neulab/docprompting-conala huggingface dataset:
numpy.polynomial.polynomial.Polynomial.integ method   polynomial.polynomial.Polynomial.integ(m=1, k=[], lbnd=None)[source]
 
Integrate. Return a series instance that is the definite integral of the current series.  Parameters 
 
mnon-negative int


The number of integrations to perform.  
karray_like


Integration constants. The first constant is applied to the first integration, the second to the second, and so on. The list of values must less than or equal to m in length and any missing values are set to zero.  
lbndScalar


The lower bound of the definite integral.    Returns 
 
new_seriesseries


A new series representing the integral. The domain is the same as the domain of the integrated series.
numpy.polynomial.polynomial.Polynomial.linspace method   polynomial.polynomial.Polynomial.linspace(n=100, domain=None)[source]
 
Return x, y values at equally spaced points in domain. Returns the x, y values at n linearly spaced points across the domain. Here y is the value of the polynomial at the points x. By default the domain is the same as that of the series instance. This method is intended mostly as a plotting aid.  New in version 1.5.0.   Parameters 
 
nint, optional


Number of point pairs to return. The default value is 100.  
domain{None, array_like}, optional


If not None, the specified domain is used instead of that of the calling instance. It should be of the form [beg,end]. The default is None which case the class domain is used.    Returns 
 
x, yndarray


x is equal to linspace(self.domain[0], self.domain[1], n) and y is the series evaluated at element of x.
numpy.polynomial.polynomial.Polynomial.mapparms method   polynomial.polynomial.Polynomial.mapparms()[source]
 
Return the mapping parameters. The returned values define a linear map off + scl*x that is applied to the input arguments before the series is evaluated. The map depends on the domain and window; if the current domain is equal to the window the resulting map is the identity. If the coefficients of the series instance are to be used by themselves outside this class, then the linear function must be substituted for the x in the standard representation of the base polynomials.  Returns 
 
off, sclfloat or complex


The mapping function is defined by off + scl*x.     Notes If the current domain is the interval [l1, r1] and the window is [l2, r2], then the linear mapping function L is defined by the equations: L(l1) = l2
L(r1) = r2
numpy.polynomial.polynomial.Polynomial.roots method   polynomial.polynomial.Polynomial.roots()[source]
 
Return the roots of the series polynomial. Compute the roots for the series. Note that the accuracy of the roots decrease the further outside the domain they lie.  Returns 
 
rootsndarray


Array containing the roots of the series.
numpy.polynomial.polynomial.Polynomial.trim method   polynomial.polynomial.Polynomial.trim(tol=0)[source]
 
Remove trailing coefficients Remove trailing coefficients until a coefficient is reached whose absolute value greater than tol or the beginning of the series is reached. If all the coefficients would be removed the series is set to [0]. A new series instance is returned with the new coefficients. The current instance remains unchanged.  Parameters 
 
tolnon-negative number.


All trailing coefficients less than tol will be removed.    Returns 
 
new_seriesseries


New instance of series with trimmed coefficients.
numpy.polynomial.polynomial.Polynomial.truncate method   polynomial.polynomial.Polynomial.truncate(size)[source]
 
Truncate series to length size. Reduce the series to length size by discarding the high degree terms. The value of size must be a positive integer. This can be useful in least squares where the coefficients of the high degree terms may be very small.  Parameters 
 
sizepositive int


The series is reduced to length size by discarding the high degree terms. The value of size must be a positive integer.    Returns 
 
new_seriesseries


New instance of series with truncated coefficients.
numpy.polynomial.polynomial.polypow   polynomial.polynomial.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.  Parameters 
 
carray_like


1-D array of array of series coefficients ordered from low to high degree.  
powinteger


Power to which the series will be raised  
maxpowerinteger, optional


Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16    Returns 
 
coefndarray


Power series of power.      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.])
numpy.polynomial.polynomial.polyroots   polynomial.polynomial.polyroots(c)[source]
 
Compute the roots of a polynomial. Return the roots (a.k.a. “zeros”) of the polynomial  \[p(x) = \sum_i c[i] * x^i.\]  Parameters 
 
c1-D array_like


1-D array of polynomial coefficients.    Returns 
 
outndarray


Array of the roots of the polynomial. If all the roots are real, then out is also real, otherwise it is complex.      See also  numpy.polynomial.chebyshev.chebroots
numpy.polynomial.legendre.legroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
  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
numpy.polynomial.polynomial.polysub   polynomial.polynomial.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.  Parameters 
 
c1, c2array_like


1-D arrays of polynomial coefficients ordered from low to high.    Returns 
 
outndarray


Of coefficients representing their difference.      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.])
numpy.polynomial.polynomial.polytrim   polynomial.polynomial.polytrim(c, tol=0)[source]
 
Remove “small” “trailing” coefficients from a polynomial. “Small” means “small in absolute value” and is controlled by the parameter tol; “trailing” means highest order coefficient(s), e.g., in [0, 1, 1, 0, 0] (which represents 0 + x + x**2 + 0*x**3 + 0*x**4) both the 3-rd and 4-th order coefficients would be “trimmed.”  Parameters 
 
carray_like


1-d array of coefficients, ordered from lowest order to highest.  
tolnumber, optional


Trailing (i.e., highest order) elements with absolute value less than or equal to tol (default value is zero) are removed.    Returns 
 
trimmedndarray


1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.    Raises 
 ValueError

If tol < 0      See also  trimseq
  Examples >>> from numpy.polynomial import polyutils as pu
>>> pu.trimcoef((0,0,3,0,5,0,0))
array([0.,  0.,  3.,  0.,  5.])
>>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
array([0.])
>>> i = complex(0,1) # works for complex
>>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
array([0.0003+0.j   , 0.001 -0.001j])
numpy.polynomial.polynomial.polyval   polynomial.polynomial.polyval(x, c, tensor=True)[source]
 
