numpy.polynomial.polynomial

module documentation

(source)

This module provides a number of objects (mostly functions) useful for dealing with polynomials, including a Polynomial class that encapsulates the usual arithmetic operations. (General information on how this module represents and works with polynomial objects is in the docstring for its "parent" sub-package, numpy.polynomial).

Classes

Constants

Arithmetic

Calculus

Misc Functions

See Also

None

Class	`Polynomial`	A power series class.
Function	`polyadd`	Add one polynomial to another.
Function	`polycompanion`	Return the companion matrix of c.
Function	`polyder`	Differentiate a polynomial.
Function	`polydiv`	Divide one polynomial by another.
Function	`polyfit`	Least-squares fit of a polynomial to data.
Function	`polyfromroots`	Generate a monic polynomial with given roots.
Function	`polygrid2d`	Evaluate a 2-D polynomial on the Cartesian product of x and y.
Function	`polygrid3d`	Evaluate a 3-D polynomial on the Cartesian product of x, y and z.
Function	`polyint`	Integrate a polynomial.
Function	`polyline`	Returns an array representing a linear polynomial.
Function	`polymul`	Multiply one polynomial by another.
Function	`polymulx`	Multiply a polynomial by x.
Function	`polypow`	Raise a polynomial to a power.
Function	`polyroots`	Compute the roots of a polynomial.
Function	`polysub`	Subtract one polynomial from another.
Function	`polyval`	Evaluate a polynomial at points x.
Function	`polyval2d`	Evaluate a 2-D polynomial at points (x, y).
Function	`polyval3d`	Evaluate a 3-D polynomial at points (x, y, z).
Function	`polyvalfromroots`	Evaluate a polynomial specified by its roots at points x.
Function	`polyvander`	Vandermonde matrix of given degree.
Function	`polyvander2d`	Pseudo-Vandermonde matrix of given degrees.
Function	`polyvander3d`	Pseudo-Vandermonde matrix of given degrees.
Variable	`polydomain`	Undocumented
Variable	`polyone`	Undocumented
Variable	`polyx`	Undocumented
Variable	`polyzero`	Undocumented

def polyadd(c1, c2): (source) ¶

Add one polynomial to another.

Returns the sum of two polynomials c1 + c2. The arguments are sequences of coefficients from lowest order term to highest, i.e., [1,2,3] represents the polynomial 1 + 2*x + 3*x**2.

See Also

polysub, polymulx, polymul, polydiv, polypow

Examples

>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([4.,  4.,  4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0

Parameters
c1:`array_like`	1-D arrays of polynomial coefficients ordered from low to high.
c2:`array_like`	1-D arrays of polynomial coefficients ordered from low to high.
Returns
`ndarray`	out - The coefficient array representing their sum.

def polycompanion(c): (source) ¶

Return the companion matrix of c.

The companion matrix for power series cannot be made symmetric by scaling the basis, so this function differs from those for the orthogonal polynomials.

Notes

New in version 1.7.0.

Parameters
c:`array_like`	1-D array of polynomial coefficients ordered from low to high degree.
Returns
`ndarray`	mat - Companion matrix of dimensions (deg, deg).

def polyder(c, m=1, scl=1, axis=0): (source) ¶

Differentiate a polynomial.

Returns the polynomial coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the polynomial 1 + 2*x + 3*x**2 while [[1,2],[1,2]] represents 1 + 1*x + 2*y + 2*x*y if axis=0 is x and axis=1 is y.

See Also

polyint

Examples

>>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3
>>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2
array([  2.,   6.,  12.])
>>> P.polyder(c,3) # (d**3/dx**3)(c) = 24
array([24.])
>>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2
array([ -2.,  -6., -12.])
>>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x
array([  6.,  24.])

Parameters
c:`array_like`	Array of polynomial coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
m:`int`, optional	Number of derivatives taken, must be non-negative. (Default: 1)
scl:`scalar`, optional	Each differentiation is multiplied by `scl`. The end result is multiplication by `scl**m`. This is for use in a linear change of variable. (Default: 1)
axis:`int`, optional	Axis over which the derivative is taken. (Default: 0). New in version 1.7.0.
Returns
`ndarray`	der - Polynomial coefficients of the derivative.

def polydiv(c1, c2): (source) ¶

Divide one polynomial by another.

