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""" Functions to operate on polynomials.
"""
'polysub', 'polymul', 'polydiv', 'polyval', 'poly1d', 'polyfit', 'RankWarning']
ones)
overrides.array_function_dispatch, module='numpy')
""" Issued by `polyfit` when the Vandermonde matrix is rank deficient.
For more information, a way to suppress the warning, and an example of `RankWarning` being issued, see `polyfit`.
"""
return seq_of_zeros
def poly(seq_of_zeros): """ Find the coefficients of a polynomial with the given sequence of roots.
Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned.
Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object.
Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree:
``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]`` where c[0] always equals 1.
Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array).
See Also -------- polyval : Compute polynomial values. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class.
Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1]_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.)
The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by
:math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`,
where **I** is the `n`-by-`n` identity matrix. [2]_
References ---------- .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry, Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996.
.. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition," Academic Press, pg. 182, 1980.
Examples -------- Given a sequence of a polynomial's zeros:
>>> np.poly((0, 0, 0)) # Multiple root example array([1, 0, 0, 0])
The line above represents z**3 + 0*z**2 + 0*z + 0.
>>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ])
The line above represents z**3 - z/4
>>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) #random
Given a square array object:
>>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([ 1. , 0. , 0.16666667])
Note how in all cases the leading coefficient is always 1.
""" seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) == 1: dt = seq_of_zeros.dtype # Let object arrays slip through, e.g. for arbitrary precision if dt != object: seq_of_zeros = seq_of_zeros.astype(mintypecode(dt.char)) else: raise ValueError("input must be 1d or non-empty square 2d array.")
if len(seq_of_zeros) == 0: return 1.0 dt = seq_of_zeros.dtype a = ones((1,), dtype=dt) for k in range(len(seq_of_zeros)): a = NX.convolve(a, array([1, -seq_of_zeros[k]], dtype=dt), 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) if NX.all(NX.sort(roots) == NX.sort(roots.conjugate())): a = a.real.copy()
return a
return p
def roots(p): """ Return the roots of a polynomial with coefficients given 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 described by::
p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n]
Parameters ---------- p : array_like Rank-1 array of polynomial coefficients.
Returns ------- out : ndarray An array containing the roots of the polynomial.
Raises ------ ValueError When `p` cannot be converted to a rank-1 array.
See also -------- poly : Find the coefficients of a polynomial with a given sequence of roots. polyval : Compute polynomial values. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class.
Notes ----- The algorithm relies on computing the eigenvalues of the companion matrix [1]_.
References ---------- .. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*. Cambridge, UK: Cambridge University Press, 1999, pp. 146-7.
Examples -------- >>> coeff = [3.2, 2, 1] >>> np.roots(coeff) array([-0.3125+0.46351241j, -0.3125-0.46351241j])
""" # If input is scalar, this makes it an array p = atleast_1d(p) if p.ndim != 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
return (p,)
""" Return an antiderivative (indefinite integral) of a polynomial.
The returned order `m` antiderivative `P` of polynomial `p` satisfies :math:`\\frac{d^m}{dx^m}P(x) = p(x)` and is defined up to `m - 1` integration constants `k`. The constants determine the low-order polynomial part
.. math:: \\frac{k_{m-1}}{0!} x^0 + \\ldots + \\frac{k_0}{(m-1)!}x^{m-1}
of `P` so that :math:`P^{(j)}(0) = k_{m-j-1}`.
Parameters ---------- p : array_like or poly1d Polynomial to integrate. A sequence is interpreted as polynomial coefficients, see `poly1d`. m : int, optional Order of the antiderivative. (Default: 1) k : list of `m` scalars or scalar, optional Integration constants. They are given in the order of integration: those corresponding to highest-order terms come first.
If ``None`` (default), all constants are assumed to be zero. If `m = 1`, a single scalar can be given instead of a list.
