 = NX.nonzero(NX.ravel(p))[0] # Return an empty array if polynomial is all zeros if len(non_zero) == 0: return NX.array([]) # 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 not issubclass(p.dtype.type, (NX.floating, NX.complexfloating)): p = p.astype(float) N = len(p) if N > 1: # build companion matrix and find its eigenvalues (the roots) A = diag(NX.ones((N-2,), p.dtype), -1) A[0, :] = -p[1:] / p[0] roots = _eigvals(A) else: roots = NX.array([]) # tack any zeros onto the back of the array roots = hstack((roots, NX.zeros(trailing_zeros, roots.dtype))) 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 = NX.zeros(m, float) k = atleast_1d(k) if len(k) == 1 and m > 1: k = k[0]*NX.ones(m, float) 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 = NX.asarray(p) y = NX.zeros(len(p)+1, float) y[:-1] = p*1.0/NX.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 = NX.asarray(p) n = len(p)-1 y = p[:-1] * NX.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 polyfit(x, y, deg, rcond=None, full=False): """Least squares polynomial fit. Do a best fit polynomial of degree 'deg' of 'x' to 'y'. Return value is a vector of polynomial coefficients [pk ... p1 p0]. Eg, for n=2 p2*x0^2 + p1*x0 + p0 = y1 p2*x1^2 + p1*x1 + p0 = y1 p2*x2^2 + p1*x2 + p0 = y2 ..... p2*xk^2 + p1*xk + p0 = yk Parameters ---------- x : array_like 1D vector of sample points. y : array_like 1D vector or 2D array of values to fit. The values should run down the columes in the 2D case. deg : integer Degree of the fitting polynomial rcond: {None, float}, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The defaul value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : {False, boolean}, optional Switch determining nature of return value. When it is False just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. Returns ------- coefficients, [residuals, rank, singular_values, rcond] : variable When full=False, only the coefficients are returned, running down the appropriate colume when y is a 2D array. When full=True, the rank of the scaled Vandermonde matrix, it's effective rank in light of the rcond value, its singular values, and the specified value of rcond are also returned. Warns ----- RankWarning : if rank is reduced and not full output The warnings can be turned off by: >>> import warnings >>> warnings.simplefilter('ignore',np.RankWarning) See Also -------- polyval : computes polynomial values. Notes ----- If X is a the Vandermonde Matrix computed from x (see http://mathworld.wolfram.com/VandermondeMatrix.html), then the polynomial least squares solution is given by the 'p' in X*p = y where X.shape is a matrix of dimensions (len(x), deg + 1), p is a vector of dimensions (deg + 1, 1), and y is a vector of dimensions (len(x), 1). This equation can be solved as p = (XT*X)^-1 * XT * y where XT is the transpose of X and -1 denotes the inverse. However, this method is susceptible to rounding errors and generally the singular value decomposition of the matrix X is preferred and that is what is done here. The singular value method takes a paramenter, 'rcond', which sets a limit on the relative size of the smallest singular value to be used in solving the equation. This may result in lowering the rank of the Vandermonde matrix, in which case a RankWarning is issued. If polyfit issues a RankWarning, try a fit of lower degree or replace x by x - x.mean(), both of which will generally improve the condition number. The routine already normalizes the vector x by its maximum absolute value to help in this regard. The rcond parameter can be set to a value smaller than its default, but the resulting fit may be spurious. The current default value of rcond is len(x)*eps, where eps is the relative precision of the floating type being used, generally around 1e-7 and 2e-16 for IEEE single and double precision respectively. This value of rcond is fairly conservative but works pretty well when x - x.mean() is used in place