Returns the quotient-with-remainder of two polynomials c1 / c2. The arguments are sequences of coefficients, from lowest order term to highest, e.g., [1,2,3] represents 1 + 2*x + 3*x**2.

See Also

polyadd, polysub, polymulx, polymul, polypow

Examples

>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polydiv(c1,c2)
(array([3.]), array([-8., -4.]))
>>> P.polydiv(c2,c1)
(array([ 0.33333333]), array([ 2.66666667,  1.33333333])) # may vary

Parameters
c1:`array_like`	1-D arrays of polynomial coefficients ordered from low to high.
c2:`array_like`	1-D arrays of polynomial coefficients ordered from low to high.
Returns
`ndarrays`	[quo, rem] - Of coefficient series representing the quotient and remainder.

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

Least-squares fit of a polynomial to data.

Return the coefficients of a polynomial of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form

p(x) = c₀ + c₁*x + ... + c_n*xⁿ,

where n is deg.

polyval: Evaluates a polynomial.
polyvander: Vandermonde matrix for powers.
numpy.linalg.lstsq: Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline: Computes spline fits.

Notes

The solution is the coefficients of the polynomial p that minimizes the sum of the weighted squared errors

E = ∑_jw²_j*|y_j − p(x_j)|²,

where the w_j are the weights. This problem is solved by setting up the (typically) over-determined matrix equation:

V(x)*c = w*y,

where V is the weighted pseudo Vandermonde matrix of x, c are the coefficients to be solved for, w are the weights, and y are the observed values. This equation is then solved using the singular value decomposition of V.

If some of the singular values of V are so small that they are neglected (and full == False), a RankWarning will be raised. This means that the coefficient values may be poorly determined. Fitting to a lower order polynomial will usually get rid of the warning (but may not be what you want, of course; if you have independent reason(s) for choosing the degree which isn't working, you may have to: a) reconsider those reasons, and/or b) reconsider the quality of your data). The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error.

Polynomial fits using double precision tend to "fail" at about (polynomial) degree 20. Fits using Chebyshev or Legendre series are generally better conditioned, but much can still depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate, splines may be a good alternative.

Examples

>>> np.random.seed(123)
>>> from numpy.polynomial import polynomial as P
>>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1]
>>> y = x**3 - x + np.random.randn(len(x))  # x^3 - x + Gaussian noise
>>> c, stats = P.polyfit(x,y,3,full=True)
>>> np.random.seed(123)
>>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1
array([ 0.01909725, -1.30598256, -0.00577963,  1.02644286]) # may vary
>>> stats # note the large SSR, explaining the rather poor results
 [array([ 38.06116253]), 4, array([ 1.38446749,  1.32119158,  0.50443316, # may vary
          0.28853036]), 1.1324274851176597e-014]

Same thing without the added noise

>>> y = x**3 - x
>>> c, stats = P.polyfit(x,y,3,full=True)
>>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1
array([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16,  1.00000000e+00])
>>> stats # note the minuscule SSR
[array([  7.46346754e-31]), 4, array([ 1.38446749,  1.32119158, # may vary
           0.50443316,  0.28853036]), 1.1324274851176597e-014]

Parameters
x:`array_like`, shape(`M`, )	x-coordinates of the `M` sample (data) points `(x[i], y[i])`.
y:`array_like`, shape(`M`, ) or (`M`, `K`)	y-coordinates of the sample points. Several sets of sample points sharing the same x-coordinates can be (independently) fit with one call to `polyfit` by passing in for `y` a 2-D array that contains one data set per column.
deg:`int` or 1-D array_like	Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
rcond:`float`, optional	Relative condition number of the fit. Singular values smaller than `rcond`, relative to the largest singular value, will be ignored. The default value is `len(x)*eps`, where `eps` is the relative precision of the platform's float type, about 2e-16 in most cases.
full:`bool`, optional	Switch determining the nature of the return value. When `False` (the default) just the coefficients are returned; when `True`, diagnostic information from the singular value decomposition (used to solve the fit's matrix equation) is also returned.
w:`array_like`, shape(`M`, ), optional	Weights. If not None, the weight `w[i]` applies to the unsquared residual `y[i] - y_hat[i]` at `x[i]`. Ideally the weights are chosen so that the errors of the products `w[i]*y[i]` all have the same variance. When using inverse-variance weighting, use `w[i] = 1/sigma(y[i])`. The default value is None. New in version 1.5.0.
Returns
coef: `ndarray`, shape (`deg` + 1, ) or (`deg` + 1, `K`) - Polynomial coefficients ordered from low to high. If `y` was 2-D, the coefficients in column `k` of `coef` represent the polynomial fit to the data in `y`'s `k`-th column. [residuals, rank, singular_values, rcond]: `list` - These values are only returned if `full == True` residuals -- sum of squared residuals of the least squares fit rank -- the numerical rank of the scaled Vandermonde matrix singular_values -- singular values of the scaled Vandermonde matrix rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`.
Raises
`RankWarning`	Raised if the matrix in the least-squares fit is rank deficient. The warning is only raised if `full == False`. The warnings can be turned off by: >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning)