See Also -------- polyder : derivative of a polynomial poly1d.integ : equivalent method
Examples -------- The defining property of the antiderivative:
>>> p = np.poly1d([1,1,1]) >>> P = np.polyint(p) >>> P poly1d([ 0.33333333, 0.5 , 1. , 0. ]) >>> np.polyder(P) == p True
The integration constants default to zero, but can be specified:
>>> P = np.polyint(p, 3) >>> P(0) 0.0 >>> np.polyder(P)(0) 0.0 >>> np.polyder(P, 2)(0) 0.0 >>> P = np.polyint(p, 3, k=[6,5,3]) >>> P poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ])
Note that 3 = 6 / 2!, and that the constants are given in the order of integrations. Constant of the highest-order polynomial term comes first:
>>> np.polyder(P, 2)(0) 6.0 >>> np.polyder(P, 1)(0) 5.0 >>> P(0) 3.0
""" 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.")
truepoly = isinstance(p, poly1d) p = NX.asarray(p) if m == 0: if truepoly: return poly1d(p) return p else: # Note: this must work also with object and integer arrays y = NX.concatenate((p.__truediv__(NX.arange(len(p), 0, -1)), [k[0]])) val = polyint(y, m - 1, k=k[1:]) if truepoly: return poly1d(val) return val
return (p,)
""" Return the derivative of the specified order of a polynomial.
Parameters ---------- p : poly1d or sequence Polynomial to differentiate. A sequence is interpreted as polynomial coefficients, see `poly1d`. m : int, optional Order of differentiation (default: 1)
Returns ------- der : poly1d A new polynomial representing the derivative.
See Also -------- polyint : Anti-derivative of a polynomial. poly1d : Class for one-dimensional polynomials.
Examples -------- The derivative of the polynomial :math:`x^3 + x^2 + x^1 + 1` is:
>>> p = np.poly1d([1,1,1,1]) >>> p2 = np.polyder(p) >>> p2 poly1d([3, 2, 1])
which evaluates to:
>>> p2(2.) 17.0
We can verify this, approximating the derivative with ``(f(x + h) - f(x))/h``:
>>> (p(2. + 0.001) - p(2.)) / 0.001 17.007000999997857
The fourth-order derivative of a 3rd-order polynomial is zero:
>>> np.polyder(p, 2) poly1d([6, 2]) >>> np.polyder(p, 3) poly1d([6]) >>> np.polyder(p, 4) poly1d([ 0.])
""" m = int(m) if m < 0: raise ValueError("Order of derivative must be positive (see polyint)")
truepoly = isinstance(p, poly1d) p = NX.asarray(p) n = len(p) - 1 y = p[:-1] * NX.arange(n, 0, -1) if m == 0: val = p else: val = polyder(y, m - 1) if truepoly: val = poly1d(val) return val
return (x, y, w)
""" Least squares polynomial fit.
Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg` to points `(x, y)`. Returns a vector of coefficients `p` that minimises the squared error in the order `deg`, `deg-1`, ... `0`.
The `Polynomial.fit <numpy.polynomial.polynomial.Polynomial.fit>` class method is recommended for new code as it is more stable numerically. See the documentation of the method for more information.
Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int Degree of the fitting polynomial rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (M,), optional Weights to apply to the y-coordinates of the sample points. For gaussian uncertainties, use 1/sigma (not 1/sigma**2). cov : bool or str, optional If given and not `False`, return not just the estimate but also its covariance matrix. By default, the covariance are scaled by chi2/sqrt(N-dof), i.e., the weights are presumed to be unreliable except in a relative sense and everything is scaled such that the reduced chi2 is unity. This scaling is omitted if ``cov='unscaled'``, as is relevant for the case that the weights are 1/sigma**2, with sigma known to be a reliable estimate of the uncertainty.
Returns ------- p : ndarray, shape (deg + 1,) or (deg + 1, K) Polynomial coefficients, highest power first. If `y` was 2-D, the coefficients for `k`-th data set are in ``p[:,k]``.
residuals, rank, singular_values, rcond Present only if `full` = True. Residuals of the least-squares fit, the effective rank of the scaled Vandermonde coefficient matrix, its singular values, and the specified value of `rcond`. For more details, see `linalg.lstsq`.