Evaluate a polynomial at points x. If c is of length n + 1, this function returns the value  \[p(x) = c_0 + c_1 * x + ... + c_n * x^n\] 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.  Parameters 
 
xarray_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.  
carray_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.  
tensorboolean, 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 
 
valuesndarray, compatible object


The shape of the returned array is described above.      See also  
polyval2d, polygrid2d, polyval3d, polygrid3d

  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.])
numpy.polynomial.polynomial.polyval2d   polynomial.polynomial.polyval2d(x, y, c)[source]
 
Evaluate a 2-D polynomial at points (x, y). This function returns the value  \[p(x,y) = \sum_{i,j} c_{i,j} * x^i * y^j\] 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.  Parameters 
 
x, yarray_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.  
carray_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 
 
valuesndarray, compatible object


The values of the two dimensional polynomial at points formed with pairs of corresponding values from x and y.      See also  
polyval, polygrid2d, polyval3d, polygrid3d

  Notes  New in version 1.7.0.
numpy.polynomial.polynomial.polyval3d   polynomial.polynomial.polyval3d(x, y, z, c)[source]
 
Evaluate a 3-D polynomial at points (x, y, z). This function returns the values:  \[p(x,y,z) = \sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k\] 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.  Parameters 
 
x, y, zarray_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.  
carray_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 
 
valuesndarray, compatible object


The values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z.      See also  
polyval, polyval2d, polygrid2d, polygrid3d

  Notes  New in version 1.7.0.
numpy.polynomial.polynomial.polyvalfromroots   polynomial.polynomial.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  \[p(x) = \prod_{n=1}^{N} (x - r_n)\] 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 r. 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.   Parameters 
 
xarray_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.  
rarray_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.  
tensorboolean, 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 
 
valuesndarray, compatible object


The shape of the returned array is described above.      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.])
numpy.polynomial.polynomial.polyvander   polynomial.polynomial.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  \[V[..., i] = x^i,\] 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.  Parameters 
 
xarray_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.  
degint


Degree of the resulting matrix.    Returns 
 
vanderndarray.


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.      See also  
polyvander2d, polyvander3d
numpy.polynomial.polynomial.polyvander2d   polynomial.polynomial.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  \[V[..., (deg[1] + 1)*i + j] = x^i * y^j,\] 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  \[c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...\] 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, yarray_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.  
deglist of ints


List of maximum degrees of the form [x_deg, y_deg].    Returns 
 
vander2dndarray


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.      See also  
polyvander, polyvander3d, polyval2d, polyval3d
numpy.polynomial.polynomial.polyvander3d   polynomial.polynomial.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  \[V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k,\] 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  \[c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...\] 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.  Parameters 
 
x, y, zarray_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.  
deglist of ints


List of maximum degrees of the form [x_deg, y_deg, z_deg].    Returns 
 
vander3dndarray


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.      See also  
polyvander, polyvander3d, polyval2d, polyval3d

  Notes  New in version 1.7.0.
numpy.polynomial.polyutils.as_series   polynomial.polyutils.as_series(alist, trim=True)[source]
 
Return argument as a list of 1-d arrays. The returned list contains array(s) of dtype double, complex double, or object. A 1-d argument of shape (N,) is parsed into N arrays of size one; a 2-d argument of shape (M,N) is parsed into M arrays of size N (i.e., is “parsed by row”); and a higher dimensional array raises a Value Error if it is not first reshaped into either a 1-d or 2-d array.  Parameters 
 
alistarray_like


A 1- or 2-d array_like  
trimboolean, optional


When True, trailing zeros are removed from the inputs. When False, the inputs are passed through intact.    Returns 
 
[a1, a2,…]list of 1-D arrays


A copy of the input data as a list of 1-d arrays.    Raises 
 ValueError

Raised when as_series cannot convert its input to 1-d arrays, or at least one of the resulting arrays is empty.     Examples >>> from numpy.polynomial import polyutils as pu
>>> a = np.arange(4)
>>> pu.as_series(a)
[array([0.]), array([1.]), array([2.]), array([3.])]
>>> b = np.arange(6).reshape((2,3))
>>> pu.as_series(b)
[array([0., 1., 2.]), array([3., 4., 5.])]
 >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16)))
[array([1.]), array([0., 1., 2.]), array([0., 1.])]
 >>> pu.as_series([2, [1.1, 0.]])
[array([2.]), array([1.1])]
 >>> pu.as_series([2, [1.1, 0.]], trim=False)
[array([2.]), array([1.1, 0. ])]
numpy.polynomial.polyutils.getdomain   polynomial.polyutils.getdomain(x)[source]
 
Return a domain suitable for given abscissae. Find a domain suitable for a polynomial or Chebyshev series defined at the values supplied.  Parameters 
 
xarray_like


1-d array of abscissae whose domain will be determined.    Returns 
 
domainndarray


1-d array containing two values. If the inputs are complex, then the two returned points are the lower left and upper right corners of the smallest rectangle (aligned with the axes) in the complex plane containing the points x. If the inputs are real, then the two points are the ends of the smallest interval containing the points x.      See also  
mapparms, mapdomain

  Examples >>> from numpy.polynomial import polyutils as pu
>>> points = np.arange(4)**2 - 5; points
array([-5, -4, -1,  4])
>>> pu.getdomain(points)
array([-5.,  4.])
>>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle
>>> pu.getdomain(c)
array([-1.-1.j,  1.+1.j])