of x. DISCLAIMER: Power series fits are full of pitfalls for the unwary once the degree of the fit becomes large or the interval of sample points is badly centered. The problem is that the powers x**n are generally a poor basis for the polynomial functions on the sample interval, resulting in a Vandermonde matrix is ill conditioned and coefficients sensitive to rounding erros. The computation of the polynomial values will also sensitive to rounding errors. Consequently, the quality of the polynomial fit should be checked against the data whenever the condition number is large. The quality of polynomial fits *can not* be taken for granted. If all you want to do is draw a smooth curve through the y values and polyfit is not doing the job, try centering the sample range or look into scipy.interpolate, which includes some nice spline fitting functions that may be of use. For more info, see http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html, but note that the k's and n's in the superscripts and subscripts on that page. The linear algebra is correct, however. """ order = int(deg) + 1 x = NX.asarray(x) + 0.0 y = NX.asarray(y) + 0.0 # check arguments. if deg < 0 : raise ValueError, "expected deg >= 0" if x.ndim != 1: raise TypeError, "expected 1D vector for x" if x.size == 0: raise TypeError, "expected non-empty vector for x" if y.ndim < 1 or y.ndim > 2 : raise TypeError, "expected 1D or 2D array for y" if x.shape[0] != y.shape[0] : raise TypeError, "expected x and y to have same length" # set rcond if rcond is None : xtype = x.dtype if xtype == NX.single or xtype == NX.csingle : rcond = len(x)*_single_eps else : rcond = len(x)*_double_eps # scale x to improve condition number scale = abs(x).max() if scale != 0 : x /= scale # solve least squares equation for powers of x v = vander(x, order) c, resids, rank, s = _lstsq(v, y, rcond) # warn on rank reduction, which indicates an ill conditioned matrix if rank != order and not full: msg = "Polyfit may be poorly conditioned" warnings.warn(msg, RankWarning) # scale returned coefficients if scale != 0 : if c.ndim == 1 : c /= vander([scale], order)[0] else : c /= vander([scale], order).T if full : return c, resids, rank, s, rcond else : return c def polyval(p, x): """Evaluate the polynomial p at x. 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] If x is a sequence then p(x) will be returned for all elements of x. If x is another polynomial then the composite polynomial p(x) will be returned. Parameters ---------- p : {array_like, poly1d} 1D array of polynomial coefficients from highest degree to zero or an instance of poly1d. x : {array_like, poly1d} A number, a 1D array of numbers, or an instance of poly1d. Returns ------- values : {array, poly1d} If either p or x is an instance of poly1d, then an instance of poly1d is returned, otherwise a 1D array is returned. In the case where x is a poly1d, the result is the composition of the two polynomials, i.e., substitution is used. Notes ----- Horners method is used to evaluate the polynomial. Even so, for polynomial if high degree the values may be inaccurate due to rounding errors. Use carefully. """ p = NX.asarray(p) if isinstance(x, poly1d): y = 0 else: x = NX.asarray(x) y = NX.zeros_like(x) for i in range(len(p)): y = x * y + p[i] return y def polyadd(a1, a2): """Adds two polynomials represented as sequences """ truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d)) a1 = atleast_1d(a1) a2 = atleast_1d(a2) diff = len(a2) - len(a1) if diff == 0: val = a1 + a2 elif diff > 0: zr = NX.zeros(diff, a1.dtype) val = NX.concatenate((zr, a1)) + a2 else: zr = NX.zeros(abs(diff), a2.dtype) val = a1 + NX.concatenate((zr, a2)) if truepoly: val = poly1d(val) return val def polysub(a1, a2): """Subtracts two polynomials represented as sequences """ truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d)) a1 = atleast_1d(a1) a2 = atleast_1d(a2) diff = len(a2) - len(a1) if diff == 0: val = a1 - a2 elif diff > 0: zr = NX.zeros(diff, a1.dtype) val = NX.concatenate((zr, a1)) - a2 else: zr = NX.zeros(abs(diff), a2.dtype) val = a1 - NX.concatenate((zr, a2)) if truepoly: val = poly1d(val) return val def polymul(a1, a2): """Multiplies two polynomials represented as sequences. """ truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d)) a1,a2 = poly1d(a1),poly1d(a2) val = NX.convolve(a1, a2) if truepoly: val = poly1d(val) return val def polydiv(u, v): """Computes q and r polynomials so that u(s) = q(s)*v(s) + r(s) and deg r < deg v. """ truepoly = (isinstance(u, poly1d) or isinstance(u, poly1d)) u = atleast_1d(u) v = atleast_1d(v) m = len(u) - 1 n = len(v) - 1 scale = 1. / v[0] q = NX.zeros((max(m-n+1,1),), float) r = u.copy() for k in range(0, m-n+1): d = scale * r[k] q[k] = d r[k:k+n+1] -= d*v while NX.allclose(r[0], 0, rtol=1e-14) and (r.shape[-1] > 1): r = r[1:] if truepoly: q = poly1d(q) r = poly1d(r) return q, r _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:] class poly1d(object): """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. p = poly1d([1,2,3], variable='lambda') will use lambda in the string representation of p. """ coeffs = None order = None variable = None def __init__(self, c_or_r, r=0, variable=None): if isinstance(c_or_r, poly1d): for key in c_or_r.__dict__.keys(): self.__dict__[key] = c_or_r.__dict__[key] if variable is not None: self.__dict__['variable'] = variable 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 = NX.array([0.]) self.__dict__['coeffs'] = c_or_r self.__dict__['order'] = len(c_or_r) - 1 if variable is None: variable = 'x' self.__dict__['variable'] = variable def __array__(self, t=None): if t: return NX.asarray(self.coeffs, t) else: return NX.asarray(self.coeffs) def __repr__(self): vals = repr(self.coeffs) vals = vals[6:-1] return "poly1d(%s)" % vals def __len__(self): return self.order def __str__(self): thestr = "0" var = self.variable # Remove leading zeros coeffs = self.coeffs[NX.logical_or.accumulate(self.coeffs != 0)] N = len(coeffs)-1 for k in range(len(coeffs)): coefstr ='%.4g' % abs(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 == 'b': newstr = var else: newstr = '%s %s' % (coefstr, var) else: if coefstr == '0': newstr = '' elif coefstr == 'b': newstr = '%s**%d' % (var, power,) else: newstr = '%s %s**%d' % (coefstr, var, power) if k > 0: if newstr != '': if coeffs[k] < 0: thestr = "%s - %s" % (thestr, newstr) else: thestr = "%s + %s" % (thestr, newstr) elif (k == 0) and (newstr != '') and (coeffs[k] < 0): thestr = "-%s" % (newstr,) else: thestr = newstr return _raise_power(thestr) def __call__(self, val): return polyval(self.coeffs, val) def __neg__(self): return poly1d(-self.coeffs) def __pos__(self): return self 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 _ 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 polydiv(self, other) def __rdiv__(self, other): if isscalar(other): return poly1d(other/self.coeffs) else: other = poly1d(other) return polydiv(other, self) def __eq__(self, other): return NX.alltrue(self.coeffs == other.coeffs) def __ne__(self, other): return NX.any(self.coeffs != other.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: try: return self.__dict__[key] except KeyError: raise AttributeError("'%s' has no attribute '%s'" % (self.__class__, 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 = NX.zeros(key-self.order, self.coeffs.dtype) self.__dict__['coeffs'] = NX.concatenate((zr, self.coeffs)) self.__dict__['order'] = key ind = 0 self.__dict__['coeffs'][ind] = val return def __iter__(self): return iter(self.coeffs) def integ(self, m=1, k=0): """Return the mth analytical integral of this polynomial. See the documentation for polyint. """ return poly1d(polyint(self.coeffs, m=m, k=k)) def deriv(self, m=1): """Return the mth derivative of this polynomial. """ return poly1d(polyder(self.coeffs, m=m)) # Stuff to do on module import warnings.simplefilter('always',RankWarning)