V : ndarray, shape (M,M) or (M,M,K) Present only if `full` = False and `cov`=True. The covariance matrix of the polynomial coefficient estimates. The diagonal of this matrix are the variance estimates for each coefficient. If y is a 2-D array, then the covariance matrix for the `k`-th data set are in ``V[:,:,k]``
Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if `full` = False.
The warnings can be turned off by
>>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning)
See Also -------- polyval : Compute polynomial values. linalg.lstsq : Computes a least-squares fit. scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes ----- The solution minimizes the squared error
.. math :: E = \\sum_{j=0}^k |p(x_j) - y_j|^2
in the equations::
x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0] x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1] ... x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k]
The coefficient matrix of the coefficients `p` is a Vandermonde matrix.
`polyfit` issues a `RankWarning` when the least-squares fit is badly conditioned. This implies that the best fit is not well-defined due to numerical error. The results may be improved by lowering the polynomial degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious: including contributions from the small singular values can add numerical noise to the result.
Note that fitting polynomial coefficients is inherently badly conditioned when the degree of the polynomial is large or the interval of sample points is badly centered. The quality of the fit should always be checked in these cases. When polynomial fits are not satisfactory, splines may be a good alternative.
References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting .. [2] Wikipedia, "Polynomial interpolation", https://en.wikipedia.org/wiki/Polynomial_interpolation
Examples -------- >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0]) >>> z = np.polyfit(x, y, 3) >>> z array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
It is convenient to use `poly1d` objects for dealing with polynomials:
>>> p = np.poly1d(z) >>> p(0.5) 0.6143849206349179 >>> p(3.5) -0.34732142857143039 >>> p(10) 22.579365079365115
High-order polynomials may oscillate wildly:
>>> p30 = np.poly1d(np.polyfit(x, y, 30)) /... RankWarning: Polyfit may be poorly conditioned... >>> p30(4) -0.80000000000000204 >>> p30(5) -0.99999999999999445 >>> p30(4.5) -0.10547061179440398
Illustration:
>>> import matplotlib.pyplot as plt >>> xp = np.linspace(-2, 6, 100) >>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--') >>> plt.ylim(-2,2) (-2, 2) >>> plt.show()
""" 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: rcond = len(x)*finfo(x.dtype).eps
# set up least squares equation for powers of x lhs = vander(x, order) rhs = y
# apply weighting if w is not None: w = NX.asarray(w) + 0.0 if w.ndim != 1: raise TypeError("expected a 1-d array for weights") if w.shape[0] != y.shape[0]: raise TypeError("expected w and y to have the same length") lhs *= w[:, NX.newaxis] if rhs.ndim == 2: rhs *= w[:, NX.newaxis] else: rhs *= w
# scale lhs to improve condition number and solve scale = NX.sqrt((lhs*lhs).sum(axis=0)) lhs /= scale c, resids, rank, s = lstsq(lhs, rhs, rcond) c = (c.T/scale).T # broadcast scale coefficients
# 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, stacklevel=2)
if full: return c, resids, rank, s, rcond elif cov: Vbase = inv(dot(lhs.T, lhs)) Vbase /= NX.outer(scale, scale) if cov == "unscaled": fac = 1 else: if len(x) <= order: raise ValueError("the number of data points must exceed order " "to scale the covariance matrix") # note, this used to be: fac = resids / (len(x) - order - 2.0) # it was deciced that the "- 2" (originally justified by "Bayesian # uncertainty analysis") is not was the user expects # (see gh-11196 and gh-11197) fac = resids / (len(x) - order) if y.ndim == 1: return c, Vbase * fac else: return c, Vbase[:,:, NX.newaxis] * fac else: return c
return (p, x)
def polyval(p, x): """ Evaluate a polynomial at specific values.
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)` is returned for each element of `x`. If `x` is another polynomial then the composite polynomial `p(x(t))` is returned.
Parameters ---------- p : array_like or poly1d object 1D array of polynomial coefficients (including coefficients equal to zero) from highest degree to the constant term, or an instance of poly1d. x : array_like or poly1d object A number, an array of numbers, or an instance of poly1d, at which to evaluate `p`.
