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""" 

Functions to operate on polynomials. 

 

""" 

from __future__ import division, absolute_import, print_function 

 

__all__ = ['poly', 'roots', 'polyint', 'polyder', 'polyadd', 

'polysub', 'polymul', 'polydiv', 'polyval', 'poly1d', 

'polyfit', 'RankWarning'] 

 

import functools 

import re 

import warnings 

import numpy.core.numeric as NX 

 

from numpy.core import (isscalar, abs, finfo, atleast_1d, hstack, dot, array, 

ones) 

from numpy.core import overrides 

from numpy.core.overrides import set_module 

from numpy.lib.twodim_base import diag, vander 

from numpy.lib.function_base import trim_zeros 

from numpy.lib.type_check import iscomplex, real, imag, mintypecode 

from numpy.linalg import eigvals, lstsq, inv 

 

 

array_function_dispatch = functools.partial( 

overrides.array_function_dispatch, module='numpy') 

 

 

@set_module('numpy') 

class RankWarning(UserWarning): 

""" 

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`. 

 

""" 

pass 

 

 

def _poly_dispatcher(seq_of_zeros): 

return seq_of_zeros 

 

 

@array_function_dispatch(_poly_dispatcher) 

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 

 

 

def _roots_dispatcher(p): 

return p 

 

 

@array_function_dispatch(_roots_dispatcher) 

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 

 

 

def _polyint_dispatcher(p, m=None, k=None): 

return (p,) 

 

 

@array_function_dispatch(_polyint_dispatcher) 

def polyint(p, m=1, k=None): 

""" 

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 

 

 

def _polyder_dispatcher(p, m=None): 

return (p,) 

 

 

@array_function_dispatch(_polyder_dispatcher) 

def polyder(p, m=1): 

""" 

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 

 

 

def _polyfit_dispatcher(x, y, deg, rcond=None, full=None, w=None, cov=None): 

return (x, y, w) 

 

 

@array_function_dispatch(_polyfit_dispatcher) 

def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False): 

""" 

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 

 

 

def _polyval_dispatcher(p, x): 

return (p, x) 

 

 

@array_function_dispatch(_polyval_dispatcher) 

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 

 

 

def _binary_op_dispatcher(a1, a2): 

return (a1, a2) 

 

 

@array_function_dispatch(_binary_op_dispatcher) 

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 

 

 

@array_function_dispatch(_binary_op_dispatcher) 

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 

 

 

@array_function_dispatch(_binary_op_dispatcher) 

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 

 

 

def _polydiv_dispatcher(u, v): 

return (u, v) 

 

 

@array_function_dispatch(_polydiv_dispatcher) 

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 

 

_poly_mat = re.compile(r"[*][*]([0-9]*)") 

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:] 

 

 

@set_module('numpy') 

class poly1d(object): 

""" 

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]) 

 

""" 

__hash__ = None 

 

@property 

def coeffs(self): 

""" A copy of the polynomial coefficients """ 

return self._coeffs.copy() 

 

@property 

def variable(self): 

""" The name of the polynomial variable """ 

return self._variable 

 

# calculated attributes 

@property 

def order(self): 

""" The order or degree of the polynomial """ 

return len(self._coeffs) - 1 

 

@property 

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. 

@property 

def _coeffs(self): 

return self.__dict__['coeffs'] 

@_coeffs.setter 

def _coeffs(self, coeffs): 

self.__dict__['coeffs'] = coeffs 

 

# alias attributes 

r = roots 

c = coef = coefficients = coeffs 

o = order 

 

def __init__(self, c_or_r, r=False, variable=None): 

if isinstance(c_or_r, poly1d): 

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 

if r: 

c_or_r = poly(c_or_r) 

c_or_r = atleast_1d(c_or_r) 

if c_or_r.ndim > 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._coeffs = c_or_r 

if variable is None: 

variable = 'x' 

self._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 

 

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) 

 

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) 

 

__truediv__ = __div__ 

 

def __rdiv__(self, other): 

if isscalar(other): 

return poly1d(other/self.coeffs) 

else: 

other = poly1d(other) 

return polydiv(other, self) 

 

__rtruediv__ = __rdiv__ 

 

def __eq__(self, other): 

if not isinstance(other, poly1d): 

return NotImplemented 

if self.coeffs.shape != other.coeffs.shape: 

return False 

return (self.coeffs == other.coeffs).all() 

 

def __ne__(self, other): 

if not isinstance(other, poly1d): 

return NotImplemented 

return not self.__eq__(other) 

 

 

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._coeffs = NX.concatenate((zr, self.coeffs)) 

ind = 0 

self._coeffs[ind] = val 

return 

 

def __iter__(self): 

return iter(self.coeffs) 

 

def integ(self, m=1, k=0): 

""" 

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)) 

 

def deriv(self, m=1): 

""" 

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 

 

warnings.simplefilter('always', RankWarning)