def polyfromroots(roots): (source) ¶

Generate a monic polynomial with given roots.

Return the coefficients of the polynomial

p(x) = (x − r₀)*(x − r₁)*...*(x − r_n),

where the r_n are the roots specified in roots. If a zero has multiplicity n, then it must appear in roots n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then roots looks something like [2, 2, 2, 3, 3]. The roots can appear in any order.

If the returned coefficients are c, then

p(x) = c₀ + c₁*x + ... + xⁿ

The coefficient of the last term is 1 for monic polynomials in this form.

Notes

The coefficients are determined by multiplying together linear factors of the form (x - r_i), i.e.

p(x) = (x − r₀)(x − r₁)...(x − r_n)

where n == len(roots) - 1; note that this implies that 1 is always returned for a_n.

Examples

>>> from numpy.polynomial import polynomial as P
>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
array([ 0., -1.,  0.,  1.])
>>> j = complex(0,1)
>>> P.polyfromroots((-j,j)) # complex returned, though values are real
array([1.+0.j,  0.+0.j,  1.+0.j])

Parameters
roots:`array_like`	Sequence containing the roots.
Returns
`ndarray`	out - 1-D array of the polynomial's coefficients If all the roots are real, then `out` is also real, otherwise it is complex. (see Examples below).

def polygrid2d(x, y, c): (source) ¶

Evaluate a 2-D polynomial on the Cartesian product of x and y.

This function returns the values:

p(a, b) = ∑_i, jc_i, j*aⁱ*b^j

where the points (a, b) consist of all pairs formed by taking a from x and b from y. The resulting points form a grid with x in the first dimension and y in the second.

The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c.

If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape.

Notes

New in version 1.7.0.

Parameters
x:`array_like`, compatible objects	The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
y:`array_like`, compatible objects	The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
c:`array_like`	Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
Returns
`ndarray`, compatible object	values - The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`.

def polygrid3d(x, y, z, c): (source) ¶

Evaluate a 3-D polynomial on the Cartesian product of x, y and z.

This function returns the values:

p(a, b, c) = ∑_i, j, kc_i, j, k*aⁱ*b^j*c^k

where the points (a, b, c) consist of all triples formed by taking a from x, b from y, and c from z. The resulting points form a grid with x in the first dimension, y in the second, and z in the third.

The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c.

If c has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

Notes

New in version 1.7.0.

Parameters
x:`array_like`, compatible objects	The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
y:`array_like`, compatible objects	The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
z:`array_like`, compatible objects	The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
c:`array_like`	Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
Returns
`ndarray`, compatible object	values - The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`.

def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0): (source) ¶

Integrate a polynomial.

Returns the polynomial coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument c is an array of coefficients, from low to high degree along each axis, e.g., [1,2,3] represents the polynomial 1 + 2*x + 3*x**2 while [[1,2],[1,2]] represents 1 + 1*x + 2*y + 2*x*y if axis=0 is x and axis=1 is y.

See Also

polyder

Notes

Note that the result of each integration is multiplied by scl. Why is this important to note? Say one is making a linear change of variable u = ax + b in an integral relative to x. Then dx = du ⁄ a, so one will need to set scl equal to 1 ⁄ a - perhaps not what one would have first thought.