Returns ------- values : ndarray or poly1d If `x` is a poly1d instance, the result is the composition of the two polynomials, i.e., `x` is "substituted" in `p` and the simplified result is returned. In addition, the type of `x` - array_like or poly1d - governs the type of the output: `x` array_like => `values` array_like, `x` a poly1d object => `values` is also.
See Also -------- poly1d: A polynomial class.
Notes ----- Horner's scheme [1]_ is used to evaluate the polynomial. Even so, for polynomials of high degree the values may be inaccurate due to rounding errors. Use carefully.
References ---------- .. [1] I. N. Bronshtein, K. A. Semendyayev, and K. A. Hirsch (Eng. trans. Ed.), *Handbook of Mathematics*, New York, Van Nostrand Reinhold Co., 1985, pg. 720.
Examples -------- >>> np.polyval([3,0,1], 5) # 3 * 5**2 + 0 * 5**1 + 1 76 >>> np.polyval([3,0,1], np.poly1d(5)) poly1d([ 76.]) >>> np.polyval(np.poly1d([3,0,1]), 5) 76 >>> np.polyval(np.poly1d([3,0,1]), np.poly1d(5)) poly1d([ 76.])
""" 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 = y * x + p[i] return y
return (a1, a2)
def polyadd(a1, a2): """ Find the sum of two polynomials.
Returns the polynomial resulting from the sum of two input polynomials. Each input must be either a poly1d object or a 1D sequence of polynomial coefficients, from highest to lowest degree.
Parameters ---------- a1, a2 : array_like or poly1d object Input polynomials.
Returns ------- out : ndarray or poly1d object The sum of the inputs. If either input is a poly1d object, then the output is also a poly1d object. Otherwise, it is a 1D array of polynomial coefficients from highest to lowest degree.
See Also -------- poly1d : A one-dimensional polynomial class. poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, polyval
Examples -------- >>> np.polyadd([1, 2], [9, 5, 4]) array([9, 6, 6])
Using poly1d objects:
>>> p1 = np.poly1d([1, 2]) >>> p2 = np.poly1d([9, 5, 4]) >>> print(p1) 1 x + 2 >>> print(p2) 2 9 x + 5 x + 4 >>> print(np.polyadd(p1, p2)) 2 9 x + 6 x + 6
""" 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): """ Difference (subtraction) of two polynomials.
Given two polynomials `a1` and `a2`, returns ``a1 - a2``. `a1` and `a2` can be either array_like sequences of the polynomials' coefficients (including coefficients equal to zero), or `poly1d` objects.
Parameters ---------- a1, a2 : array_like or poly1d Minuend and subtrahend polynomials, respectively.
Returns ------- out : ndarray or poly1d Array or `poly1d` object of the difference polynomial's coefficients.
See Also -------- polyval, polydiv, polymul, polyadd
Examples -------- .. math:: (2 x^2 + 10 x - 2) - (3 x^2 + 10 x -4) = (-x^2 + 2)
>>> np.polysub([2, 10, -2], [3, 10, -4]) array([-1, 0, 2])
""" 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): """ Find the product of two polynomials.
Finds the polynomial resulting from the multiplication of the two input polynomials. Each input must be either a poly1d object or a 1D sequence of polynomial coefficients, from highest to lowest degree.
Parameters ---------- a1, a2 : array_like or poly1d object Input polynomials.
Returns ------- out : ndarray or poly1d object The polynomial resulting from the multiplication of the inputs. If either inputs is a poly1d object, then the output is also a poly1d object. Otherwise, it is a 1D array of polynomial coefficients from highest to lowest degree.
See Also -------- poly1d : A one-dimensional polynomial class. poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, polyval convolve : Array convolution. Same output as polymul, but has parameter for overlap mode.
Examples -------- >>> np.polymul([1, 2, 3], [9, 5, 1]) array([ 9, 23, 38, 17, 3])
Using poly1d objects:
>>> p1 = np.poly1d([1, 2, 3]) >>> p2 = np.poly1d([9, 5, 1]) >>> print(p1) 2 1 x + 2 x + 3 >>> print(p2) 2 9 x + 5 x + 1 >>> print(np.polymul(p1, p2)) 4 3 2 9 x + 23 x + 38 x + 17 x + 3
""" 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
return (u, v)
def polydiv(u, v): """ Returns the quotient and remainder of polynomial division.
The input arrays are the coefficients (including any coefficients equal to zero) of the "numerator" (dividend) and "denominator" (divisor) polynomials, respectively.
Parameters ---------- u : array_like or poly1d Dividend polynomial's coefficients.
v : array_like or poly1d Divisor polynomial's coefficients.
Returns ------- q : ndarray Coefficients, including those equal to zero, of the quotient. r : ndarray Coefficients, including those equal to zero, of the remainder.
See Also -------- poly, polyadd, polyder, polydiv, polyfit, polyint, polymul, polysub, polyval
Notes ----- Both `u` and `v` must be 0-d or 1-d (ndim = 0 or 1), but `u.ndim` need not equal `v.ndim`. In other words, all four possible combinations - ``u.ndim = v.ndim = 0``, ``u.ndim = v.ndim = 1``, ``u.ndim = 1, v.ndim = 0``, and ``u.ndim = 0, v.ndim = 1`` - work.
Examples -------- .. math:: \\frac{3x^2 + 5x + 2}{2x + 1} = 1.5x + 1.75, remainder 0.25
>>> x = np.array([3.0, 5.0, 2.0]) >>> y = np.array([2.0, 1.0]) >>> np.polydiv(x, y) (array([ 1.5 , 1.75]), array([ 0.25]))
""" truepoly = (isinstance(u, poly1d) or isinstance(u, poly1d)) u = atleast_1d(u) + 0.0 v = atleast_1d(v) + 0.0 # w has the common type w = u[0] + v[0] m = len(u) - 1 n = len(v) - 1 scale = 1. / v[0] q = NX.zeros((max(m - n + 1, 1),), w.dtype) r = u.astype(w.dtype) 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: return poly1d(q), poly1d(r) return q, r
def _raise_power(astr, wrap=70): n = 0 line1 = '' line2 = '' output = ' ' while True: 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:]
""" A one-dimensional polynomial class.
A convenience class, used to encapsulate "natural" operations on polynomials so that said operations may take on their customary form in code (see Examples).
Parameters ---------- c_or_r : array_like The polynomial's coefficients, in decreasing powers, or if the value of the second parameter is True, the polynomial's roots (values where the polynomial evaluates to 0). For example, ``poly1d([1, 2, 3])`` returns an object that represents :math:`x^2 + 2x + 3`, whereas ``poly1d([1, 2, 3], True)`` returns one that represents :math:`(x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x -6`. r : bool, optional If True, `c_or_r` specifies the polynomial's roots; the default is False. variable : str, optional Changes the variable used when printing `p` from `x` to `variable` (see Examples).
Examples -------- Construct the polynomial :math:`x^2 + 2x + 3`:
>>> p = np.poly1d([1, 2, 3]) >>> print(np.poly1d(p)) 2 1 x + 2 x + 3
Evaluate the polynomial at :math:`x = 0.5`:
>>> p(0.5) 4.25
Find the roots:
>>> p.r array([-1.+1.41421356j, -1.-1.41421356j]) >>> p(p.r) array([ -4.44089210e-16+0.j, -4.44089210e-16+0.j])
These numbers in the previous line represent (0, 0) to machine precision
Show the coefficients:
>>> p.c array([1, 2, 3])
Display the order (the leading zero-coefficients are removed):
>>> p.order 2
Show the coefficient of the k-th power in the polynomial (which is equivalent to ``p.c[-(i+1)]``):
>>> p[1] 2
Polynomials can be added, subtracted, multiplied, and divided (returns quotient and remainder):
>>> p * p poly1d([ 1, 4, 10, 12, 9])
>>> (p**3 + 4) / p (poly1d([ 1., 4., 10., 12., 9.]), poly1d([ 4.]))