Examples

>>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3)
>>> P.polyint(c) # should return array([0, 1, 1, 1])
array([0.,  1.,  1.,  1.])
>>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20])
 array([ 0.        ,  0.        ,  0.        ,  0.16666667,  0.08333333, # may vary
         0.05      ])
>>> P.polyint(c,k=3) # should return array([3, 1, 1, 1])
array([3.,  1.,  1.,  1.])
>>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1])
array([6.,  1.,  1.,  1.])
>>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2])
array([ 0., -2., -2., -2.])

Parameters
c:`array_like`	1-D array of polynomial coefficients, ordered from low to high.
m:`int`, optional	Order of integration, must be positive. (Default: 1)
k:{[], list, scalar}, optional	Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If `k == []` (the default), all constants are set to zero. If `m == 1`, a single scalar can be given instead of a list.
lbnd:`scalar`, optional	The lower bound of the integral. (Default: 0)
scl:`scalar`, optional	Following each integration the result is multiplied by `scl` before the integration constant is added. (Default: 1)
axis:`int`, optional	Axis over which the integral is taken. (Default: 0). New in version 1.7.0.
Returns
`ndarray`	S - Coefficient array of the integral.
Raises
`ValueError`	If `m < 1`, `len(k) > m`, `np.ndim(lbnd) != 0`, or `np.ndim(scl) != 0`.

def polyline(off, scl): (source) ¶

Returns an array representing a linear polynomial.

Examples

>>> from numpy.polynomial import polynomial as P
>>> P.polyline(1,-1)
array([ 1, -1])
>>> P.polyval(1, P.polyline(1,-1)) # should be 0
0.0

Parameters
off:`scalars`	The "y-intercept" and "slope" of the line, respectively.
scl:`scalars`	The "y-intercept" and "slope" of the line, respectively.
Returns
`ndarray`	y - This module's representation of the linear polynomial `off + scl*x`.

def polymul(c1, c2): (source) ¶

Multiply one polynomial by another.

Returns the product of two polynomials c1 * c2. The arguments are sequences of coefficients, from lowest order term to highest, e.g., [1,2,3] represents the polynomial 1 + 2*x + 3*x**2.

See Also

polyadd, polysub, polymulx, polydiv, polypow

Examples

>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polymul(c1,c2)
array([  3.,   8.,  14.,   8.,   3.])

Parameters
c1:`array_like`	1-D arrays of coefficients representing a polynomial, relative to the "standard" basis, and ordered from lowest order term to highest.
c2:`array_like`	1-D arrays of coefficients representing a polynomial, relative to the "standard" basis, and ordered from lowest order term to highest.
Returns
`ndarray`	out - Of the coefficients of their product.

def polymulx(c): (source) ¶

Multiply a polynomial by x.

Multiply the polynomial c by x, where x is the independent variable.

See Also

polyadd, polysub, polymul, polydiv, polypow

Notes

New in version 1.5.0.

Parameters
c:`array_like`	1-D array of polynomial coefficients ordered from low to high.
Returns
`ndarray`	out - Array representing the result of the multiplication.

def polypow(c, pow, maxpower=None): (source) ¶

Raise a polynomial to a power.

Returns the polynomial c raised to the power pow. The argument c is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series 1 + 2*x + 3*x**2.

See Also

polyadd, polysub, polymulx, polymul, polydiv

Examples

>>> from numpy.polynomial import polynomial as P
>>> P.polypow([1,2,3], 2)
array([ 1., 4., 10., 12., 9.])

Parameters
c:`array_like`	1-D array of array of series coefficients ordered from low to high degree.
pow:`integer`	Power to which the series will be raised
maxpower:`integer`, optional	Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16
Returns
`ndarray`	coef - Power series of power.

def polyroots(c): (source) ¶

Compute the roots of a polynomial.

Return the roots (a.k.a. "zeros") of the polynomial

p(x) = ∑_ic[i]*xⁱ.

Notes

The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the power series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method.

Examples

>>> import numpy.polynomial.polynomial as poly
>>> poly.polyroots(poly.polyfromroots((-1,0,1)))
array([-1.,  0.,  1.])
>>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> poly.polyroots(poly.polyfromroots((-j,0,j)))
array([  0.00000000e+00+0.j,   0.00000000e+00+1.j,   2.77555756e-17-1.j]) # may vary

Parameters
c:1-D array_like	1-D array of polynomial coefficients.
Returns
`ndarray`	out - Array of the roots of the polynomial. If all the roots are real, then `out` is also real, otherwise it is complex.

def polysub(c1, c2): (source) ¶

Subtract one polynomial from another.

Returns the difference of two polynomials c1 - c2. The arguments are sequences of coefficients from lowest order term to highest, i.e., [1,2,3] represents the polynomial 1 + 2*x + 3*x**2.

See Also

polyadd, polymulx, polymul, polydiv, polypow

Examples

>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2.,  0.,  2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2.,  0., -2.])

Parameters
c1:`array_like`	1-D arrays of polynomial coefficients ordered from low to high.
c2:`array_like`	1-D arrays of polynomial coefficients ordered from low to high.
Returns
`ndarray`	out - Of coefficients representing their difference.

def polyval(x, c, tensor=True): (source) ¶

Evaluate a polynomial at points x.

If c is of length n + 1, this function returns the value

p(x) = c₀ + c₁*x + ... + c_n*xⁿ

The parameter x is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either x or its elements must support multiplication and addition both with themselves and with the elements of c.

If c is a 1-D array, then p(x) will have the same shape as x. If c is multidimensional, then the shape of the result depends on the value of tensor. If tensor is true the shape will be c.shape[1:] + x.shape. If tensor is false the shape will be c.shape[1:]. Note that scalars have shape (,).

Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern.

Notes

The evaluation uses Horner's method.

Examples

>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
       [2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1.,   6.],
       [17.,  34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
       [2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[2.,  4.],
       [4.,  7.]])
>>> polyval([1,2], coef, tensor=False)
array([2.,  7.])

Parameters
x:`array_like`, compatible object	If `x` is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, `x` or its elements must support addition and multiplication with with themselves and with the elements of `c`.
c:`array_like`	Array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If `c` is multidimensional the remaining indices enumerate multiple polynomials. In the two dimensional case the coefficients may be thought of as stored in the columns of `c`.
tensor:`boolean`, optional	If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of `x`. Scalars have dimension 0 for this action. The result is that every column of coefficients in `c` is evaluated for every element of `x`. If False, `x` is broadcast over the columns of `c` for the evaluation. This keyword is useful when `c` is multidimensional. The default value is True. New in version 1.7.0.
Returns
`ndarray`, compatible object	values - The shape of the returned array is described above.

def polyval2d(x, y, c): (source) ¶

Evaluate a 2-D polynomial at points (x, y).

This function returns the value

p(x, y) = ∑_i, jc_i, j*xⁱ*y^j

The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c.

If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape.

Notes

New in version 1.7.0.

Parameters
x:`array_like`, compatible objects	The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
y:`array_like`, compatible objects	The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar.
c:`array_like`	Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
Returns
`ndarray`, compatible object	values - The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`.

def polyval3d(x, y, z, c): (source) ¶

Evaluate a 3-D polynomial at points (x, y, z).

This function returns the values:

p(x, y, z) = ∑_i, j, kc_i, j, k*xⁱ*y^j*z^k

The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c.

If c has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape.

Notes

New in version 1.7.0.

Parameters
x:`array_like`, compatible object	The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar.
y:`array_like`, compatible object	The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar.
z:`array_like`, compatible object	The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar.
c:`array_like`	Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in `c[i,j,k]`. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients.
Returns
`ndarray`, compatible object	values - The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`.

def polyvalfromroots(x, r, tensor=True): (source) ¶

Evaluate a polynomial specified by its roots at points x.

If r is of length N, this function returns the value

p(x) = ^N∏_n = 1(x − r_n)

If r is a 1-D array, then p(x) will have the same shape as x. If r is multidimensional, then the shape of the result depends on the value of tensor. If tensor is True the shape will be r.shape[1:] + x.shape; that is, each polynomial is evaluated at every value of x. If tensor is False, the shape will be r.shape[1:]; that is, each polynomial is evaluated only for the corresponding broadcast value of x. Note that scalars have shape (,).

New in version 1.12.