``asarray(p)`` gives the coefficient array, so polynomials can be used in all functions that accept arrays:
>>> p**2 # square of polynomial poly1d([ 1, 4, 10, 12, 9])
>>> np.square(p) # square of individual coefficients array([1, 4, 9])
The variable used in the string representation of `p` can be modified, using the `variable` parameter:
>>> p = np.poly1d([1,2,3], variable='z') >>> print(p) 2 1 z + 2 z + 3
Construct a polynomial from its roots:
>>> np.poly1d([1, 2], True) poly1d([ 1, -3, 2])
This is the same polynomial as obtained by:
>>> np.poly1d([1, -1]) * np.poly1d([1, -2]) poly1d([ 1, -3, 2])
"""
def coeffs(self): """ A copy of the polynomial coefficients """ return self._coeffs.copy()
def variable(self): """ The name of the polynomial variable """ return self._variable
# calculated attributes def order(self): """ The order or degree of the polynomial """ return len(self._coeffs) - 1
def roots(self): """ The roots of the polynomial, where self(x) == 0 """ return roots(self._coeffs)
# our internal _coeffs property need to be backed by __dict__['coeffs'] for # scipy to work correctly. def _coeffs(self): return self.__dict__['coeffs'] def _coeffs(self, coeffs):
# alias attributes
self._variable = c_or_r._variable self._coeffs = c_or_r._coeffs
if set(c_or_r.__dict__) - set(self.__dict__): msg = ("In the future extra properties will not be copied " "across when constructing one poly1d from another") warnings.warn(msg, FutureWarning, stacklevel=2) self.__dict__.update(c_or_r.__dict__)
if variable is not None: self._variable = variable return c_or_r = poly(c_or_r) raise ValueError("Polynomial must be 1d only.") c_or_r = NX.array([0.])
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
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
def fmt_float(q): s = '%.4g' % q if s.endswith('.0000'): s = s[:-5] return s
for k in range(len(coeffs)): if not iscomplex(coeffs[k]): coefstr = fmt_float(real(coeffs[k])) elif real(coeffs[k]) == 0: coefstr = '%sj' % fmt_float(imag(coeffs[k])) else: coefstr = '(%s + %sj)' % (fmt_float(real(coeffs[k])), fmt_float(imag(coeffs[k])))
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 newstr.startswith('-'): thestr = "%s - %s" % (thestr, newstr[1:]) else: thestr = "%s + %s" % (thestr, newstr) else: thestr = newstr return _raise_power(thestr)
return polyval(self.coeffs, val)
return poly1d(-self.coeffs)
return self
if isscalar(other): return poly1d(self.coeffs * other) else: other = poly1d(other) return poly1d(polymul(self.coeffs, other.coeffs))
if isscalar(other): return poly1d(other * self.coeffs) else: other = poly1d(other) return poly1d(polymul(self.coeffs, other.coeffs))
other = poly1d(other) return poly1d(polyadd(self.coeffs, other.coeffs))
other = poly1d(other) return poly1d(polyadd(self.coeffs, other.coeffs))
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)
other = poly1d(other) return poly1d(polysub(self.coeffs, other.coeffs))
other = poly1d(other) return poly1d(polysub(other.coeffs, self.coeffs))
if isscalar(other): return poly1d(self.coeffs/other) else: other = poly1d(other) return polydiv(self, other)
if isscalar(other): return poly1d(other/self.coeffs) else: other = poly1d(other) return polydiv(other, self)
if not isinstance(other, poly1d): return NotImplemented if self.coeffs.shape != other.coeffs.shape: return False return (self.coeffs == other.coeffs).all()
if not isinstance(other, poly1d): return NotImplemented return not self.__eq__(other)
ind = self.order - val if val > self.order: return 0 if val < 0: return 0 return self.coeffs[ind]
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._coeffs = NX.concatenate((zr, self.coeffs)) ind = 0 self._coeffs[ind] = val return
return iter(self.coeffs)
""" Return an antiderivative (indefinite integral) of this polynomial.
Refer to `polyint` for full documentation.
See Also -------- polyint : equivalent function
""" return poly1d(polyint(self.coeffs, m=m, k=k))
""" Return a derivative of this polynomial.
Refer to `polyder` for full documentation.
See Also -------- polyder : equivalent function
""" return poly1d(polyder(self.coeffs, m=m))
# Stuff to do on module import
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