See Also

polyroots, polyfromroots, polyval

Examples

>>> from numpy.polynomial.polynomial import polyvalfromroots
>>> polyvalfromroots(1, [1,2,3])
0.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
       [2, 3]])
>>> polyvalfromroots(a, [-1, 0, 1])
array([[-0.,   0.],
       [ 6.,  24.]])
>>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients
>>> r # each column of r defines one polynomial
array([[-2, -1],
       [ 0,  1]])
>>> b = [-2, 1]
>>> polyvalfromroots(b, r, tensor=True)
array([[-0.,  3.],
       [ 3., 0.]])
>>> polyvalfromroots(b, r, tensor=False)
array([-0.,  0.])

Parameters
x:`array_like`, compatible object	If `x` is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, `x` or its elements must support addition and multiplication with with themselves and with the elements of `r`.
r:`array_like`	Array of roots. If `r` is multidimensional the first index is the root index, while the remaining indices enumerate multiple polynomials. For instance, in the two dimensional case the roots of each polynomial may be thought of as stored in the columns of `r`.
tensor:`boolean`, optional	If True, the shape of the roots array is extended with ones on the right, one for each dimension of `x`. Scalars have dimension 0 for this action. The result is that every column of coefficients in `r` is evaluated for every element of `x`. If False, `x` is broadcast over the columns of `r` for the evaluation. This keyword is useful when `r` is multidimensional. The default value is True.
Returns
`ndarray`, compatible object	values - The shape of the returned array is described above.

def polyvander(x, deg): (source) ¶

Vandermonde matrix of given degree.

Returns the Vandermonde matrix of degree deg and sample points x. The Vandermonde matrix is defined by

V[..., i] = xⁱ,

where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the power of x.

If c is a 1-D array of coefficients of length n + 1 and V is the matrix V = polyvander(x, n), then np.dot(V, c) and polyval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of polynomials of the same degree and sample points.

See Also

polyvander2d, polyvander3d

Parameters
x:`array_like`	Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If `x` is scalar it is converted to a 1-D array.
deg:`int`	Degree of the resulting matrix.
Returns
`ndarray.`	vander - The Vandermonde matrix. The shape of the returned matrix is `x.shape + (deg + 1,)`, where the last index is the power of `x`. The dtype will be the same as the converted `x`.

def polyvander2d(x, y, deg): (source) ¶

Pseudo-Vandermonde matrix of given degrees.

Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by

V[..., (deg[1] + 1)*i + j] = xⁱ*y^j,

where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the powers of x and y.

If V = polyvander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order

c₀₀, c₀₁, c₀₂..., c₁₀, c₁₁, c₁₂...

and np.dot(V, c.flat) and polyval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D polynomials of the same degrees and sample points.

Parameters
x:`array_like`	Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
y:`array_like`	Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
deg:`list` of `ints`	List of maximum degrees of the form [x_deg, y_deg].
Returns
`ndarray`	vander2d - The shape of the returned matrix is `x.shape + (order,)`, where order = (deg[0] + 1)(deg*([1] + 1). The dtype will be the same as the converted `x` and `y`.

def polyvander3d(x, y, z, deg): (source) ¶

Pseudo-Vandermonde matrix of given degrees.

Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). If l, m, n are the given degrees in x, y, z, then The pseudo-Vandermonde matrix is defined by

V[..., (m + 1)(n + 1)i + (n + 1)j + k] = xⁱ*y^j*z^k,

where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading indices of V index the points (x, y, z) and the last index encodes the powers of x, y, and z.

If V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns of V correspond to the elements of a 3-D coefficient array c of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order

c₀₀₀, c₀₀₁, c₀₀₂, ..., c₀₁₀, c₀₁₁, c₀₁₂, ...

and np.dot(V, c.flat) and polyval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D polynomials of the same degrees and sample points.

Notes

New in version 1.7.0.

Parameters
x:`array_like`	Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
y:`array_like`	Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
z:`array_like`	Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
deg:`list` of `ints`	List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
`ndarray`	vander3d - The shape of the returned matrix is `x.shape + (order,)`, where order = (deg[0] + 1)(deg([1] + 1)(deg[2] + 1). The dtype will be the same as the converted `x`, `y`, and `z`.

polydomain = (source) ¶

Undocumented

polyone = (source) ¶

Undocumented

polyx = (source) ¶

Undocumented

polyzero = (source) ¶

Undocumented